csg.js

/*
## License

Copyright (c) 2013 Eduard Bespalov (edwbes@gmail.com): .solidFromSlices()
Copyright (c) 2013 Rene K. Mueller (http://OpenJSCAD.org): AMF export added, CSG.center([flag,flag,flag]);
Copyright (c) 2012 Joost Nieuwenhuijse (joost@newhouse.nl)
Copyright (c) 2012 Alexandre Girard (https://github.com/alx)
Copyright (c) 2011 Evan Wallace (http://evanw.github.com/csg.js/) -- original csg.js

All code released under MIT license

## Overview

For an overview of the CSG process see the original csg.js code:
http://evanw.github.com/csg.js/

CSG operations through BSP trees suffer from one problem: heavy fragmentation
of polygons. If two CSG solids of n polygons are unified, the resulting solid may have
in the order of n*n polygons, because each polygon is split by the planes of all other
polygons. After a few operations the number of polygons explodes.

This version of CSG.js solves the problem in 3 ways:

1. Every polygon split is recorded in a tree (CSG.PolygonTreeNode). This is a separate
tree, not to be confused with the CSG tree. If a polygon is split into two parts but in
the end both fragments have not been discarded by the CSG operation, we can retrieve
the original unsplit polygon from the tree, instead of the two fragments.

This does not completely solve the issue though: if a polygon is split multiple times
the number of fragments depends on the order of subsequent splits, and we might still
end up with unncessary splits:
Suppose a polygon is first split into A and B, and then into A1, B1, A2, B2. Suppose B2 is
discarded. We will end up with 2 polygons: A and B1. Depending on the actual split boundaries
we could still have joined A and B1 into one polygon. Therefore a second approach is used as well:

2. After CSG operations all coplanar polygon fragments are joined by a retesselating
operation. See CSG.reTesselated(). Retesselation is done through a
linear sweep over the polygon surface. The sweep line passes over the y coordinates
of all vertices in the polygon. Polygons are split at each sweep line, and the fragments
are joined horizontally and vertically into larger polygons (making sure that we
will end up with convex polygons).
This still doesn't solve the problem completely: due to floating point imprecisions
we may end up with small gaps between polygons, and polygons may not be exactly coplanar
anymore, and as a result the retesselation algorithm may fail to join those polygons.
Therefore:

3. A canonicalization algorithm is implemented: it looks for vertices that have
approximately the same coordinates (with a certain tolerance, say 1e-5) and replaces
them with the same vertex. If polygons share a vertex they will actually point to the
same CSG.Vertex instance. The same is done for polygon planes. See CSG.canonicalized().


Performance improvements to the original CSG.js:

Replaced the flip() and invert() methods by flipped() and inverted() which don't
modify the source object. This allows to get rid of all clone() calls, so that
multiple polygons can refer to the same CSG.Plane instance etc.

The original union() used an extra invert(), clipTo(), invert() sequence just to remove the
coplanar front faces from b; this is now combined in a single b.clipTo(a, true) call.

Detection whether a polygon is in front or in back of a plane: for each polygon
we are caching the coordinates of the bounding sphere. If the bounding sphere is
in front or in back of the plane we don't have to check the individual vertices
anymore.


Other additions to the original CSG.js:

CSG.Vector class has been renamed into CSG.Vector3D

Classes for 3D lines, 2D vectors, 2D lines, and methods to find the intersection of
a line and a plane etc.

Transformations: CSG.transform(), CSG.translate(), CSG.rotate(), CSG.scale()

Expanding or contracting a solid: CSG.expand() and CSG.contract(). Creates nice
smooth corners.

The vertex normal has been removed since it complicates retesselation. It's not needed
for solid CAD anyway.

*/

(function(module){

var _CSGDEBUG = false;

function fnNumberSort(a, b) {
	return a - b;
}

// # class CSG
// Holds a binary space partition tree representing a 3D solid. Two solids can
// be combined using the `union()`, `subtract()`, and `intersect()` methods.
var CSG = function() {
	this.polygons = [];
	this.properties = new CSG.Properties();
	this.isCanonicalized = true;
	this.isRetesselated = true;
};

CSG.defaultResolution2D = 32;
CSG.defaultResolution3D = 12;

// Construct a CSG solid from a list of `CSG.Polygon` instances.
CSG.fromPolygons = function(polygons) {
	var csg = new CSG();
	csg.polygons = polygons;
	csg.isCanonicalized = false;
	csg.isRetesselated = false;
	return csg;
};

// Construct a CSG solid from generated slices.
// Look at CSG.Polygon.prototype.solidFromSlices for details
CSG.fromSlices = function(options) {
	return (new CSG.Polygon.createFromPoints([
			[0, 0, 0], [1, 0, 0], [1, 1, 0], [0, 1, 0]
	])).solidFromSlices(options);
};

// create from an untyped object with identical property names:
CSG.fromObject = function(obj) {
	var polygons = obj.polygons.map(function(p) {
		return CSG.Polygon.fromObject(p);
	});
	var csg = CSG.fromPolygons(polygons);
	csg = csg.canonicalized();
	return csg;
};

// Reconstruct a CSG from the output of toCompactBinary()
CSG.fromCompactBinary = function(bin) {
	if(bin['class'] != "CSG") throw new Error("Not a CSG");
	var planes = [],
		planeData = bin.planeData,
		numplanes = planeData.length / 4,
		arrayindex = 0,
		x, y, z, w, normal, plane;
	for(var planeindex = 0; planeindex < numplanes; planeindex++) {
		x = planeData[arrayindex++];
		y = planeData[arrayindex++];
		z = planeData[arrayindex++];
		w = planeData[arrayindex++];
		normal = new CSG.Vector3D(x, y, z);
		plane = new CSG.Plane(normal, w);
		planes.push(plane);
	}

	var vertices = [],
		vertexData = bin.vertexData,
		numvertices = vertexData.length / 3,
		pos, vertex;
	arrayindex = 0;
	for(var vertexindex = 0; vertexindex < numvertices; vertexindex++) {
		x = vertexData[arrayindex++];
		y = vertexData[arrayindex++];
		z = vertexData[arrayindex++];
		pos = new CSG.Vector3D(x, y, z);
		vertex = new CSG.Vertex(pos);
		vertices.push(vertex);
	}

	var shareds = bin.shared.map(function(shared) {
		return CSG.Polygon.Shared.fromObject(shared);
	});

	var polygons = [],
		numpolygons = bin.numPolygons,
		numVerticesPerPolygon = bin.numVerticesPerPolygon,
		polygonVertices = bin.polygonVertices,
		polygonPlaneIndexes = bin.polygonPlaneIndexes,
		polygonSharedIndexes = bin.polygonSharedIndexes,
		numpolygonvertices, polygonvertices, shared, polygon;//already defined plane,
	arrayindex = 0;
	for(var polygonindex = 0; polygonindex < numpolygons; polygonindex++) {
		numpolygonvertices = numVerticesPerPolygon[polygonindex];
		polygonvertices = [];
		for(var i = 0; i < numpolygonvertices; i++) {
			polygonvertices.push(vertices[polygonVertices[arrayindex++]]);
		}
		plane = planes[polygonPlaneIndexes[polygonindex]];
		shared = shareds[polygonSharedIndexes[polygonindex]];
		polygon = new CSG.Polygon(polygonvertices, shared, plane);
		polygons.push(polygon);
	}
	var csg = CSG.fromPolygons(polygons);
	csg.isCanonicalized = true;
	csg.isRetesselated = true;
	return csg;
};

CSG.prototype = {
	toPolygons: function() {
		return this.polygons;
	},

	// Return a new CSG solid representing space in either this solid or in the
	// solid `csg`. Neither this solid nor the solid `csg` are modified.
	//
	//     A.union(B)
	//
	//     +-------+            +-------+
	//     |       |            |       |
	//     |   A   |            |       |
	//     |    +--+----+   =   |       +----+
	//     +----+--+    |       +----+       |
	//          |   B   |            |       |
	//          |       |            |       |
	//          +-------+            +-------+
	//
	union: function(csg) {
		var csgs;
		if(csg instanceof Array) {
			csgs = csg;
		} else {
			csgs = [csg];
		}
		var result = this;
		for(var i = 0; i < csgs.length; i++) {
			var islast = (i == (csgs.length - 1));
			result = result.unionSub(csgs[i], islast, islast);
		}
		return result;
	},

	unionSub: function(csg, retesselate, canonicalize) {
		if(!this.mayOverlap(csg)) {
			return this.unionForNonIntersecting(csg);
		} else {
			var a = new CSG.Tree(this.polygons);
			var b = new CSG.Tree(csg.polygons);
			a.clipTo(b, false);

			// b.clipTo(a, true); // ERROR: this doesn't work
			b.clipTo(a);
			b.invert();
			b.clipTo(a);
			b.invert();

			var newpolygons = a.allPolygons().concat(b.allPolygons());
			var result = CSG.fromPolygons(newpolygons);
			result.properties = this.properties._merge(csg.properties);
			if(retesselate) result = result.reTesselated();
			if(canonicalize) result = result.canonicalized();
			return result;
		}
	},

	// Like union, but when we know that the two solids are not intersecting
	// Do not use if you are not completely sure that the solids do not intersect!
	unionForNonIntersecting: function(csg) {
		var newpolygons = this.polygons.concat(csg.polygons);
		var result = CSG.fromPolygons(newpolygons);
		result.properties = this.properties._merge(csg.properties);
		result.isCanonicalized = this.isCanonicalized && csg.isCanonicalized;
		result.isRetesselated = this.isRetesselated && csg.isRetesselated;
		return result;
	},

	// Return a new CSG solid representing space in this solid but not in the
	// solid `csg`. Neither this solid nor the solid `csg` are modified.
	//
	//     A.subtract(B)
	//
	//     +-------+            +-------+
	//     |       |            |       |
	//     |   A   |            |       |
	//     |    +--+----+   =   |    +--+
	//     +----+--+    |       +----+
	//          |   B   |
	//          |       |
	//          +-------+
	//
	subtract: function(csg) {
		var csgs;
		if(csg instanceof Array) {
			csgs = csg;
		} else {
			csgs = [csg];
		}
		var result = this;
		for(var i = 0; i < csgs.length; i++) {
			var islast = (i == (csgs.length - 1));
			result = result.subtractSub(csgs[i], islast, islast);
		}
		return result;
	},

	subtractSub: function(csg, retesselate, canonicalize) {
		var a = new CSG.Tree(this.polygons);
		var b = new CSG.Tree(csg.polygons);
		a.invert();
		a.clipTo(b);
		b.clipTo(a, true);
		a.addPolygons(b.allPolygons());
		a.invert();
		var result = CSG.fromPolygons(a.allPolygons());
		result.properties = this.properties._merge(csg.properties);
		if(retesselate) result = result.reTesselated();
		if(canonicalize) result = result.canonicalized();
		return result;
	},

	// Return a new CSG solid representing space both this solid and in the
	// solid `csg`. Neither this solid nor the solid `csg` are modified.
	//
	//     A.intersect(B)
	//
	//     +-------+
	//     |       |
	//     |   A   |
	//     |    +--+----+   =   +--+
	//     +----+--+    |       +--+
	//          |   B   |
	//          |       |
	//          +-------+
	//
	intersect: function(csg) {
		var csgs;
		if(csg instanceof Array) {
			csgs = csg;
		} else {
			csgs = [csg];
		}
		var result = this;
		for(var i = 0; i < csgs.length; i++) {
			var islast = (i == (csgs.length - 1));
			result = result.intersectSub(csgs[i], islast, islast);
		}
		return result;
	},

	intersectSub: function(csg, retesselate, canonicalize) {
		var a = new CSG.Tree(this.polygons);
		var b = new CSG.Tree(csg.polygons);
		a.invert();
		b.clipTo(a);
		b.invert();
		a.clipTo(b);
		b.clipTo(a);
		a.addPolygons(b.allPolygons());
		a.invert();
		var result = CSG.fromPolygons(a.allPolygons());
		result.properties = this.properties._merge(csg.properties);
		if(retesselate) result = result.reTesselated();
		if(canonicalize) result = result.canonicalized();
		return result;
	},

	// Return a new CSG solid with solid and empty space switched. This solid is
	// not modified.
	inverse: function() {
		var flippedpolygons = this.polygons.map(function(p) {
			return p.flipped();
		});
		return CSG.fromPolygons(flippedpolygons);
		// TODO: flip properties
	},

	// Affine transformation of CSG object. Returns a new CSG object
	transform1: function(matrix4x4) {
		var newpolygons = this.polygons.map(function(p) {
			return p.transform(matrix4x4);
		});
		var result = CSG.fromPolygons(newpolygons);
		result.properties = this.properties._transform(matrix4x4);
		result.isRetesselated = this.isRetesselated;
		return result;
	},

	transform: function(matrix4x4) {
		var ismirror = matrix4x4.isMirroring();
		var transformedvertices = {};
		var transformedplanes = {};
		var newpolygons = this.polygons.map(function(p) {
			var newplane;
			var plane = p.plane;
			var planetag = plane.getTag();
			if(planetag in transformedplanes) {
				newplane = transformedplanes[planetag];
			} else {
				newplane = plane.transform(matrix4x4);
				transformedplanes[planetag] = newplane;
			}
			var newvertices = p.vertices.map(function(v) {
				var newvertex;
				var vertextag = v.getTag();
				if(vertextag in transformedvertices) {
					newvertex = transformedvertices[vertextag];
				} else {
					newvertex = v.transform(matrix4x4);
					transformedvertices[vertextag] = newvertex;
				}
				return newvertex;
			});
			if(ismirror) newvertices.reverse();
			return new CSG.Polygon(newvertices, p.shared, newplane);
		});
		var result = CSG.fromPolygons(newpolygons);
		result.properties = this.properties._transform(matrix4x4);
		result.isRetesselated = this.isRetesselated;
		result.isCanonicalized = this.isCanonicalized;
		return result;
	},

	toStlString: function() {
		var result = "solid csg.js\n";
		this.polygons.map(function(p) {
			result += p.toStlString();
		});
		result += "endsolid csg.js\n";
		return result;
	},

	toAMFString: function(m) {
		var result = "<?xml version=\"1.0\" encoding=\"UTF-8\"?>\n<amf"+(m&&m.unit?" unit=\"+m.unit\"":"")+">\n";
		for(var k in m) {
			result += "<metadata type=\""+k+"\">"+m[k]+"</metadata>\n";
		}
		result += "<object id=\"0\">\n<mesh>\n<vertices>\n";

		this.polygons.map(function(p) {                  // first we dump all vertices of all polygons
         for(var i=0; i<p.vertices.length; i++) {
            result += p.vertices[i].toAMFString();
         }
      });
      result += "</vertices>\n";

      var n = 0;
		this.polygons.map(function(p) {                  // then we dump all polygons
         result += "<volume>\n";
         if(p.vertices.length<3) 
            return;
			var r = 1, g = 0.4, b = 1, a = 1, colorSet = false;
			if(p.shared && p.shared.color) {
				r = p.shared.color[0];
				g = p.shared.color[1];
				b = p.shared.color[2];
            a = p.shared.color[3];
				colorSet = true;
			} else if(p.color) {
            r = p.color[0];
            g = p.color[1];
            b = p.color[2];
            if(p.color.length()>3) a = p.color[3];
				colorSet = true;
         }

   		result += "<color><r>"+r+"</r><g>"+g+"</g><b>"+b+"</b>"+(a!==undefined?"<a>"+a+"</a>":"")+"</color>";

         for(var i=0; i<p.vertices.length-2; i++) {      // making sure they are all triangles (triangular polygons)
            result += "<triangle>";
            result += "<v1>" + (n) + "</v1>";
            result += "<v2>" + (n+i+1) + "</v2>";
            result += "<v3>" + (n+i+2) + "</v3>";
            result += "</triangle>\n";
         }
         n += p.vertices.length;
         result += "</volume>\n";
		});
		result += "</mesh>\n</object>\n";
		result += "</amf>\n";
		return result;
	},

	toX3D: function() {
		// materialPolygonLists
		// key: a color string (e.g. "0 1 1" for yellow)
		// value: an array of strings specifying polygons of this color
		//        (as space-separated indices into vertexCoords)
		var materialPolygonLists = {},
		// list of coordinates (as "x y z" strings)
			vertexCoords = [],
		// map to look up the index in vertexCoords of a given vertex
			vertexTagToCoordIndexMap = {};

		this.polygons.map(function(p) {
			var red = 0,
				green = 0,
				blue = 1; // default color is blue
			if(p.shared && p.shared.color) {
				red = p.shared.color[0];
				green = p.shared.color[1];
				blue = p.shared.color[2];
			}

			var polygonVertexIndices = [],
				numvertices = p.vertices.length,
				vertex;
			for(var i = 0; i < numvertices; i++) {
				vertex = p.vertices[i];
				if(!(vertex.getTag() in vertexTagToCoordIndexMap)) {
					vertexCoords.push(vertex.pos._x.toString() + " " +
						vertex.pos._y.toString() + " " +
						vertex.pos._z.toString()
					);
					vertexTagToCoordIndexMap[vertex.getTag()] = vertexCoords.length - 1;
				}
				polygonVertexIndices.push(vertexTagToCoordIndexMap[vertex.getTag()]);
			}

			var polygonString = polygonVertexIndices.join(" ");

			var colorString = red.toString() + " " + green.toString() + " " + blue.toString();
			if(!(colorString in materialPolygonLists)) {
				materialPolygonLists[colorString] = [];
			}
			// add this polygonString to the list of colorString-colored polygons
			materialPolygonLists[colorString].push(polygonString);
		});


		// create output document
		var docType = document.implementation.createDocumentType("X3D",
			'ISO//Web3D//DTD X3D 3.1//EN" "http://www.web3d.org/specifications/x3d-3.1.dtd', null);
		var exportDoc = document.implementation.createDocument(null, "X3D", docType);
		exportDoc.insertBefore(
			exportDoc.createProcessingInstruction('xml', 'version="1.0" encoding="UTF-8"'),
			exportDoc.doctype);

		var exportRoot = exportDoc.getElementsByTagName("X3D")[0];
		exportRoot.setAttribute("profile", "Interchange");
		exportRoot.setAttribute("version", "3.1");
		exportRoot.setAttribute("xsd:noNamespaceSchemaLocation","http://www.web3d.org/specifications/x3d-3.1.xsd");
		exportRoot.setAttribute("xmlns:xsd", "http://www.w3.org/2001/XMLSchema-instance");

		var exportScene = exportDoc.createElement("Scene");
		exportRoot.appendChild(exportScene);

		/*
	  For each color, create a shape made of an appropriately colored
	  material which contains all polygons that are this color.

	  The first shape will contain the definition of all vertices,
	  (<Coordinate DEF="coords_mesh"/>), which will be referenced by
	  subsequent shapes.
	*/
		var coordsMeshDefined = false;
		for(var colorString in materialPolygonLists) {
			var polygonList = materialPolygonLists[colorString];
			var shape = exportDoc.createElement("Shape");
			exportScene.appendChild(shape);

			var appearance = exportDoc.createElement("Appearance");
			shape.appendChild(appearance);

			var material = exportDoc.createElement("Material");
			appearance.appendChild(material);
			material.setAttribute("diffuseColor", colorString);
			material.setAttribute("ambientIntensity", "1.0");

			var ifs = exportDoc.createElement("IndexedFaceSet");
			shape.appendChild(ifs);
			ifs.setAttribute("solid", "true");
			ifs.setAttribute("coordIndex", polygonList.join(" -1 ") + " -1");

			var coordinate = exportDoc.createElement("Coordinate");
			ifs.appendChild(coordinate);
			if(coordsMeshDefined) {
				coordinate.setAttribute("USE", "coords_mesh");
			} else {
				coordinate.setAttribute("DEF", "coords_mesh");
				coordinate.setAttribute("point", vertexCoords.join(" "));
				coordsMeshDefined = true;
			}
		}

		var x3dstring = (new XMLSerializer()).serializeToString(exportDoc);
		return new Blob([x3dstring], {
			type: "model/x3d+xml"
		});
	},

	// see http://en.wikipedia.org/wiki/STL_%28file_format%29#Binary_STL
	toStlBinary: function(p) {
		// first check if the host is little-endian:
		var buffer = new ArrayBuffer(4);
		var int32buffer = new Int32Array(buffer, 0, 1);
		var int8buffer = new Int8Array(buffer, 0, 4);
		int32buffer[0] = 0x11223344;
		if(int8buffer[0] != 0x44) {
			throw new Error("Binary STL output is currently only supported on little-endian (Intel) processors");
		}

		var numtriangles = 0;
		this.polygons.map(function(p) {
			var numvertices = p.vertices.length;
			var thisnumtriangles = (numvertices >= 3) ? numvertices - 2 : 0;
			numtriangles += thisnumtriangles;
		});
		var headerarray = new Uint8Array(80);
		for(var i = 0; i < 80; i++) {
			headerarray[i] = 65;
		}
		var ar1 = new Uint32Array(1);
		ar1[0] = numtriangles;
		// write the triangles to allTrianglesBuffer:
		var allTrianglesBuffer = new ArrayBuffer(50 * numtriangles);
		var allTrianglesBufferAsInt8 = new Int8Array(allTrianglesBuffer);
		// a tricky problem is that a Float32Array must be aligned at 4-byte boundaries (at least in certain browsers)
		// while each triangle takes 50 bytes. Therefore we write each triangle to a temporary buffer, and copy that
		// into allTrianglesBuffer:
		var triangleBuffer = new ArrayBuffer(50);
		var triangleBufferAsInt8 = new Int8Array(triangleBuffer);
		// each triangle consists of 12 floats:
		var triangleFloat32array = new Float32Array(triangleBuffer, 0, 12);
		// and one uint16:
		var triangleUint16array = new Uint16Array(triangleBuffer, 48, 1);
		var byteoffset = 0;
		this.polygons.map(function(p) {
			var numvertices = p.vertices.length;
			for(var i = 0; i < numvertices - 2; i++) {
				var normal = p.plane.normal;
				triangleFloat32array[0] = normal._x;
				triangleFloat32array[1] = normal._y;
				triangleFloat32array[2] = normal._z;
				var arindex = 3;
				for(var v = 0; v < 3; v++) {
					var vv = v + ((v > 0) ? i : 0);
					var vertexpos = p.vertices[vv].pos;
					triangleFloat32array[arindex++] = vertexpos._x;
					triangleFloat32array[arindex++] = vertexpos._y;
					triangleFloat32array[arindex++] = vertexpos._z;
				}
				triangleUint16array[0] = 0;
				// copy the triangle into allTrianglesBuffer:
				allTrianglesBufferAsInt8.set(triangleBufferAsInt8, byteoffset);
				byteoffset += 50;
			}
		});
      if(p&&p.webBlob) {      // -- want a blob direct
			return new Blob([headerarray.buffer, ar1.buffer, allTrianglesBuffer], {
			   type: "application/sla"
			});
      } else {
         // we differentiate, as binary string blobbing gives bad blob in web, we need binary string for CLI
         //    perhaps there is a way to make it working (see openjscad for stlb)\
         //
         // concat 3 buffers together -- don't make blob so early, we want data (non-blob) for nodejs too
         //    must be string data direct to write
         var stl = new Uint8Array(headerarray.buffer.byteLength + ar1.buffer.byteLength + allTrianglesBuffer.byteLength);
         var j = 0;
         for(var i=0; i<headerarray.buffer.byteLength; i++) { stl.buffer[j++] = headerarray.buffer[i]; }
         for(var i=0; i<ar1.buffer.byteLength; i++) { stl.buffer[j++] = ar1.buffer[i]; }
         for(var i=0; i<allTrianglesBuffer.byteLength; i++) { stl.buffer[j++] = allTrianglesBuffer[i]; }
         return String.fromCharCode.apply(null, new Uint8Array(stl.buffer)); 
         
      }   
	},

	toString: function() {
		var result = "CSG solid:\n";
		this.polygons.map(function(p) {
			result += p.toString();
		});
		return result;
	},

   center: function(c) {
      if(!c.length) c = [c,c,c];
      var b = this.getBounds();
      return this.translate([
         c[0]?-(b[1].x-b[0].x)/2-b[0].x:0,
         c[1]?-(b[1].y-b[0].y)/2-b[0].y:0,
         c[2]?-(b[1].z-b[0].z)/2-b[0].z:0]);
   },

	// Expand the solid
	// resolution: number of points per 360 degree for the rounded corners
	expand: function(radius, resolution) {
		var result = this.expandedShell(radius, resolution, true);
		result = result.reTesselated();
		result.properties = this.properties; // keep original properties
		return result;
	},

	// Contract the solid
	// resolution: number of points per 360 degree for the rounded corners
	contract: function(radius, resolution) {
		var expandedshell = this.expandedShell(radius, resolution, false);
		var result = this.subtract(expandedshell);
		result = result.reTesselated();
		result.properties = this.properties; // keep original properties
		return result;
	},

	// Create the expanded shell of the solid:
	// All faces are extruded to get a thickness of 2*radius
	// Cylinders are constructed around every side
	// Spheres are placed on every vertex
	// unionWithThis: if true, the resulting solid will be united with 'this' solid;
	//   the result is a true expansion of the solid
	//   If false, returns only the shell
	expandedShell: function(radius, resolution, unionWithThis) {
		var csg = this.reTesselated();
		var result;
		if(unionWithThis) {
			result = csg;
		} else {
			result = new CSG();
		}

		// first extrude all polygons:
		csg.polygons.map(function(polygon) {
			var extrudevector = polygon.plane.normal.unit().times(2 * radius);
			var translatedpolygon = polygon.translate(extrudevector.times(-0.5));
			var extrudedface = translatedpolygon.extrude(extrudevector);
			result = result.unionSub(extrudedface, false, false);
		});

		// Make a list of all unique vertex pairs (i.e. all sides of the solid)
		// For each vertex pair we collect the following:
		//   v1: first coordinate
		//   v2: second coordinate
		//   planenormals: array of normal vectors of all planes touching this side
		var vertexpairs = {}; // map of 'vertex pair tag' to {v1, v2, planenormals}
		csg.polygons.map(function(polygon) {
			var numvertices = polygon.vertices.length;
			var prevvertex = polygon.vertices[numvertices - 1];
			var prevvertextag = prevvertex.getTag();
			for(var i = 0; i < numvertices; i++) {
				var vertex = polygon.vertices[i];
				var vertextag = vertex.getTag();
				var vertextagpair;
				if(vertextag < prevvertextag) {
					vertextagpair = vertextag + "-" + prevvertextag;
				} else {
					vertextagpair = prevvertextag + "-" + vertextag;
				}
				var obj;
				if(vertextagpair in vertexpairs) {
					obj = vertexpairs[vertextagpair];
				} else {
					obj = {
						v1: prevvertex,
						v2: vertex,
						planenormals: []
					};
					vertexpairs[vertextagpair] = obj;
				}
				obj.planenormals.push(polygon.plane.normal);

				prevvertextag = vertextag;
				prevvertex = vertex;
			}
		});

		// now construct a cylinder on every side
		// The cylinder is always an approximation of a true cylinder: it will have <resolution> polygons
		// around the sides. We will make sure though that the cylinder will have an edge at every
		// face that touches this side. This ensures that we will get a smooth fill even
		// if two edges are at, say, 10 degrees and the resolution is low.
		// Note: the result is not retesselated yet but it really should be!
		for(var vertextagpair in vertexpairs) {
			var vertexpair = vertexpairs[vertextagpair],
				startpoint = vertexpair.v1.pos,
				endpoint = vertexpair.v2.pos,
			// our x,y and z vectors:
				zbase = endpoint.minus(startpoint).unit(),
				xbase = vertexpair.planenormals[0].unit(),
				ybase = xbase.cross(zbase),

			// make a list of angles that the cylinder should traverse:
				angles = [];

			// first of all equally spaced around the cylinder:
			for(var i = 0; i < resolution; i++) {
				angles.push(i * Math.PI * 2 / resolution);
			}

			// and also at every normal of all touching planes:
			for(var i = 0, iMax = vertexpair.planenormals.length; i < iMax; i++) {
				var planenormal = vertexpair.planenormals[i],
					si = ybase.dot(planenormal),
					co = xbase.dot(planenormal),
					angle = Math.atan2(si, co);

				if(angle < 0) angle += Math.PI * 2;
				angles.push(angle);
				angle = Math.atan2(-si, -co);
				if(angle < 0) angle += Math.PI * 2;
				angles.push(angle);
			}

			// this will result in some duplicate angles but we will get rid of those later.
			// Sort:
			angles = angles.sort(fnNumberSort);

			// Now construct the cylinder by traversing all angles:
			var numangles = angles.length,
				prevp1, prevp2,
				startfacevertices = [],
				endfacevertices = [],
				polygons = [];
			for(var i = -1; i < numangles; i++) {
				var angle = angles[(i < 0) ? (i + numangles) : i],
					si = Math.sin(angle),
					co = Math.cos(angle),
					p = xbase.times(co * radius).plus(ybase.times(si * radius)),
					p1 = startpoint.plus(p),
					p2 = endpoint.plus(p),
					skip = false;
				if(i >= 0) {
					if(p1.distanceTo(prevp1) < 1e-5) {
						skip = true;
					}
				}
				if(!skip) {
					if(i >= 0) {
						startfacevertices.push(new CSG.Vertex(p1));
						endfacevertices.push(new CSG.Vertex(p2));
						var polygonvertices = [
							new CSG.Vertex(prevp2),
							new CSG.Vertex(p2),
							new CSG.Vertex(p1),
							new CSG.Vertex(prevp1)];
						var polygon = new CSG.Polygon(polygonvertices);
						polygons.push(polygon);
					}
					prevp1 = p1;
					prevp2 = p2;
				}
			}
			endfacevertices.reverse();
			polygons.push(new CSG.Polygon(startfacevertices));
			polygons.push(new CSG.Polygon(endfacevertices));
			var cylinder = CSG.fromPolygons(polygons);
			result = result.unionSub(cylinder, false, false);
		}

		// make a list of all unique vertices
		// For each vertex we also collect the list of normals of the planes touching the vertices
		var vertexmap = {};
		csg.polygons.map(function(polygon) {
			polygon.vertices.map(function(vertex) {
				var vertextag = vertex.getTag();
				var obj;
				if(vertextag in vertexmap) {
					obj = vertexmap[vertextag];
				} else {
					obj = {
						pos: vertex.pos,
						normals: []
					};
					vertexmap[vertextag] = obj;
				}
				obj.normals.push(polygon.plane.normal);
			});
		});

		// and build spheres at each vertex
		// We will try to set the x and z axis to the normals of 2 planes
		// This will ensure that our sphere tesselation somewhat matches 2 planes
		for(var vertextag in vertexmap) {
			var vertexobj = vertexmap[vertextag];
			// use the first normal to be the x axis of our sphere:
			var xaxis = vertexobj.normals[0].unit();
			// and find a suitable z axis. We will use the normal which is most perpendicular to the x axis:
			var bestzaxis = null;
			var bestzaxisorthogonality = 0;
			for(var i = 1; i < vertexobj.normals.length; i++) {
				var normal = vertexobj.normals[i].unit();
				var cross = xaxis.cross(normal);
				var crosslength = cross.length();
				if(crosslength > 0.05) {
					if(crosslength > bestzaxisorthogonality) {
						bestzaxisorthogonality = crosslength;
						bestzaxis = normal;
					}
				}
			}
			if(!bestzaxis) {
				bestzaxis = xaxis.randomNonParallelVector();
			}
			var yaxis = xaxis.cross(bestzaxis).unit();
			var zaxis = yaxis.cross(xaxis);
			var sphere = CSG.sphere({
				center: vertexobj.pos,
				radius: radius,
				resolution: resolution,
				axes: [xaxis, yaxis, zaxis]
			});
			result = result.unionSub(sphere, false, false);
		}

		return result;
	},

	canonicalized: function() {
		if(this.isCanonicalized) {
			return this;
		} else {
			var factory = new CSG.fuzzyCSGFactory();
			var result = factory.getCSG(this);
			result.isCanonicalized = true;
			result.isRetesselated = this.isRetesselated;
			result.properties = this.properties; // keep original properties
			return result;
		}
	},

	reTesselated: function() {
		if(this.isRetesselated) {
			return this;
		} else {
			var csg = this.canonicalized();
			var polygonsPerPlane = {};
			csg.polygons.map(function(polygon) {
				var planetag = polygon.plane.getTag();
				var sharedtag = polygon.shared.getTag();
				planetag += "/" + sharedtag;
				if(!(planetag in polygonsPerPlane)) {
					polygonsPerPlane[planetag] = [];
				}
				polygonsPerPlane[planetag].push(polygon);
			});
			var destpolygons = [];
			for(var planetag in polygonsPerPlane) {
				var sourcepolygons = polygonsPerPlane[planetag];
				if(sourcepolygons.length < 2) {
					destpolygons = destpolygons.concat(sourcepolygons);
				} else {
					var retesselayedpolygons = [];
					CSG.reTesselateCoplanarPolygons(sourcepolygons, retesselayedpolygons);
					destpolygons = destpolygons.concat(retesselayedpolygons);
				}
			}
			var result = CSG.fromPolygons(destpolygons);
			result.isRetesselated = true;
			result = result.canonicalized();
			//      result.isCanonicalized = true;
			result.properties = this.properties; // keep original properties
			return result;
		}
	},

	// returns an array of two CSG.Vector3Ds (minimum coordinates and maximum coordinates)
	getBounds: function() {
		if(!this.cachedBoundingBox) {
			var minpoint = new CSG.Vector3D(0, 0, 0);
			var maxpoint = new CSG.Vector3D(0, 0, 0);
			var polygons = this.polygons;
			var numpolygons = polygons.length;
			for(var i = 0; i < numpolygons; i++) {
				var polygon = polygons[i];
				var bounds = polygon.boundingBox();
				if(i === 0) {
					minpoint = bounds[0];
					maxpoint = bounds[1];
				} else {
					minpoint = minpoint.min(bounds[0]);
					maxpoint = maxpoint.max(bounds[1]);
				}
			}
			this.cachedBoundingBox = [minpoint, maxpoint];
		}
		return this.cachedBoundingBox;
	},

	// returns true if there is a possibility that the two solids overlap
	// returns false if we can be sure that they do not overlap
	mayOverlap: function(csg) {
		if((this.polygons.length === 0) || (csg.polygons.length === 0)) {
			return false;
		} else {
			var mybounds = this.getBounds();
			var otherbounds = csg.getBounds();
         // [0].x/y  
         //    +-----+
         //    |     |
         //    |     |
         //    +-----+ 
         //          [1].x/y
         //return false;
         //echo(mybounds,"=",otherbounds);
			if(mybounds[1].x < otherbounds[0].x) return false;
			if(mybounds[0].x > otherbounds[1].x) return false;
			if(mybounds[1].y < otherbounds[0].y) return false;
			if(mybounds[0].y > otherbounds[1].y) return false;
			if(mybounds[1].z < otherbounds[0].z) return false;
			if(mybounds[0].z > otherbounds[1].z) return false;
			return true;
		}
	},

	// Cut the solid by a plane. Returns the solid on the back side of the plane
	cutByPlane: function(plane) {
		if(this.polygons.length === 0) {
			return new CSG();
		}
		// Ideally we would like to do an intersection with a polygon of inifinite size
		// but this is not supported by our implementation. As a workaround, we will create
		// a cube, with one face on the plane, and a size larger enough so that the entire
		// solid fits in the cube.
		// find the max distance of any vertex to the center of the plane:
		var planecenter = plane.normal.times(plane.w);
		var maxdistance = 0;
		this.polygons.map(function(polygon) {
			polygon.vertices.map(function(vertex) {
				var distance = vertex.pos.distanceToSquared(planecenter);
				if(distance > maxdistance) maxdistance = distance;
			});
		});
		maxdistance = Math.sqrt(maxdistance);
		maxdistance *= 1.01; // make sure it's really larger
		// Now build a polygon on the plane, at any point farther than maxdistance from the plane center:
		var vertices = [];
		var orthobasis = new CSG.OrthoNormalBasis(plane);
		vertices.push(new CSG.Vertex(orthobasis.to3D(new CSG.Vector2D(maxdistance, -maxdistance))));
		vertices.push(new CSG.Vertex(orthobasis.to3D(new CSG.Vector2D(-maxdistance, -maxdistance))));
		vertices.push(new CSG.Vertex(orthobasis.to3D(new CSG.Vector2D(-maxdistance, maxdistance))));
		vertices.push(new CSG.Vertex(orthobasis.to3D(new CSG.Vector2D(maxdistance, maxdistance))));
		var polygon = new CSG.Polygon(vertices, null, plane.flipped());

		// and extrude the polygon into a cube, backwards of the plane:
		var cube = polygon.extrude(plane.normal.times(-maxdistance));

		// Now we can do the intersection:
		var result = this.intersect(cube);
		result.properties = this.properties; // keep original properties
		return result;
	},

	// Connect a solid to another solid, such that two CSG.Connectors become connected
	//   myConnector: a CSG.Connector of this solid
	//   otherConnector: a CSG.Connector to which myConnector should be connected
	//   mirror: false: the 'axis' vectors of the connectors should point in the same direction
	//           true: the 'axis' vectors of the connectors should point in opposite direction
	//   normalrotation: degrees of rotation between the 'normal' vectors of the two
	//                   connectors
	connectTo: function(myConnector, otherConnector, mirror, normalrotation) {
		var matrix = myConnector.getTransformationTo(otherConnector, mirror, normalrotation);
		return this.transform(matrix);
	},

	// set the .shared property of all polygons
	// Returns a new CSG solid, the original is unmodified!
	setShared: function(shared) {
		var polygons = this.polygons.map(function(p) {
			return new CSG.Polygon(p.vertices, shared, p.plane);
		});
		var result = CSG.fromPolygons(polygons);
		result.properties = this.properties; // keep original properties
		result.isRetesselated = this.isRetesselated;
		result.isCanonicalized = this.isCanonicalized;
		return result;
	},

	/**
	 * @param {Array} color [red, green, blue] color values are float numbers 0..1
	 * @return {CSG} new CSG instance
	 */
	setColor: function(red, green, blue, alpha) { //for backward compatibility
		var color = red instanceof Array ? red : [red||0, green||0, blue||0, isNaN(alpha) ? 1. : alpha];
		var newshared = new CSG.Polygon.Shared(color);
		return this.setShared(newshared);
	},

	toCompactBinary: function() {
		var csg = this.canonicalized(),
			numpolygons = csg.polygons.length,
			numpolygonvertices = 0,
			numvertices = 0,
			vertexmap = {},
			vertices = [],
			numplanes = 0,
			planemap = {},
			polygonindex = 0,
			planes = [],
			shareds = [],
			sharedmap = {},
			numshared = 0;
		// for (var i = 0, iMax = csg.polygons.length; i < iMax; i++) {
		// 	var p = csg.polygons[i];
		// 	for (var j = 0, jMax = p.length; j < jMax; j++) {
		// 		++numpolygonvertices;
		// 		var vertextag = p[j].getTag();
		// 		if(!(vertextag in vertexmap)) {
		// 			vertexmap[vertextag] = numvertices++;
		// 			vertices.push(p[j]);
		// 		}
		// 	}
		csg.polygons.map(function(p) {
			p.vertices.map(function(v) {
				++numpolygonvertices;
				var vertextag = v.getTag();
				if(!(vertextag in vertexmap)) {
					vertexmap[vertextag] = numvertices++;
					vertices.push(v);
				}
			});

			var planetag = p.plane.getTag();
			if(!(planetag in planemap)) {
				planemap[planetag] = numplanes++;
				planes.push(p.plane);
			}
			var sharedtag = p.shared.getTag();
			if(!(sharedtag in sharedmap)) {
				sharedmap[sharedtag] = numshared++;
				shareds.push(p.shared);
			}
		});
		var numVerticesPerPolygon = new Uint32Array(numpolygons),
			polygonSharedIndexes = new Uint32Array(numpolygons),
			polygonVertices = new Uint32Array(numpolygonvertices),
			polygonPlaneIndexes = new Uint32Array(numpolygons),
			vertexData = new Float64Array(numvertices * 3),
			planeData = new Float64Array(numplanes * 4),
			polygonVerticesIndex = 0;
		for(var polygonindex = 0; polygonindex < numpolygons; ++polygonindex) {
			var p = csg.polygons[polygonindex];
			numVerticesPerPolygon[polygonindex] = p.vertices.length;
			p.vertices.map(function(v) {
				var vertextag = v.getTag();
				var vertexindex = vertexmap[vertextag];
				polygonVertices[polygonVerticesIndex++] = vertexindex;
			});
			var planetag = p.plane.getTag();
			var planeindex = planemap[planetag];
			polygonPlaneIndexes[polygonindex] = planeindex;
			var sharedtag = p.shared.getTag();
			var sharedindex = sharedmap[sharedtag];
			polygonSharedIndexes[polygonindex] = sharedindex;
		}
		var verticesArrayIndex = 0;
		vertices.map(function(v) {
			var pos = v.pos;
			vertexData[verticesArrayIndex++] = pos._x;
			vertexData[verticesArrayIndex++] = pos._y;
			vertexData[verticesArrayIndex++] = pos._z;
		});
		var planesArrayIndex = 0;
		planes.map(function(p) {
			var normal = p.normal;
			planeData[planesArrayIndex++] = normal._x;
			planeData[planesArrayIndex++] = normal._y;
			planeData[planesArrayIndex++] = normal._z;
			planeData[planesArrayIndex++] = p.w;
		});
		var result = {
			"class": "CSG",
			numPolygons: numpolygons,
			numVerticesPerPolygon: numVerticesPerPolygon,
			polygonPlaneIndexes: polygonPlaneIndexes,
			polygonSharedIndexes: polygonSharedIndexes,
			polygonVertices: polygonVertices,
			vertexData: vertexData,
			planeData: planeData,
			shared: shareds
		};
		return result;
	},

	// For debugging
	// Creates a new solid with a tiny cube at every vertex of the source solid
	toPointCloud: function(cuberadius) {
		var csg = this.reTesselated();

		var result = new CSG();

		// make a list of all unique vertices
		// For each vertex we also collect the list of normals of the planes touching the vertices
		var vertexmap = {};
		csg.polygons.map(function(polygon) {
			polygon.vertices.map(function(vertex) {
				vertexmap[vertex.getTag()] = vertex.pos;
			});
		});

		for(var vertextag in vertexmap) {
			var pos = vertexmap[vertextag];
			var cube = CSG.cube({
				center: pos,
				radius: cuberadius
			});
			result = result.unionSub(cube, false, false);
		}
		result = result.reTesselated();
		return result;
	},

	// Get the transformation that transforms this CSG such that it is lying on the z=0 plane,
	// as flat as possible (i.e. the least z-height).
	// So that it is in an orientation suitable for CNC milling
	getTransformationToFlatLying: function() {
		if(this.polygons.length === 0) {
			return new CSG.Matrix4x4(); // unity
		} else {
			// get a list of unique planes in the CSG:
			var csg = this.canonicalized();
			var planemap = {};
			csg.polygons.map(function(polygon) {
				planemap[polygon.plane.getTag()] = polygon.plane;
			});
			// try each plane in the CSG and find the plane that, when we align it flat onto z=0,
			// gives the least height in z-direction.
			// If two planes give the same height, pick the plane that originally had a normal closest
			// to [0,0,-1].
			var xvector = new CSG.Vector3D(1, 0, 0);
			var yvector = new CSG.Vector3D(0, 1, 0);
			var zvector = new CSG.Vector3D(0, 0, 1);
			var z0connectorx = new CSG.Connector([0, 0, 0], [0, 0, -1], xvector);
			var z0connectory = new CSG.Connector([0, 0, 0], [0, 0, -1], yvector);
			var isfirst = true;
			var minheight = 0;
			var maxdotz = 0;
			var besttransformation;
			for(var planetag in planemap) {
				var plane = planemap[planetag];
				var pointonplane = plane.normal.times(plane.w);
				var transformation;
				// We need a normal vecrtor for the transformation
				// determine which is more perpendicular to the plane normal: x or y?
				// we will align this as much as possible to the x or y axis vector
				var xorthogonality = plane.normal.cross(xvector).length();
				var yorthogonality = plane.normal.cross(yvector).length();
				if(xorthogonality > yorthogonality) {
					// x is better:
					var planeconnector = new CSG.Connector(pointonplane, plane.normal, xvector);
					transformation = planeconnector.getTransformationTo(z0connectorx, false, 0);
				} else {
					// y is better:
					var planeconnector = new CSG.Connector(pointonplane, plane.normal, yvector);
					transformation = planeconnector.getTransformationTo(z0connectory, false, 0);
				}
				var transformedcsg = csg.transform(transformation);
				var dotz = -plane.normal.dot(zvector);
				var bounds = transformedcsg.getBounds();
				var zheight = bounds[1].z - bounds[0].z;
				var isbetter = isfirst;
				if(!isbetter) {
					if(zheight < minheight) {
						isbetter = true;
					} else if(zheight == minheight) {
						if(dotz > maxdotz) isbetter = true;
					}
				}
				if(isbetter) {
					// translate the transformation around the z-axis and onto the z plane:
					var translation = [
						-0.5 * (bounds[1].x + bounds[0].x),
						-0.5 * (bounds[1].y + bounds[0].y),
						-bounds[0].z];
					transformation = transformation.multiply(CSG.Matrix4x4.translation(translation));
					minheight = zheight;
					maxdotz = dotz;
					besttransformation = transformation;
				}
				isfirst = false;
			}
			return besttransformation;
		}
	},

	lieFlat: function() {
		var transformation = this.getTransformationToFlatLying();
		return this.transform(transformation);
	},

	// project the 3D CSG onto a plane
	// This returns a 2D CAG with the 'shadow' shape of the 3D solid when projected onto the
	// plane represented by the orthonormal basis
	projectToOrthoNormalBasis: function(orthobasis) {
		var cags = [];
		this.polygons.map(function(polygon) {
			var cag = polygon.projectToOrthoNormalBasis(orthobasis);
			if(cag.sides.length > 0) {
				cags.push(cag);
			}
		});
		var result = new CAG().union(cags);
		return result;
	},

	sectionCut: function(orthobasis) {
		var plane1 = orthobasis.plane;
		var plane2 = orthobasis.plane.flipped();
		plane1 = new CSG.Plane(plane1.normal, plane1.w + 1e-4);
		plane2 = new CSG.Plane(plane2.normal, plane2.w + 1e-4);
		var cut3d = this.cutByPlane(plane1);
		cut3d = cut3d.cutByPlane(plane2);
		return cut3d.projectToOrthoNormalBasis(orthobasis);
	},

	/*
  fixTJunctions:

  Suppose we have two polygons ACDB and EDGF:

   A-----B
   |     |
   |     E--F
   |     |  |
   C-----D--G

  Note that vertex E forms a T-junction on the side BD. In this case some STL slicers will complain
  that the solid is not watertight. This is because the watertightness check is done by checking if
  each side DE is matched by another side ED.

  This function will return a new solid with ACDB replaced by ACDEB

  Note that this can create polygons that are slightly non-convex (due to rounding errors). Therefore the result should
  not be used for further CSG operations!
  */
	fixTJunctions: function() {
		var csg = this.canonicalized();
		var sidemap = {};
		for(var polygonindex = 0; polygonindex < csg.polygons.length; polygonindex++) {
			var polygon = csg.polygons[polygonindex];
			var numvertices = polygon.vertices.length;
			if(numvertices >= 3) // should be true
			{
				var vertex = polygon.vertices[0];
				var vertextag = vertex.getTag();
				for(var vertexindex = 0; vertexindex < numvertices; vertexindex++) {
					var nextvertexindex = vertexindex + 1;
					if(nextvertexindex == numvertices) nextvertexindex = 0;
					var nextvertex = polygon.vertices[nextvertexindex];
					var nextvertextag = nextvertex.getTag();
					var sidetag = vertextag + "/" + nextvertextag;
					var reversesidetag = nextvertextag + "/" + vertextag;
					if(reversesidetag in sidemap) {
						// this side matches the same side in another polygon. Remove from sidemap:
						var ar = sidemap[reversesidetag];
						ar.splice(-1, 1);
						if(ar.length === 0) {
							delete sidemap[reversesidetag];
						}
					} else {
						var sideobj = {
							vertex0: vertex,
							vertex1: nextvertex,
							polygonindex: polygonindex
						};
						if(!(sidetag in sidemap)) {
							sidemap[sidetag] = [sideobj];
						} else {
							sidemap[sidetag].push(sideobj);
						}
					}
					vertex = nextvertex;
					vertextag = nextvertextag;
				}
			}
		}
		// now sidemap contains 'unmatched' sides
		// i.e. side AB in one polygon does not have a matching side BA in another polygon
		var vertextag2sidestart = {};
		var vertextag2sideend = {};
		var sidestocheck = {};
		var sidemapisempty = true;
		for(var sidetag in sidemap) {
			sidemapisempty = false;
			sidestocheck[sidetag] = true;
			sidemap[sidetag].map(function(sideobj) {
				var starttag = sideobj.vertex0.getTag();
				var endtag = sideobj.vertex1.getTag();
				if(starttag in vertextag2sidestart) {
					vertextag2sidestart[starttag].push(sidetag);
				} else {
					vertextag2sidestart[starttag] = [sidetag];
				}
				if(endtag in vertextag2sideend) {
					vertextag2sideend[endtag].push(sidetag);
				} else {
					vertextag2sideend[endtag] = [sidetag];
				}
			});
		}

		if(!sidemapisempty) {
			// make a copy of the polygons array, since we are going to modify it:
			var polygons = csg.polygons.slice(0);

			function addSide (vertex0, vertex1, polygonindex) {
				var starttag = vertex0.getTag();
				var endtag = vertex1.getTag();
				if(starttag == endtag) throw new Error("Assertion failed");
				var newsidetag = starttag + "/" + endtag;
				var reversesidetag = endtag + "/" + starttag;
				if(reversesidetag in sidemap) {
					// we have a matching reverse oriented side.
					// Instead of adding the new side, cancel out the reverse side:
					// console.log("addSide("+newsidetag+") has reverse side:");
					deleteSide(vertex1, vertex0, null);
					return null;
				}
				//  console.log("addSide("+newsidetag+")");
				var newsideobj = {
					vertex0: vertex0,
					vertex1: vertex1,
					polygonindex: polygonindex
				};
				if(!(newsidetag in sidemap)) {
					sidemap[newsidetag] = [newsideobj];
				} else {
					sidemap[newsidetag].push(newsideobj);
				}
				if(starttag in vertextag2sidestart) {
					vertextag2sidestart[starttag].push(newsidetag);
				} else {
					vertextag2sidestart[starttag] = [newsidetag];
				}
				if(endtag in vertextag2sideend) {
					vertextag2sideend[endtag].push(newsidetag);
				} else {
					vertextag2sideend[endtag] = [newsidetag];
				}
				return newsidetag;
			}

			function deleteSide (vertex0, vertex1, polygonindex) {
				var starttag = vertex0.getTag();
				var endtag = vertex1.getTag();
				var sidetag = starttag + "/" + endtag;
				// console.log("deleteSide("+sidetag+")");
				if(!(sidetag in sidemap)) throw new Error("Assertion failed");
				var idx = -1;
				var sideobjs = sidemap[sidetag];
				for(var i = 0; i < sideobjs.length; i++) {
					var sideobj = sideobjs[i];
					if(sideobj.vertex0 != vertex0) continue;
					if(sideobj.vertex1 != vertex1) continue;
					if(polygonindex !== null) {
						if(sideobj.polygonindex != polygonindex) continue;
					}
					idx = i;
					break;
				}
				if(idx < 0) throw new Error("Assertion failed");
				sideobjs.splice(idx, 1);
				if(sideobjs.length === 0) {
					delete sidemap[sidetag];
				}
				idx = vertextag2sidestart[starttag].indexOf(sidetag);
				if(idx < 0) throw new Error("Assertion failed");
				vertextag2sidestart[starttag].splice(idx, 1);
				if(vertextag2sidestart[starttag].length === 0) {
					delete vertextag2sidestart[starttag];
				}

				idx = vertextag2sideend[endtag].indexOf(sidetag);
				if(idx < 0) throw new Error("Assertion failed");
				vertextag2sideend[endtag].splice(idx, 1);
				if(vertextag2sideend[endtag].length === 0) {
					delete vertextag2sideend[endtag];
				}
			}


			while(true) {
				var sidemapisempty = true;
				for(var sidetag in sidemap) {
					sidemapisempty = false;
					sidestocheck[sidetag] = true;
				}
				if(sidemapisempty) break;
				var donesomething = false;
				while(true) {
					var sidetagtocheck = null;
					for(var sidetag in sidestocheck) {
						sidetagtocheck = sidetag;
						break;
					}
					if(sidetagtocheck === null) break; // sidestocheck is empty, we're done!
					var donewithside = true;
					if(sidetagtocheck in sidemap) {
						var sideobjs = sidemap[sidetagtocheck];
						if(sideobjs.length === 0) throw new Error("Assertion failed");
						var sideobj = sideobjs[0];
						for(var directionindex = 0; directionindex < 2; directionindex++) {
							var startvertex = (directionindex === 0) ? sideobj.vertex0 : sideobj.vertex1;
							var endvertex = (directionindex === 0) ? sideobj.vertex1 : sideobj.vertex0;
							var startvertextag = startvertex.getTag();
							var endvertextag = endvertex.getTag();
							var matchingsides = [];
							if(directionindex === 0) {
								if(startvertextag in vertextag2sideend) {
									matchingsides = vertextag2sideend[startvertextag];
								}
							} else {
								if(startvertextag in vertextag2sidestart) {
									matchingsides = vertextag2sidestart[startvertextag];
								}
							}
							for(var matchingsideindex = 0; matchingsideindex < matchingsides.length; matchingsideindex++) {
								var matchingsidetag = matchingsides[matchingsideindex];
								var matchingside = sidemap[matchingsidetag][0];
								var matchingsidestartvertex = (directionindex === 0) ? matchingside.vertex0 : matchingside.vertex1;
								var matchingsideendvertex = (directionindex === 0) ? matchingside.vertex1 : matchingside.vertex0;
								var matchingsidestartvertextag = matchingsidestartvertex.getTag();
								var matchingsideendvertextag = matchingsideendvertex.getTag();
								if(matchingsideendvertextag != startvertextag) throw new Error("Assertion failed");
								if(matchingsidestartvertextag == endvertextag) {
									// matchingside cancels sidetagtocheck
									deleteSide(startvertex, endvertex, null);
									deleteSide(endvertex, startvertex, null);
									donewithside = false;
									directionindex = 2; // skip reverse direction check
									donesomething = true;
									break;
								} else {
									var startpos = startvertex.pos;
									var endpos = endvertex.pos;
									var checkpos = matchingsidestartvertex.pos;
									var direction = checkpos.minus(startpos);
									// Now we need to check if endpos is on the line startpos-checkpos:
									var t = endpos.minus(startpos).dot(direction) / direction.dot(direction);
									if((t > 0) && (t < 1)) {
										var closestpoint = startpos.plus(direction.times(t));
										var distancesquared = closestpoint.distanceToSquared(endpos);
										if(distancesquared < 1e-10) {
											// Yes it's a t-junction! We need to split matchingside in two:
											var polygonindex = matchingside.polygonindex;
											var polygon = polygons[polygonindex];
											// find the index of startvertextag in polygon:
											var insertionvertextag = matchingside.vertex1.getTag();
											var insertionvertextagindex = -1;
											for(var i = 0; i < polygon.vertices.length; i++) {
												if(polygon.vertices[i].getTag() == insertionvertextag) {
													insertionvertextagindex = i;
													break;
												}
											}
											if(insertionvertextagindex < 0) throw new Error("Assertion failed");
											// split the side by inserting the vertex:
											var newvertices = polygon.vertices.slice(0);
											newvertices.splice(insertionvertextagindex, 0, endvertex);
											var newpolygon = new CSG.Polygon(newvertices, polygon.shared /*polygon.plane*/ );
											polygons[polygonindex] = newpolygon;

											// remove the original sides from our maps:
											// deleteSide(sideobj.vertex0, sideobj.vertex1, null);
											deleteSide(matchingside.vertex0, matchingside.vertex1, polygonindex);
											var newsidetag1 = addSide(matchingside.vertex0, endvertex, polygonindex);
											var newsidetag2 = addSide(endvertex, matchingside.vertex1, polygonindex);
											if(newsidetag1 !== null) sidestocheck[newsidetag1] = true;
											if(newsidetag2 !== null) sidestocheck[newsidetag2] = true;
											donewithside = false;
											directionindex = 2; // skip reverse direction check
											donesomething = true;
											break;
										} // if(distancesquared < 1e-10)
									} // if( (t > 0) && (t < 1) )
								} // if(endingstidestartvertextag == endvertextag)
							} // for matchingsideindex
						} // for directionindex
					} // if(sidetagtocheck in sidemap)
					if(donewithside) {
						delete sidestocheck[sidetag];
					}
				}
				if(!donesomething) break;
			}
			var newcsg = CSG.fromPolygons(polygons);
			newcsg.properties = csg.properties;
			newcsg.isCanonicalized = true;
			newcsg.isRetesselated = true;
			csg = newcsg;
		} // if(!sidemapisempty)
		var sidemapisempty = true;
		for(var sidetag in sidemap) {
			sidemapisempty = false;
			break;
		}
		if(!sidemapisempty) {
			throw new Error("!sidemapisempty");
		}
		return csg;
	}
};

// Parse an option from the options object
// If the option is not present, return the default value
CSG.parseOption = function(options, optionname, defaultvalue) {
	var result = defaultvalue;
	if(options) {
		if(optionname in options) {
			result = options[optionname];
		}
	}
	return result;
};

// Parse an option and force into a CSG.Vector3D. If a scalar is passed it is converted
// into a vector with equal x,y,z
CSG.parseOptionAs3DVector = function(options, optionname, defaultvalue) {
	var result = CSG.parseOption(options, optionname, defaultvalue);
	result = new CSG.Vector3D(result);
	return result;
};

// Parse an option and force into a CSG.Vector2D. If a scalar is passed it is converted
// into a vector with equal x,y
CSG.parseOptionAs2DVector = function(options, optionname, defaultvalue) {
	var result = CSG.parseOption(options, optionname, defaultvalue);
	result = new CSG.Vector2D(result);
	return result;
};

CSG.parseOptionAsFloat = function(options, optionname, defaultvalue) {
	var result = CSG.parseOption(options, optionname, defaultvalue);
	if(typeof(result) == "string") {
		result = Number(result);
	}
	if(isNaN(result) || typeof(result) != "number") {
		throw new Error("Parameter " + optionname + " should be a number");
	}
	return result;
};

CSG.parseOptionAsInt = function(options, optionname, defaultvalue) {
	var result = CSG.parseOption(options, optionname, defaultvalue);
	result = Number(Math.floor(result));
	if (isNaN(result)) {
		throw new Error("Parameter " + optionname + " should be a number");
	}
	return result;
};

CSG.parseOptionAsBool = function(options, optionname, defaultvalue) {
	var result = CSG.parseOption(options, optionname, defaultvalue);
	if(typeof(result) == "string") {
		if(result == "true") result = true;
		else if(result == "false") result = false;
		else if(result == 0) result = false;
	}
	result = !! result;
	return result;
};

// Construct an axis-aligned solid cuboid.
// Parameters:
//   center: center of cube (default [0,0,0])
//   radius: radius of cube (default [1,1,1]), can be specified as scalar or as 3D vector
//
// Example code:
//
//     var cube = CSG.cube({
//       center: [0, 0, 0],
//       radius: 1
//     });
CSG.cube = function(options) {
	var c = CSG.parseOptionAs3DVector(options, "center", [0, 0, 0]);
	var r = CSG.parseOptionAs3DVector(options, "radius", [1, 1, 1]);
	var result = CSG.fromPolygons([
		[
			[0, 4, 6, 2],
			[-1, 0, 0]
		],
		[
			[1, 3, 7, 5],
			[+1, 0, 0]
		],
		[
			[0, 1, 5, 4],
			[0, -1, 0]
		],
		[
			[2, 6, 7, 3],
			[0, +1, 0]
		],
		[
			[0, 2, 3, 1],
			[0, 0, -1]
		],
		[
			[4, 5, 7, 6],
			[0, 0, +1]
		]
	].map(function(info) {
		//var normal = new CSG.Vector3D(info[1]);
		//var plane = new CSG.Plane(normal, 1);
		var vertices = info[0].map(function(i) {
			var pos = new CSG.Vector3D(
			c.x + r.x * (2 * !! (i & 1) - 1), c.y + r.y * (2 * !! (i & 2) - 1), c.z + r.z * (2 * !! (i & 4) - 1));
			return new CSG.Vertex(pos);
		});
		return new CSG.Polygon(vertices, null /* , plane */ );
	}));
	result.properties.cube = new CSG.Properties();
	result.properties.cube.center = new CSG.Vector3D(c);
	// add 6 connectors, at the centers of each face:
	result.properties.cube.facecenters = [
		new CSG.Connector(new CSG.Vector3D([r.x, 0, 0]).plus(c), [1, 0, 0], [0, 0, 1]),
		new CSG.Connector(new CSG.Vector3D([-r.x, 0, 0]).plus(c), [-1, 0, 0], [0, 0, 1]),
		new CSG.Connector(new CSG.Vector3D([0, r.y, 0]).plus(c), [0, 1, 0], [0, 0, 1]),
		new CSG.Connector(new CSG.Vector3D([0, -r.y, 0]).plus(c), [0, -1, 0], [0, 0, 1]),
		new CSG.Connector(new CSG.Vector3D([0, 0, r.z]).plus(c), [0, 0, 1], [1, 0, 0]),
		new CSG.Connector(new CSG.Vector3D([0, 0, -r.z]).plus(c), [0, 0, -1], [1, 0, 0])
	];
	return result;
};

// Construct a solid sphere
//
// Parameters:
//   center: center of sphere (default [0,0,0])
//   radius: radius of sphere (default 1), must be a scalar
//   resolution: determines the number of polygons per 360 degree revolution (default 12)
//   axes: (optional) an array with 3 vectors for the x, y and z base vectors
//
// Example usage:
//
//     var sphere = CSG.sphere({
//       center: [0, 0, 0],
//       radius: 2,
//       resolution: 32,
//     });
CSG.sphere = function(options) {
	options = options || {};
	var center = CSG.parseOptionAs3DVector(options, "center", [0, 0, 0]);
	var radius = CSG.parseOptionAsFloat(options, "radius", 1);
	var resolution = CSG.parseOptionAsInt(options, "resolution", CSG.defaultResolution3D);
	var xvector, yvector, zvector;
	if('axes' in options) {
		xvector = options.axes[0].unit().times(radius);
		yvector = options.axes[1].unit().times(radius);
		zvector = options.axes[2].unit().times(radius);
	} else {
		xvector = new CSG.Vector3D([1, 0, 0]).times(radius);
		yvector = new CSG.Vector3D([0, -1, 0]).times(radius);
		zvector = new CSG.Vector3D([0, 0, 1]).times(radius);
	}
	if(resolution < 4) resolution = 4;
	var qresolution = Math.round(resolution / 4);
	var prevcylinderpoint;
	var polygons = [];
	for(var slice1 = 0; slice1 <= resolution; slice1++) {
		var angle = Math.PI * 2.0 * slice1 / resolution;
		var cylinderpoint = xvector.times(Math.cos(angle)).plus(yvector.times(Math.sin(angle)));
		if(slice1 > 0) {
			// cylinder vertices:
			var vertices = [];
			var prevcospitch, prevsinpitch;
			for(var slice2 = 0; slice2 <= qresolution; slice2++) {
				var pitch = 0.5 * Math.PI * slice2 / qresolution;
				var cospitch = Math.cos(pitch);
				var sinpitch = Math.sin(pitch);
				if(slice2 > 0) {
					vertices = [];
					vertices.push(new CSG.Vertex(center.plus(prevcylinderpoint.times(prevcospitch).minus(zvector.times(prevsinpitch)))));
					vertices.push(new CSG.Vertex(center.plus(cylinderpoint.times(prevcospitch).minus(zvector.times(prevsinpitch)))));
					if(slice2 < qresolution) {
						vertices.push(new CSG.Vertex(center.plus(cylinderpoint.times(cospitch).minus(zvector.times(sinpitch)))));
					}
					vertices.push(new CSG.Vertex(center.plus(prevcylinderpoint.times(cospitch).minus(zvector.times(sinpitch)))));
					polygons.push(new CSG.Polygon(vertices));
					vertices = [];
					vertices.push(new CSG.Vertex(center.plus(prevcylinderpoint.times(prevcospitch).plus(zvector.times(prevsinpitch)))));
					vertices.push(new CSG.Vertex(center.plus(cylinderpoint.times(prevcospitch).plus(zvector.times(prevsinpitch)))));
					if(slice2 < qresolution) {
						vertices.push(new CSG.Vertex(center.plus(cylinderpoint.times(cospitch).plus(zvector.times(sinpitch)))));
					}
					vertices.push(new CSG.Vertex(center.plus(prevcylinderpoint.times(cospitch).plus(zvector.times(sinpitch)))));
					vertices.reverse();
					polygons.push(new CSG.Polygon(vertices));
				}
				prevcospitch = cospitch;
				prevsinpitch = sinpitch;
			}
		}
		prevcylinderpoint = cylinderpoint;
	}
	var result = CSG.fromPolygons(polygons);
	result.properties.sphere = new CSG.Properties();
	result.properties.sphere.center = new CSG.Vector3D(center);
	result.properties.sphere.facepoint = center.plus(xvector);
	return result;
};

// Construct a solid cylinder.
//
// Parameters:
//   start: start point of cylinder (default [0, -1, 0])
//   end: end point of cylinder (default [0, 1, 0])
//   radius: radius of cylinder (default 1), must be a scalar
//   resolution: determines the number of polygons per 360 degree revolution (default 12)
//
// Example usage:
//
//     var cylinder = CSG.cylinder({
//       start: [0, -1, 0],
//       end: [0, 1, 0],
//       radius: 1,
//       resolution: 16
//     });
CSG.cylinder = function(options) {
	var s = CSG.parseOptionAs3DVector(options, "start", [0, -1, 0]);
	var e = CSG.parseOptionAs3DVector(options, "end", [0, 1, 0]);
	var r = CSG.parseOptionAsFloat(options, "radius", 1);
	var rEnd = CSG.parseOptionAsFloat(options, "radiusEnd", r);
	var rStart = CSG.parseOptionAsFloat(options, "radiusStart", r);

	if((rEnd < 0) || (rStart < 0)) {
		throw new Error("Radius should be non-negative");
	}
	if((rEnd === 0) && (rStart === 0)) {
		throw new Error("Either radiusStart or radiusEnd should be positive");
	}

	var slices = CSG.parseOptionAsInt(options, "resolution", CSG.defaultResolution2D);
	var ray = e.minus(s);
	var axisZ = ray.unit(); //, isY = (Math.abs(axisZ.y) > 0.5);
	var axisX = axisZ.randomNonParallelVector().unit();

	//  var axisX = new CSG.Vector3D(isY, !isY, 0).cross(axisZ).unit();
	var axisY = axisX.cross(axisZ).unit();
	var start = new CSG.Vertex(s);
	var end = new CSG.Vertex(e);
	var polygons = [];

	function point(stack, slice, radius) {
		var angle = slice * Math.PI * 2;
		var out = axisX.times(Math.cos(angle)).plus(axisY.times(Math.sin(angle)));
		var pos = s.plus(ray.times(stack)).plus(out.times(radius));
		return new CSG.Vertex(pos);
	}
	for(var i = 0; i < slices; i++) {
		var t0 = i / slices,
			t1 = (i + 1) / slices;
		if(rEnd == rStart) {
			polygons.push(new CSG.Polygon([start, point(0, t0, rEnd), point(0, t1, rEnd)]));
			polygons.push(new CSG.Polygon([point(0, t1, rEnd), point(0, t0, rEnd), point(1, t0, rEnd), point(1, t1, rEnd)]));
			polygons.push(new CSG.Polygon([end, point(1, t1, rEnd), point(1, t0, rEnd)]));
		} else {
			if(rStart > 0) {
				polygons.push(new CSG.Polygon([start, point(0, t0, rStart), point(0, t1, rStart)]));
				polygons.push(new CSG.Polygon([point(0, t0, rStart), point(1, t0, rEnd), point(0, t1, rStart)]));
			}
			if(rEnd > 0) {
				polygons.push(new CSG.Polygon([end, point(1, t1, rEnd), point(1, t0, rEnd)]));
				polygons.push(new CSG.Polygon([point(1, t0, rEnd), point(1, t1, rEnd), point(0, t1, rStart)]));
			}
		}
	}
	var result = CSG.fromPolygons(polygons);
	result.properties.cylinder = new CSG.Properties();
	result.properties.cylinder.start = new CSG.Connector(s, axisZ.negated(), axisX);
	result.properties.cylinder.end = new CSG.Connector(e, axisZ, axisX);
	result.properties.cylinder.facepoint = s.plus(axisX.times(rStart));
	return result;
};

// Like a cylinder, but with rounded ends instead of flat
//
// Parameters:
//   start: start point of cylinder (default [0, -1, 0])
//   end: end point of cylinder (default [0, 1, 0])
//   radius: radius of cylinder (default 1), must be a scalar
//   resolution: determines the number of polygons per 360 degree revolution (default 12)
//   normal: a vector determining the starting angle for tesselation. Should be non-parallel to start.minus(end)
//
// Example usage:
//
//     var cylinder = CSG.roundedCylinder({
//       start: [0, -1, 0],
//       end: [0, 1, 0],
//       radius: 1,
//       resolution: 16
//     });
CSG.roundedCylinder = function(options) {
	var p1 = CSG.parseOptionAs3DVector(options, "start", [0, -1, 0]);
	var p2 = CSG.parseOptionAs3DVector(options, "end", [0, 1, 0]);
	var radius = CSG.parseOptionAsFloat(options, "radius", 1);
	var direction = p2.minus(p1);
	var defaultnormal;
	if(Math.abs(direction.x) > Math.abs(direction.y)) {
		defaultnormal = new CSG.Vector3D(0, 1, 0);
	} else {
		defaultnormal = new CSG.Vector3D(1, 0, 0);
	}
	var normal = CSG.parseOptionAs3DVector(options, "normal", defaultnormal);
	var resolution = CSG.parseOptionAsInt(options, "resolution", CSG.defaultResolution3D);
	if(resolution < 4) resolution = 4;
	var polygons = [];
	var qresolution = Math.floor(0.25 * resolution);
	var length = direction.length();
	if(length < 1e-10) {
		return CSG.sphere({
			center: p1,
			radius: radius,
			resolution: resolution
		});
	}
	var zvector = direction.unit().times(radius);
	var xvector = zvector.cross(normal).unit().times(radius);
	var yvector = xvector.cross(zvector).unit().times(radius);
	var prevcylinderpoint;
	for(var slice1 = 0; slice1 <= resolution; slice1++) {
		var angle = Math.PI * 2.0 * slice1 / resolution;
		var cylinderpoint = xvector.times(Math.cos(angle)).plus(yvector.times(Math.sin(angle)));
		if(slice1 > 0) {
			// cylinder vertices:
			var vertices = [];
			vertices.push(new CSG.Vertex(p1.plus(cylinderpoint)));
			vertices.push(new CSG.Vertex(p1.plus(prevcylinderpoint)));
			vertices.push(new CSG.Vertex(p2.plus(prevcylinderpoint)));
			vertices.push(new CSG.Vertex(p2.plus(cylinderpoint)));
			polygons.push(new CSG.Polygon(vertices));
			var prevcospitch, prevsinpitch;
			for(var slice2 = 0; slice2 <= qresolution; slice2++) {
				var pitch = 0.5 * Math.PI * slice2 / qresolution;
				//var pitch = Math.asin(slice2/qresolution);
				var cospitch = Math.cos(pitch);
				var sinpitch = Math.sin(pitch);
				if(slice2 > 0) {
					vertices = [];
					vertices.push(new CSG.Vertex(p1.plus(prevcylinderpoint.times(prevcospitch).minus(zvector.times(prevsinpitch)))));
					vertices.push(new CSG.Vertex(p1.plus(cylinderpoint.times(prevcospitch).minus(zvector.times(prevsinpitch)))));
					if(slice2 < qresolution) {
						vertices.push(new CSG.Vertex(p1.plus(cylinderpoint.times(cospitch).minus(zvector.times(sinpitch)))));
					}
					vertices.push(new CSG.Vertex(p1.plus(prevcylinderpoint.times(cospitch).minus(zvector.times(sinpitch)))));
					polygons.push(new CSG.Polygon(vertices));
					vertices = [];
					vertices.push(new CSG.Vertex(p2.plus(prevcylinderpoint.times(prevcospitch).plus(zvector.times(prevsinpitch)))));
					vertices.push(new CSG.Vertex(p2.plus(cylinderpoint.times(prevcospitch).plus(zvector.times(prevsinpitch)))));
					if(slice2 < qresolution) {
						vertices.push(new CSG.Vertex(p2.plus(cylinderpoint.times(cospitch).plus(zvector.times(sinpitch)))));
					}
					vertices.push(new CSG.Vertex(p2.plus(prevcylinderpoint.times(cospitch).plus(zvector.times(sinpitch)))));
					vertices.reverse();
					polygons.push(new CSG.Polygon(vertices));
				}
				prevcospitch = cospitch;
				prevsinpitch = sinpitch;
			}
		}
		prevcylinderpoint = cylinderpoint;
	}
	var result = CSG.fromPolygons(polygons);
	var ray = zvector.unit();
	var axisX = xvector.unit();
	result.properties.roundedCylinder = new CSG.Properties();
	result.properties.roundedCylinder.start = new CSG.Connector(p1, ray.negated(), axisX);
	result.properties.roundedCylinder.end = new CSG.Connector(p2, ray, axisX);
	result.properties.roundedCylinder.facepoint = p1.plus(xvector);
	return result;
};

// Construct an axis-aligned solid rounded cuboid.
// Parameters:
//   center: center of cube (default [0,0,0])
//   radius: radius of cube (default [1,1,1]), can be specified as scalar or as 3D vector
//   roundradius: radius of rounded corners (default 0.2), must be a scalar
//   resolution: determines the number of polygons per 360 degree revolution (default 8)
//
// Example code:
//
//     var cube = CSG.roundedCube({
//       center: [0, 0, 0],
//       radius: 1,
//       roundradius: 0.2,
//       resolution: 8,
//     });
CSG.roundedCube = function(options) {
	var center = CSG.parseOptionAs3DVector(options, "center", [0, 0, 0]);
	var cuberadius = CSG.parseOptionAs3DVector(options, "radius", [1, 1, 1]);
	var resolution = CSG.parseOptionAsInt(options, "resolution", CSG.defaultResolution3D);
	if(resolution < 4) resolution = 4;
	var roundradius = CSG.parseOptionAsFloat(options, "roundradius", 0.2);
	var innercuberadius = cuberadius;
	innercuberadius = innercuberadius.minus(new CSG.Vector3D(roundradius));
	var result = CSG.cube({
		center: center,
		radius: [cuberadius.x, innercuberadius.y, innercuberadius.z]
	});
	result = result.unionSub(CSG.cube({
		center: center,
		radius: [innercuberadius.x, cuberadius.y, innercuberadius.z]
	}), false, false);
	result = result.unionSub(CSG.cube({
		center: center,
		radius: [innercuberadius.x, innercuberadius.y, cuberadius.z]
	}), false, false);
	for(var level = 0; level < 2; level++) {
		var z = innercuberadius.z;
		if(level == 1) z = -z;
		var p1 = new CSG.Vector3D(innercuberadius.x, innercuberadius.y, z).plus(center);
		var p2 = new CSG.Vector3D(innercuberadius.x, -innercuberadius.y, z).plus(center);
		var p3 = new CSG.Vector3D(-innercuberadius.x, -innercuberadius.y, z).plus(center);
		var p4 = new CSG.Vector3D(-innercuberadius.x, innercuberadius.y, z).plus(center);
		var sphere = CSG.sphere({
			center: p1,
			radius: roundradius,
			resolution: resolution
		});
		result = result.unionSub(sphere, false, false);
		sphere = CSG.sphere({
			center: p2,
			radius: roundradius,
			resolution: resolution
		});
		result = result.unionSub(sphere, false, false);
		sphere = CSG.sphere({
			center: p3,
			radius: roundradius,
			resolution: resolution
		});
		result = result.unionSub(sphere, false, false);
		sphere = CSG.sphere({
			center: p4,
			radius: roundradius,
			resolution: resolution
		});
		result = result.unionSub(sphere, false, true);
		var cylinder = CSG.cylinder({
			start: p1,
			end: p2,
			radius: roundradius,
			resolution: resolution
		});
		result = result.unionSub(cylinder, false, false);
		cylinder = CSG.cylinder({
			start: p2,
			end: p3,
			radius: roundradius,
			resolution: resolution
		});
		result = result.unionSub(cylinder, false, false);
		cylinder = CSG.cylinder({
			start: p3,
			end: p4,
			radius: roundradius,
			resolution: resolution
		});
		result = result.unionSub(cylinder, false, false);
		cylinder = CSG.cylinder({
			start: p4,
			end: p1,
			radius: roundradius,
			resolution: resolution
		});
		result = result.unionSub(cylinder, false, false);
		if(level === 0) {
			var d = new CSG.Vector3D(0, 0, -2 * z);
			cylinder = CSG.cylinder({
				start: p1,
				end: p1.plus(d),
				radius: roundradius,
				resolution: resolution
			});
			result = result.unionSub(cylinder);
			cylinder = CSG.cylinder({
				start: p2,
				end: p2.plus(d),
				radius: roundradius,
				resolution: resolution
			});
			result = result.unionSub(cylinder);
			cylinder = CSG.cylinder({
				start: p3,
				end: p3.plus(d),
				radius: roundradius,
				resolution: resolution
			});
			result = result.unionSub(cylinder);
			cylinder = CSG.cylinder({
				start: p4,
				end: p4.plus(d),
				radius: roundradius,
				resolution: resolution
			});
			result = result.unionSub(cylinder, false, true);
		}
	}
	result = result.reTesselated();
	result.properties.roundedCube = new CSG.Properties();
	result.properties.roundedCube.center = new CSG.Vertex(center);
	result.properties.roundedCube.facecenters = [
		new CSG.Connector(new CSG.Vector3D([cuberadius.x, 0, 0]).plus(center), [1, 0, 0], [0, 0, 1]),
		new CSG.Connector(new CSG.Vector3D([-cuberadius.x, 0, 0]).plus(center), [-1, 0, 0], [0, 0, 1]),
		new CSG.Connector(new CSG.Vector3D([0, cuberadius.y, 0]).plus(center), [0, 1, 0], [0, 0, 1]),
		new CSG.Connector(new CSG.Vector3D([0, -cuberadius.y, 0]).plus(center), [0, -1, 0], [0, 0, 1]),
		new CSG.Connector(new CSG.Vector3D([0, 0, cuberadius.z]).plus(center), [0, 0, 1], [1, 0, 0]),
		new CSG.Connector(new CSG.Vector3D([0, 0, -cuberadius.z]).plus(center), [0, 0, -1], [1, 0, 0])];
	return result;
};

CSG.IsFloat = function(n) {
	return(!isNaN(n)) || (n === Infinity) || (n === -Infinity);
};

// solve 2x2 linear equation:
// [ab][x] = [u]
// [cd][y]   [v]
CSG.solve2Linear = function(a, b, c, d, u, v) {
	var det = a * d - b * c;
	var invdet = 1.0 / det;
	var x = u * d - b * v;
	var y = -u * c + a * v;
	x *= invdet;
	y *= invdet;
	return [x, y];
};

// # class Vector3D
// Represents a 3D vector.
//
// Example usage:
//
//     new CSG.Vector3D(1, 2, 3);
//     new CSG.Vector3D([1, 2, 3]);
//     new CSG.Vector3D({ x: 1, y: 2, z: 3 });
//     new CSG.Vector3D(1, 2); // assumes z=0
//     new CSG.Vector3D([1, 2]); // assumes z=0
CSG.Vector3D = function(x, y, z) {
	if(arguments.length == 3) {
		this._x = parseFloat(x);
		this._y = parseFloat(y);
		this._z = parseFloat(z);
	} else if(arguments.length == 2) {
		this._x = parseFloat(x);
		this._y = parseFloat(y);
		this._z = 0;
	} else {
		var ok = true;
		if(arguments.length == 1) {
			if(typeof(x) == "object") {
				if(x instanceof CSG.Vector3D) {
					this._x = x._x;
					this._y = x._y;
					this._z = x._z;
				} else if(x instanceof CSG.Vector2D) {
					this._x = x._x;
					this._y = x._y;
					this._z = 0;
				} else if(x instanceof Array) {
					if((x.length < 2) || (x.length > 3)) {
						ok = false;
					} else {
						this._x = parseFloat(x[0]);
						this._y = parseFloat(x[1]);
						if(x.length == 3) {
							this._z = parseFloat(x[2]);
						} else {
							this._z = 0;
						}
					}
				} else if(('x' in x) && ('y' in x)) {
					this._x = parseFloat(x.x);
					this._y = parseFloat(x.y);
					if('z' in x) {
						this._z = parseFloat(x.z);
					} else {
						this._z = 0;
					}
				} else ok = false;
			} else {
				var v = parseFloat(x);
				this._x = v;
				this._y = v;
				this._z = v;
			}
		} else ok = false;
		if(ok) {
			if((!CSG.IsFloat(this._x)) || (!CSG.IsFloat(this._y)) || (!CSG.IsFloat(this._z))) ok = false;
		}
		if(!ok) {
			throw new Error("wrong arguments");
		}
	}
};

CSG.Vector3D.prototype = {
	get x() {
		return this._x;
	}, get y() {
		return this._y;
	}, get z() {
		return this._z;
	},

	set x(v) {
		throw new Error("Vector3D is immutable");
	}, set y(v) {
		throw new Error("Vector3D is immutable");
	}, set z(v) {
		throw new Error("Vector3D is immutable");
	},

	clone: function() {
		return new CSG.Vector3D(this);
	},

	negated: function() {
		return new CSG.Vector3D(-this._x, -this._y, -this._z);
	},

	abs: function() {
		return new CSG.Vector3D(Math.abs(this._x), Math.abs(this._y), Math.abs(this._z));
	},

	plus: function(a) {
		return new CSG.Vector3D(this._x + a._x, this._y + a._y, this._z + a._z);
	},

	minus: function(a) {
		return new CSG.Vector3D(this._x - a._x, this._y - a._y, this._z - a._z);
	},

	times: function(a) {
		return new CSG.Vector3D(this._x * a, this._y * a, this._z * a);
	},

	dividedBy: function(a) {
		return new CSG.Vector3D(this._x / a, this._y / a, this._z / a);
	},

	dot: function(a) {
		return this._x * a._x + this._y * a._y + this._z * a._z;
	},

	lerp: function(a, t) {
		return this.plus(a.minus(this).times(t));
	},

	lengthSquared: function() {
		return this.dot(this);
	},

	length: function() {
		return Math.sqrt(this.lengthSquared());
	},

	unit: function() {
		return this.dividedBy(this.length());
	},

	cross: function(a) {
		return new CSG.Vector3D(
		this._y * a._z - this._z * a._y, this._z * a._x - this._x * a._z, this._x * a._y - this._y * a._x);
	},

	distanceTo: function(a) {
		return this.minus(a).length();
	},

	distanceToSquared: function(a) {
		return this.minus(a).lengthSquared();
	},

	equals: function(a) {
		return(this._x == a._x) && (this._y == a._y) && (this._z == a._z);
	},

	// Right multiply by a 4x4 matrix (the vector is interpreted as a row vector)
	// Returns a new CSG.Vector3D
	multiply4x4: function(matrix4x4) {
		return matrix4x4.leftMultiply1x3Vector(this);
	},

	transform: function(matrix4x4) {
		return matrix4x4.leftMultiply1x3Vector(this);
	},

	toStlString: function() {
		return this._x + " " + this._y + " " + this._z;
	},

	toAMFString: function() {
		return "<x>" + this._x + "</x><y>" + this._y + "</y><z>" + this._z + "</z>";
	},

	toString: function() {
		return "(" + this._x.toFixed(2) + ", " + this._y.toFixed(2) + ", " + this._z.toFixed(2) + ")";
	},

	// find a vector that is somewhat perpendicular to this one
	randomNonParallelVector: function() {
		var abs = this.abs();
		if((abs._x <= abs._y) && (abs._x <= abs._z)) {
			return new CSG.Vector3D(1, 0, 0);
		} else if((abs._y <= abs._x) && (abs._y <= abs._z)) {
			return new CSG.Vector3D(0, 1, 0);
		} else {
			return new CSG.Vector3D(0, 0, 1);
		}
	},

	min: function(p) {
		return new CSG.Vector3D(
		Math.min(this._x, p._x), Math.min(this._y, p._y), Math.min(this._z, p._z));
	},

	max: function(p) {
		return new CSG.Vector3D(
		Math.max(this._x, p._x), Math.max(this._y, p._y), Math.max(this._z, p._z));
	}
};

// # class Vertex
// Represents a vertex of a polygon. Use your own vertex class instead of this
// one to provide additional features like texture coordinates and vertex
// colors. Custom vertex classes need to provide a `pos` property
// `flipped()`, and `interpolate()` methods that behave analogous to the ones
// defined by `CSG.Vertex`.
CSG.Vertex = function(pos) {
	this.pos = pos;
};

// create from an untyped object with identical property names:
CSG.Vertex.fromObject = function(obj) {
	var pos = new CSG.Vector3D(obj.pos);
	return new CSG.Vertex(pos);
};

CSG.Vertex.prototype = {
	// Return a vertex with all orientation-specific data (e.g. vertex normal) flipped. Called when the
	// orientation of a polygon is flipped.
	flipped: function() {
		return this;
	},

	getTag: function() {
		var result = this.tag;
		if(!result) {
			result = CSG.getTag();
			this.tag = result;
		}
		return result;
	},

	// Create a new vertex between this vertex and `other` by linearly
	// interpolating all properties using a parameter of `t`. Subclasses should
	// override this to interpolate additional properties.
	interpolate: function(other, t) {
		var newpos = this.pos.lerp(other.pos, t);
		return new CSG.Vertex(newpos);
	},

	// Affine transformation of vertex. Returns a new CSG.Vertex
	transform: function(matrix4x4) {
		var newpos = this.pos.multiply4x4(matrix4x4);
		return new CSG.Vertex(newpos);
	},

	toStlString: function() {
		return "vertex " + this.pos.toStlString() + "\n";
	},

	toAMFString: function() {
		return "<vertex><coordinates>" + this.pos.toAMFString() + "</coordinates></vertex>\n";
	},

	toString: function() {
		return this.pos.toString();
	}
};

// # class Plane
// Represents a plane in 3D space.
CSG.Plane = function(normal, w) {
	this.normal = normal;
	this.w = w;
};

// create from an untyped object with identical property names:
CSG.Plane.fromObject = function(obj) {
	var normal = new CSG.Vector3D(obj.normal);
	var w = parseFloat(obj.w);
	return new CSG.Plane(normal, w);
};

// `CSG.Plane.EPSILON` is the tolerance used by `splitPolygon()` to decide if a
// point is on the plane.
CSG.Plane.EPSILON = 1e-5;

CSG.Plane.fromVector3Ds = function(a, b, c) {
	var n = b.minus(a).cross(c.minus(a)).unit();
	return new CSG.Plane(n, n.dot(a));
};

// like fromVector3Ds, but allow the vectors to be on one point or one line
// in such a case a random plane through the given points is constructed
CSG.Plane.anyPlaneFromVector3Ds = function(a, b, c) {
	var v1 = b.minus(a);
	var v2 = c.minus(a);
	if(v1.length() < 1e-5) {
		v1 = v2.randomNonParallelVector();
	}
	if(v2.length() < 1e-5) {
		v2 = v1.randomNonParallelVector();
	}
	var normal = v1.cross(v2);
	if(normal.length() < 1e-5) {
		// this would mean that v1 == v2.negated()
		v2 = v1.randomNonParallelVector();
		normal = v1.cross(v2);
	}
	normal = normal.unit();
	return new CSG.Plane(normal, normal.dot(a));
};

CSG.Plane.fromPoints = function(a, b, c) {
	a = new CSG.Vector3D(a);
	b = new CSG.Vector3D(b);
	c = new CSG.Vector3D(c);
	return CSG.Plane.fromVector3Ds(a, b, c);
};

CSG.Plane.fromNormalAndPoint = function(normal, point) {
	normal = new CSG.Vector3D(normal);
	point = new CSG.Vector3D(point);
	normal = normal.unit();
	var w = point.dot(normal);
	return new CSG.Plane(normal, w);
};

CSG.Plane.prototype = {
	flipped: function() {
		return new CSG.Plane(this.normal.negated(), -this.w);
	},

	getTag: function() {
		var result = this.tag;
		if(!result) {
			result = CSG.getTag();
			this.tag = result;
		}
		return result;
	},

	equals: function(n) {
		return this.normal.equals(n.normal) && this.w == n.w;
	},

	transform: function(matrix4x4) {
		var ismirror = matrix4x4.isMirroring();
		// get two vectors in the plane:
		var r = this.normal.randomNonParallelVector();
		var u = this.normal.cross(r);
		var v = this.normal.cross(u);
		// get 3 points in the plane:
		var point1 = this.normal.times(this.w);
		var point2 = point1.plus(u);
		var point3 = point1.plus(v);
		// transform the points:
		point1 = point1.multiply4x4(matrix4x4);
		point2 = point2.multiply4x4(matrix4x4);
		point3 = point3.multiply4x4(matrix4x4);
		// and create a new plane from the transformed points:
		var newplane = CSG.Plane.fromVector3Ds(point1, point2, point3);
		if(ismirror) {
			// the transform is mirroring
			// We should mirror the plane:
			newplane = newplane.flipped();
		}
		return newplane;
	},

	// Returns object:
	// .type:
	//   0: coplanar-front
	//   1: coplanar-back
	//   2: front
	//   3: back
	//   4: spanning
	// In case the polygon is spanning, returns:
	// .front: a CSG.Polygon of the front part
	// .back: a CSG.Polygon of the back part
	splitPolygon: function(polygon) {
		var result = {
			type: null,
			front: null,
			back: null
		};
		// cache in local vars (speedup):
		var planenormal = this.normal;
		var vertices = polygon.vertices;
		var numvertices = vertices.length;
		if(polygon.plane.equals(this)) {
			result.type = 0;
		} else {
			var EPS = CSG.Plane.EPSILON;
			var thisw = this.w;
			var hasfront = false;
			var hasback = false;
			var vertexIsBack = [];
			var MINEPS = -EPS;
			for(var i = 0; i < numvertices; i++) {
				var t = planenormal.dot(vertices[i].pos) - thisw;
				var isback = (t < 0);
				vertexIsBack.push(isback);
				if(t > EPS) hasfront = true;
				if(t < MINEPS) hasback = true;
			}
			if((!hasfront) && (!hasback)) {
				// all points coplanar
				var t = planenormal.dot(polygon.plane.normal);
				result.type = (t >= 0) ? 0 : 1;
			} else if(!hasback) {
				result.type = 2;
			} else if(!hasfront) {
				result.type = 3;
			} else {
				// spanning
				result.type = 4;
				var frontvertices = [],
					backvertices = [];
				var isback = vertexIsBack[0];
				for(var vertexindex = 0; vertexindex < numvertices; vertexindex++) {
					var vertex = vertices[vertexindex];
					var nextvertexindex = vertexindex + 1;
					if(nextvertexindex >= numvertices) nextvertexindex = 0;
					var nextisback = vertexIsBack[nextvertexindex];
					if(isback == nextisback) {
						// line segment is on one side of the plane:
						if(isback) {
							backvertices.push(vertex);
						} else {
							frontvertices.push(vertex);
						}
					} else {
						// line segment intersects plane:
						var point = vertex.pos;
						var nextpoint = vertices[nextvertexindex].pos;
						var intersectionpoint = this.splitLineBetweenPoints(point, nextpoint);
						var intersectionvertex = new CSG.Vertex(intersectionpoint);
						if(isback) {
							backvertices.push(vertex);
							backvertices.push(intersectionvertex);
							frontvertices.push(intersectionvertex);
						} else {
							frontvertices.push(vertex);
							frontvertices.push(intersectionvertex);
							backvertices.push(intersectionvertex);
						}
					}
					isback = nextisback;
				} // for vertexindex
				// remove duplicate vertices:
				var EPS_SQUARED = CSG.Plane.EPSILON * CSG.Plane.EPSILON;
				if(backvertices.length >= 3) {
					var prevvertex = backvertices[backvertices.length - 1];
					for(var vertexindex = 0; vertexindex < backvertices.length; vertexindex++) {
						var vertex = backvertices[vertexindex];
						if(vertex.pos.distanceToSquared(prevvertex.pos) < EPS_SQUARED) {
							backvertices.splice(vertexindex, 1);
							vertexindex--;
						}
						prevvertex = vertex;
					}
				}
				if(frontvertices.length >= 3) {
					var prevvertex = frontvertices[frontvertices.length - 1];
					for(var vertexindex = 0; vertexindex < frontvertices.length; vertexindex++) {
						var vertex = frontvertices[vertexindex];
						if(vertex.pos.distanceToSquared(prevvertex.pos) < EPS_SQUARED) {
							frontvertices.splice(vertexindex, 1);
							vertexindex--;
						}
						prevvertex = vertex;
					}
				}
				if(frontvertices.length >= 3) {
					result.front = new CSG.Polygon(frontvertices, polygon.shared, polygon.plane);
				}
				if(backvertices.length >= 3) {
					result.back = new CSG.Polygon(backvertices, polygon.shared, polygon.plane);
				}
			}
		}
		return result;
	},

	// robust splitting of a line by a plane
	// will work even if the line is parallel to the plane
	splitLineBetweenPoints: function(p1, p2) {
		var direction = p2.minus(p1);
		var labda = (this.w - this.normal.dot(p1)) / this.normal.dot(direction);
		if(isNaN(labda)) labda = 0;
		if(labda > 1) labda = 1;
		if(labda < 0) labda = 0;
		var result = p1.plus(direction.times(labda));
		return result;
	},

	// returns CSG.Vector3D
	intersectWithLine: function(line3d) {
		return line3d.intersectWithPlane(this);
	},

	// intersection of two planes
	intersectWithPlane: function(plane) {
		return CSG.Line3D.fromPlanes(this, plane);
	},

	signedDistanceToPoint: function(point) {
		var t = this.normal.dot(point) - this.w;
		return t;
	},

	toString: function() {
		return "[normal: " + this.normal.toString() + ", w: " + this.w + "]";
	},

	mirrorPoint: function(point3d) {
		var distance = this.signedDistanceToPoint(point3d);
		var mirrored = point3d.minus(this.normal.times(distance * 2.0));
		return mirrored;
	}
};


// # class Polygon
// Represents a convex polygon. The vertices used to initialize a polygon must
// be coplanar and form a convex loop. They do not have to be `CSG.Vertex`
// instances but they must behave similarly (duck typing can be used for
// customization).
//
// Each convex polygon has a `shared` property, which is shared between all
// polygons that are clones of each other or were split from the same polygon.
// This can be used to define per-polygon properties (such as surface color).
//
// The plane of the polygon is calculated from the vertex coordinates
// To avoid unnecessary recalculation, the plane can alternatively be
// passed as the third argument
CSG.Polygon = function(vertices, shared, plane) {
	this.vertices = vertices;
	if(!shared) shared = CSG.Polygon.defaultShared;
	this.shared = shared;
	//var numvertices = vertices.length;

	if(arguments.length >= 3) {
		this.plane = plane;
	} else {
		this.plane = CSG.Plane.fromVector3Ds(vertices[0].pos, vertices[1].pos, vertices[2].pos);
	}

	if(_CSGDEBUG) {
		this.checkIfConvex();
	}
};

// create from an untyped object with identical property names:
CSG.Polygon.fromObject = function(obj) {
	var vertices = obj.vertices.map(function(v) {
		return CSG.Vertex.fromObject(v);
	});
	var shared = CSG.Polygon.Shared.fromObject(obj.shared);
	var plane = CSG.Plane.fromObject(obj.plane);
	return new CSG.Polygon(vertices, shared, plane);
};

CSG.Polygon.prototype = {
	// check whether the polygon is convex (it should be, otherwise we will get unexpected results)
	checkIfConvex: function() {
		if(!CSG.Polygon.verticesConvex(this.vertices, this.plane.normal)) {
			CSG.Polygon.verticesConvex(this.vertices, this.plane.normal);
			throw new Error("Not convex!");
		}
	},

	/**
	 * @param {Array} color [red, green, blue, alpha] color values are float numbers 0..1
	 * @return {CSG.Polygon} The current polygon
	 */
	setColor: function(red, green, blue, alpha) {
		var color = red instanceof Array ? red : [red||0, green||0, blue||0, isNaN(alpha) ? 1. : alpha];
		this.shared = new CSG.Polygon.Shared(color);
		return this;
	},

	// Extrude a polygon into the direction offsetvector
	// Returns a CSG object
	extrude: function(offsetvector) {
		var newpolygons = [];

		var polygon1 = this;
		var direction = polygon1.plane.normal.dot(offsetvector);
		if(direction > 0) {
			polygon1 = polygon1.flipped();
		}
		newpolygons.push(polygon1);
		var polygon2 = polygon1.translate(offsetvector);
		var numvertices = this.vertices.length;
		for(var i = 0; i < numvertices; i++) {
			var sidefacepoints = [];
			var nexti = (i < (numvertices - 1)) ? i + 1 : 0;
			sidefacepoints.push(polygon1.vertices[i].pos);
			sidefacepoints.push(polygon2.vertices[i].pos);
			sidefacepoints.push(polygon2.vertices[nexti].pos);
			sidefacepoints.push(polygon1.vertices[nexti].pos);
			var sidefacepolygon = CSG.Polygon.createFromPoints(sidefacepoints, this.shared);
			newpolygons.push(sidefacepolygon);
		}
		polygon2 = polygon2.flipped();
		newpolygons.push(polygon2);
		return CSG.fromPolygons(newpolygons);
	},

	translate: function(offset) {
		return this.transform(CSG.Matrix4x4.translation(offset));
	},

	// returns an array with a CSG.Vector3D (center point) and a radius
	boundingSphere: function() {
		if(!this.cachedBoundingSphere) {
			var box = this.boundingBox();
			var middle = box[0].plus(box[1]).times(0.5);
			var radius3 = box[1].minus(middle);
			var radius = radius3.length();
			this.cachedBoundingSphere = [middle, radius];
		}
		return this.cachedBoundingSphere;
	},

	// returns an array of two CSG.Vector3Ds (minimum coordinates and maximum coordinates)
	boundingBox: function() {
		if(!this.cachedBoundingBox) {
			var minpoint, maxpoint;
			var vertices = this.vertices;
			var numvertices = vertices.length;
			if(numvertices === 0) {
				minpoint = new CSG.Vector3D(0, 0, 0);
			} else {
				minpoint = vertices[0].pos;
			}
			maxpoint = minpoint;
			for(var i = 1; i < numvertices; i++) {
				var point = vertices[i].pos;
				minpoint = minpoint.min(point);
				maxpoint = maxpoint.max(point);
			}
			this.cachedBoundingBox = [minpoint, maxpoint];
		}
		return this.cachedBoundingBox;
	},

	flipped: function() {
		var newvertices = this.vertices.map(function(v) {
			return v.flipped();
		});
		newvertices.reverse();
		var newplane = this.plane.flipped();
		return new CSG.Polygon(newvertices, this.shared, newplane);
	},

	// Affine transformation of polygon. Returns a new CSG.Polygon
	transform: function(matrix4x4) {
		var newvertices = this.vertices.map(function(v) {
			return v.transform(matrix4x4);
		});
		var newplane = this.plane.transform(matrix4x4);
		var scalefactor = matrix4x4.elements[0] * matrix4x4.elements[5] * matrix4x4.elements[10];
		if(scalefactor < 0) {
			// the transformation includes mirroring. We need to reverse the vertex order
			// in order to preserve the inside/outside orientation:
			newvertices.reverse();
		}
		return new CSG.Polygon(newvertices, this.shared, newplane);
	},

	toStlString: function() {
		var result = "";
		if(this.vertices.length >= 3) // should be!
		{
			// STL requires triangular polygons. If our polygon has more vertices, create
			// multiple triangles:
			var firstVertexStl = this.vertices[0].toStlString();
			for(var i = 0; i < this.vertices.length - 2; i++) {
				result += "facet normal " + this.plane.normal.toStlString() + "\nouter loop\n";
				result += firstVertexStl;
				result += this.vertices[i + 1].toStlString();
				result += this.vertices[i + 2].toStlString();
				result += "endloop\nendfacet\n";
			}
		}
		return result;
	},

	toString: function() {
		var result = "Polygon plane: " + this.plane.toString() + "\n";
		this.vertices.map(function(vertex) {
			result += "  " + vertex.toString() + "\n";
		});
		return result;
	},

	// project the 3D polygon onto a plane
	projectToOrthoNormalBasis: function(orthobasis) {
		var points2d = this.vertices.map(function(vertex) {
			return orthobasis.to2D(vertex.pos);
		});
		var result = CAG.fromPointsNoCheck(points2d);
		var area = result.area();
		if(Math.abs(area) < 1e-5) {
			// the polygon was perpendicular to the orthnormal plane. The resulting 2D polygon would be degenerate
			// return an empty area instead:
			result = new CAG();
		} else if(area < 0) {
			result = result.flipped();
		}
		return result;
	},

	/**
	 * Creates solid from slices (CSG.Polygon) by generating walls
	 * @param {Object} options Solid generating options
	 *	- numslices {Number} Number of slices to be generated
	 *	- callback(t, slice) {Function} Callback function generating slices.
	 *			arguments: t = [0..1], slice = [0..numslices - 1]
	 *			return: CSG.Polygon or null to skip
	 *	- loop {Boolean} no flats, only walls, it's used to generate solids like a tor
	 *
	 *	by Eduard Bespalov AKA tedbeer (2013)
	 */
	solidFromSlices: function(options) {
		var polygons = [],
			csg = null,
			prev = null,
			bottom = null,
			top = null,
			numSlices = 2,
			bLoop = false,
			fnCallback,
			flipped = null;

		if (options) {
			bLoop = Boolean(options['loop']);

			if (options.numslices)
				numSlices = options.numslices;

			if (options.callback)
				fnCallback = options.callback;
		}
		if (!fnCallback) {
			var square = new CSG.Polygon.createFromPoints([
						[0, 0, 0], [1, 0, 0], [1, 1, 0], [0, 1, 0]
					]);
			fnCallback = function(t, slice) {
				return t == 0 || t == 1 ? square.translate([0,0,t]) : null;
			}
		}
		for(var i = 0, iMax = numSlices - 1; i <= iMax; i++) {
			csg = fnCallback.call(this, i / iMax, i);
			if (csg) {
				if (!(csg instanceof CSG.Polygon)) {
					throw new Error("CSG.Polygon.solidFromSlices callback error: CSG.Polygon expected");
				}
				csg.checkIfConvex();

				if (prev) {//generate walls
					if (flipped === null) {//not generated yet
						flipped = prev.plane.signedDistanceToPoint(csg.vertices[0].pos) < 0;
					}
					this._addWalls(polygons, prev, csg, flipped);

				} else {//the first - will be a bottom
					bottom = csg;
				}
				prev = csg;
			} //callback can return null to skip that slice
		}
		top = csg;

		if (bLoop) {
			var bSameTopBottom = bottom.vertices.length == top.vertices.length &&
								bottom.vertices.every(function(v, index){
									return v.pos.equals(top.vertices[index].pos)
								});
			//if top and bottom are not the same -
			//generate walls between them
			if (!bSameTopBottom) {
				this._addWalls(polygons, top, bottom, flipped);
			} //else - already generated
		} else {
			//save top and bottom
			//TODO: flip if necessary
			polygons.unshift(flipped ? bottom : bottom.flipped());
			polygons.push(flipped ? top.flipped() : top);
		}
		return CSG.fromPolygons(polygons);
	},
	/**
	 *
	 * @param walls Array of wall polygons
	 * @param bottom Bottom polygon
	 * @param top Top polygon
	 */
	_addWalls: function(walls, bottom, top, bFlipped) {
		var bottomPoints = bottom.vertices.slice(0),//make a copy
			topPoints = top.vertices.slice(0),//make a copy
			color = top.shared || null;

		//check if bottom perimeter is closed
		if (!bottomPoints[0].pos.equals(bottomPoints[bottomPoints.length - 1].pos)) {
			bottomPoints.push(bottomPoints[0]);
		}

		//check if top perimeter is closed
		if (!topPoints[0].pos.equals(topPoints[topPoints.length - 1].pos)) {
			topPoints.push(topPoints[0]);
		}
		if (bFlipped) {
			bottomPoints = bottomPoints.reverse();
			topPoints = topPoints.reverse();
		}

		var iTopLen = topPoints.length - 1,
			iBotLen = bottomPoints.length - 1,
			iExtra = iTopLen - iBotLen, //how many extra triangles we need
			bMoreTops = iExtra > 0,
			bMoreBottoms = iExtra < 0;

		var aMin = []; //indexes to start extra triangles (polygon with minimal square)
		//init - we need exactly /iExtra/ small triangles
		for (var i = Math.abs(iExtra); i > 0; i--) {
			aMin.push({len: Infinity, index: -1});
		}

		var len;
		if (bMoreBottoms) {
			for (var i = 0; i < iBotLen; i++) {
				len = bottomPoints[i].pos.distanceToSquared(bottomPoints[i+1].pos);
				//find the element to replace
				for (var j = aMin.length - 1; j >= 0; j--) {
					if (aMin[j].len > len) {
						aMin[j].len = len;
						aMin.index = j;
						break;
					}
				}//for
			}
		} else if (bMoreTops) {
			for (var i = 0; i < iTopLen; i++) {
				len = topPoints[i].pos.distanceToSquared(topPoints[i+1].pos);
				//find the element to replace
				for (var j = aMin.length - 1; j >= 0; j--) {
					if (aMin[j].len > len) {
						aMin[j].len = len;
						aMin.index = j;
						break;
					}
				}//for
			}
		}//if
		//sort by index
		aMin.sort(fnSortByIndex);
		var getTriangle = function addWallsPutTriangle (pointA, pointB, pointC, color) {
			return new CSG.Polygon([pointA, pointB, pointC], color);
			//return bFlipped ? triangle.flipped() : triangle;
		};

		var bpoint = bottomPoints[0],
			tpoint = topPoints[0],
			secondPoint,
			nBotFacet, nTopFacet; //length of triangle facet side
		for (var iB = 0, iT = 0, iMax = iTopLen + iBotLen; iB + iT < iMax;) {
			if (aMin.length) {
				if (bMoreTops && iT == aMin[0].index) {//one vertex is on the bottom, 2 - on the top
					secondPoint = topPoints[++iT];
					//console.log('<<< extra top: ' + secondPoint + ', ' + tpoint + ', bottom: ' + bpoint);
					walls.push(getTriangle(
						secondPoint, tpoint, bpoint, color
					));
					tpoint = secondPoint;
					aMin.shift();
					continue;
				} else if (bMoreBottoms && iB == aMin[0].index) {
					secondPoint = bottomPoints[++iB];
					walls.push(getTriangle(
						tpoint, bpoint, secondPoint, color
					));
					bpoint = secondPoint;
					aMin.shift();
					continue;
				}
			}
			//choose the shortest path
			if (iB < iBotLen) { //one vertex is on the top, 2 - on the bottom
				nBotFacet = tpoint.pos.distanceToSquared(bottomPoints[iB+1].pos);
			} else {
				nBotFacet = Infinity;
			}
			if (iT < iTopLen) { //one vertex is on the bottom, 2 - on the top
				nTopFacet = bpoint.pos.distanceToSquared(topPoints[iT+1].pos);
			} else {
				nTopFacet = Infinity;
			}
			if (nBotFacet <= nTopFacet) {
				secondPoint = bottomPoints[++iB];
				walls.push(getTriangle(
					tpoint, bpoint, secondPoint, color
				));
				bpoint = secondPoint;
			} else if (iT < iTopLen) { //nTopFacet < Infinity
				secondPoint = topPoints[++iT];
				//console.log('<<< top: ' + secondPoint + ', ' + tpoint + ', bottom: ' + bpoint);
				walls.push(getTriangle(
					secondPoint, tpoint, bpoint, color
				));
				tpoint = secondPoint;
			};
		}
		return walls;
	}
};

CSG.Polygon.verticesConvex = function(vertices, planenormal) {
	var numvertices = vertices.length;
	if(numvertices > 2) {
		var prevprevpos = vertices[numvertices - 2].pos;
		var prevpos = vertices[numvertices - 1].pos;
		for(var i = 0; i < numvertices; i++) {
			var pos = vertices[i].pos;
			if(!CSG.Polygon.isConvexPoint(prevprevpos, prevpos, pos, planenormal)) {
				return false;
			}
			prevprevpos = prevpos;
			prevpos = pos;
		}
	}
	return true;
};

// Create a polygon from the given points
CSG.Polygon.createFromPoints = function(points, shared, plane) {
	var normal;
	if(arguments.length < 3) {
		// initially set a dummy vertex normal:
		normal = new CSG.Vector3D(0, 0, 0);
	} else {
		normal = plane.normal;
	}
	var vertices = [];
	points.map(function(p) {
		var vec = new CSG.Vector3D(p);
		var vertex = new CSG.Vertex(vec);
		vertices.push(vertex);
	});
	var polygon;
	if(arguments.length < 3) {
		polygon = new CSG.Polygon(vertices, shared);
	} else {
		polygon = new CSG.Polygon(vertices, shared, plane);
	}
	return polygon;
};

// calculate whether three points form a convex corner
//  prevpoint, point, nextpoint: the 3 coordinates (CSG.Vector3D instances)
//  normal: the normal vector of the plane
CSG.Polygon.isConvexPoint = function(prevpoint, point, nextpoint, normal) {
	var crossproduct = point.minus(prevpoint).cross(nextpoint.minus(point));
	var crossdotnormal = crossproduct.dot(normal);
	return(crossdotnormal >= 0);
};

CSG.Polygon.isStrictlyConvexPoint = function(prevpoint, point, nextpoint, normal) {
	var crossproduct = point.minus(prevpoint).cross(nextpoint.minus(point));
	var crossdotnormal = crossproduct.dot(normal);
	return(crossdotnormal >= 1e-5);
};

// # class CSG.Polygon.Shared
// Holds the shared properties for each polygon (currently only color)
CSG.Polygon.Shared = function(color) {
	this.color = color;
};

CSG.Polygon.Shared.fromObject = function(obj) {
	return new CSG.Polygon.Shared(obj.color);
};

CSG.Polygon.Shared.prototype = {
	getTag: function() {
		var result = this.tag;
		if(!result) {
			result = CSG.getTag();
			this.tag = result;
		}
		return result;
	},
	// get a string uniquely identifying this object
	getHash: function() {
		if(!this.color) return "null";
		return "" + this.color[0] + "/" + this.color[1] + "/" + this.color[2] + "/" + this.color[3];
	}
};

CSG.Polygon.defaultShared = new CSG.Polygon.Shared(null);

// # class PolygonTreeNode
// This class manages hierarchical splits of polygons
// At the top is a root node which doesn hold a polygon, only child PolygonTreeNodes
// Below that are zero or more 'top' nodes; each holds a polygon. The polygons can be in different planes
// splitByPlane() splits a node by a plane. If the plane intersects the polygon, two new child nodes
// are created holding the splitted polygon.
// getPolygons() retrieves the polygon from the tree. If for PolygonTreeNode the polygon is split but
// the two split parts (child nodes) are still intact, then the unsplit polygon is returned.
// This ensures that we can safely split a polygon into many fragments. If the fragments are untouched,
//  getPolygons() will return the original unsplit polygon instead of the fragments.
// remove() removes a polygon from the tree. Once a polygon is removed, the parent polygons are invalidated
// since they are no longer intact.
// constructor creates the root node:
CSG.PolygonTreeNode = function() {
	this.parent = null;
	this.children = [];
	this.polygon = null;
	this.removed = false;
};

CSG.PolygonTreeNode.prototype = {
	// fill the tree with polygons. Should be called on the root node only; child nodes must
	// always be a derivate (split) of the parent node.
	addPolygons: function(polygons) {
		if(!this.isRootNode())
			// new polygons can only be added to root node; children can only be splitted polygons
			throw new Error("Assertion failed");
		var _this = this;
		polygons.map(function(polygon) {
			_this.addChild(polygon);
		});
	},

	// remove a node
	// - the siblings become toplevel nodes
	// - the parent is removed recursively
	remove: function() {
		if(!this.removed) {
			this.removed = true;

			if(_CSGDEBUG) {
				if(this.isRootNode()) throw new Error("Assertion failed"); // can't remove root node
				if(this.children.length) throw new Error("Assertion failed"); // we shouldn't remove nodes with children
			}

			// remove ourselves from the parent's children list:
			var parentschildren = this.parent.children;
			var i = parentschildren.indexOf(this);
			if(i < 0) throw new Error("Assertion failed");
			parentschildren.splice(i, 1);

			// invalidate the parent's polygon, and of all parents above it:
			this.parent.recursivelyInvalidatePolygon();
		}
	},

	isRemoved: function() {
		return this.removed;
	},

	isRootNode: function() {
		return !this.parent;
	},

	// invert all polygons in the tree. Call on the root node
	invert: function() {
		if(!this.isRootNode()) throw new Error("Assertion failed"); // can only call this on the root node
		this.invertSub();
	},

	getPolygon: function() {
		if(!this.polygon) throw new Error("Assertion failed"); // doesn't have a polygon, which means that it has been broken down
		return this.polygon;
	},

	getPolygons: function(result) {
		if(this.polygon) {
			// the polygon hasn't been broken yet. We can ignore the children and return our polygon:
			result.push(this.polygon);
		} else {
			// our polygon has been split up and broken, so gather all subpolygons from the children:
			var childpolygons = [];
			this.children.map(function(child) {
				child.getPolygons(childpolygons);
			});
			childpolygons.map(function(p) {
				result.push(p);
			});
		}
	},

	// split the node by a plane; add the resulting nodes to the frontnodes and backnodes array
	// If the plane doesn't intersect the polygon, the 'this' object is added to one of the arrays
	// If the plane does intersect the polygon, two new child nodes are created for the front and back fragments,
	//  and added to both arrays.
	splitByPlane: function(plane, coplanarfrontnodes, coplanarbacknodes, frontnodes, backnodes) {
		var children = this.children;
		var numchildren = children.length;
		if(numchildren > 0) {
			// if we have children, split the children
			for(var i = 0; i < numchildren; i++) {
				children[i].splitByPlane(plane, coplanarfrontnodes, coplanarbacknodes, frontnodes, backnodes);
			}
		} else {
			// no children. Split the polygon:
			var polygon = this.polygon;
			if(polygon) {
				var bound = polygon.boundingSphere();
				var sphereradius = bound[1] + 1e-4;
				var planenormal = plane.normal;
				var spherecenter = bound[0];
				var d = planenormal.dot(spherecenter) - plane.w;
				if(d > sphereradius) {
					frontnodes.push(this);
				} else if(d < -sphereradius) {
					backnodes.push(this);
				} else {
					var splitresult = plane.splitPolygon(polygon);
					switch(splitresult.type) {
					case 0:
						// coplanar front:
						coplanarfrontnodes.push(this);
						break;

					case 1:
						// coplanar back:
						coplanarbacknodes.push(this);
						break;

					case 2:
						// front:
						frontnodes.push(this);
						break;

					case 3:
						// back:
						backnodes.push(this);
						break;

					case 4:
						// spanning:
						if(splitresult.front) {
							var frontnode = this.addChild(splitresult.front);
							frontnodes.push(frontnode);
						}
						if(splitresult.back) {
							var backnode = this.addChild(splitresult.back);
							backnodes.push(backnode);
						}
						break;
					}
				}
			}
		}
	},


	// PRIVATE methods from here:
	// add child to a node
	// this should be called whenever the polygon is split
	// a child should be created for every fragment of the split polygon
	// returns the newly created child
	addChild: function(polygon) {
		var newchild = new CSG.PolygonTreeNode();
		newchild.parent = this;
		newchild.polygon = polygon;
		this.children.push(newchild);
		return newchild;
	},

	invertSub: function() {
		if(this.polygon) {
			this.polygon = this.polygon.flipped();
		}
		this.children.map(function(child) {
			child.invertSub();
		});
	},

	recursivelyInvalidatePolygon: function() {
		if(this.polygon) {
			this.polygon = null;
			if(this.parent) {
				this.parent.recursivelyInvalidatePolygon();
			}
		}
	}
};



// # class Tree
// This is the root of a BSP tree
// We are using this separate class for the root of the tree, to hold the PolygonTreeNode root
// The actual tree is kept in this.rootnode
CSG.Tree = function(polygons) {
	this.polygonTree = new CSG.PolygonTreeNode();
	this.rootnode = new CSG.Node(null);
	if(polygons) this.addPolygons(polygons);
};

CSG.Tree.prototype = {
	invert: function() {
		this.polygonTree.invert();
		this.rootnode.invert();
	},

	// Remove all polygons in this BSP tree that are inside the other BSP tree
	// `tree`.
	clipTo: function(tree, alsoRemovecoplanarFront) {
		alsoRemovecoplanarFront = alsoRemovecoplanarFront ? true : false;
		this.rootnode.clipTo(tree, alsoRemovecoplanarFront);
	},

	allPolygons: function() {
		var result = [];
		this.polygonTree.getPolygons(result);
		return result;
	},

	addPolygons: function(polygons) {
		var _this = this;
		var polygontreenodes = polygons.map(function(p) {
			return _this.polygonTree.addChild(p);
		});
		this.rootnode.addPolygonTreeNodes(polygontreenodes);
	}
};

// # class Node
// Holds a node in a BSP tree. A BSP tree is built from a collection of polygons
// by picking a polygon to split along.
// Polygons are not stored directly in the tree, but in PolygonTreeNodes, stored in
// this.polygontreenodes. Those PolygonTreeNodes are children of the owning
// CSG.Tree.polygonTree
// This is not a leafy BSP tree since there is
// no distinction between internal and leaf nodes.
CSG.Node = function(parent) {
	this.plane = null;
	this.front = null;
	this.back = null;
	this.polygontreenodes = [];
	this.parent = parent;
};

CSG.Node.prototype = {
	// Convert solid space to empty space and empty space to solid space.
	invert: function() {
		if(this.plane) this.plane = this.plane.flipped();
		if(this.front) this.front.invert();
		if(this.back) this.back.invert();
		var temp = this.front;
		this.front = this.back;
		this.back = temp;
	},

	// clip polygontreenodes to our plane
	// calls remove() for all clipped PolygonTreeNodes
	clipPolygons: function(polygontreenodes, alsoRemovecoplanarFront) {
		if(this.plane) {
			var backnodes = [];
			var frontnodes = [];
			var coplanarfrontnodes = alsoRemovecoplanarFront ? backnodes : frontnodes;
			var plane = this.plane;
			var numpolygontreenodes = polygontreenodes.length;
			for(var i = 0; i < numpolygontreenodes; i++) {
				var node = polygontreenodes[i];
				if(!node.isRemoved()) {
					node.splitByPlane(plane, coplanarfrontnodes, backnodes, frontnodes, backnodes);
				}
			}
			if(this.front && (frontnodes.length > 0)) {
				this.front.clipPolygons(frontnodes, alsoRemovecoplanarFront);
			}
			var numbacknodes = backnodes.length;
			if(this.back && (numbacknodes > 0)) {
				this.back.clipPolygons(backnodes, alsoRemovecoplanarFront);
			} else {
				// there's nothing behind this plane. Delete the nodes behind this plane:
				for(var i = 0; i < numbacknodes; i++) {
					backnodes[i].remove();
				}
			}
		}
	},

	// Remove all polygons in this BSP tree that are inside the other BSP tree
	// `tree`.
	clipTo: function(tree, alsoRemovecoplanarFront) {
		if(this.polygontreenodes.length > 0) {
			tree.rootnode.clipPolygons(this.polygontreenodes, alsoRemovecoplanarFront);
		}
		if(this.front) this.front.clipTo(tree, alsoRemovecoplanarFront);
		if(this.back) this.back.clipTo(tree, alsoRemovecoplanarFront);
	},

	addPolygonTreeNodes: function(polygontreenodes) {
		if(polygontreenodes.length === 0) return;
		var _this = this;
		if(!this.plane) {
			var bestplane = polygontreenodes[0].getPolygon().plane;
			/*
	  var parentnormals = [];
	  this.getParentPlaneNormals(parentnormals, 6);
//parentnormals = [];
	  var numparentnormals = parentnormals.length;
	  var minmaxnormal = 1.0;
	  polygontreenodes.map(function(polygontreenode){
		var plane = polygontreenodes[0].getPolygon().plane;
		var planenormal = plane.normal;
		var maxnormaldot = -1.0;
		parentnormals.map(function(parentnormal){
		  var dot = parentnormal.dot(planenormal);
		  if(dot > maxnormaldot) maxnormaldot = dot;
		});
		if(maxnormaldot < minmaxnormal)
		{
		  minmaxnormal = maxnormaldot;
		  bestplane = plane;
		}
	  });
*/
			this.plane = bestplane;
		}
		var frontnodes = [];
		var backnodes = [];
		polygontreenodes.map(function(polygontreenode) {
			polygontreenode.splitByPlane(_this.plane, _this.polygontreenodes, backnodes, frontnodes, backnodes);
		});
		if(frontnodes.length > 0) {
			if(!this.front) this.front = new CSG.Node(this);
			this.front.addPolygonTreeNodes(frontnodes);
		}
		if(backnodes.length > 0) {
			if(!this.back) this.back = new CSG.Node(this);
			this.back.addPolygonTreeNodes(backnodes);
		}
	},

	getParentPlaneNormals: function(normals, maxdepth) {
		if(maxdepth > 0) {
			if(this.parent) {
				normals.push(this.parent.plane.normal);
				this.parent.getParentPlaneNormals(normals, maxdepth - 1);
			}
		}
	}
};

//////////
// # class Matrix4x4:
// Represents a 4x4 matrix. Elements are specified in row order
CSG.Matrix4x4 = function(elements) {
	if(arguments.length >= 1) {
		this.elements = elements;
	} else {
		// if no arguments passed: create unity matrix
		this.elements = [1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1];
	}
};

CSG.Matrix4x4.prototype = {
	plus: function(m) {
		var r = [];
		for(var i = 0; i < 16; i++) {
			r[i] = this.elements[i] + m.elements[i];
		}
		return new CSG.Matrix4x4(r);
	},

	minus: function(m) {
		var r = [];
		for(var i = 0; i < 16; i++) {
			r[i] = this.elements[i] - m.elements[i];
		}
		return new CSG.Matrix4x4(r);
	},

	// right multiply by another 4x4 matrix:
	multiply: function(m) {
		// cache elements in local variables, for speedup:
		var this0 = this.elements[0];
		var this1 = this.elements[1];
		var this2 = this.elements[2];
		var this3 = this.elements[3];
		var this4 = this.elements[4];
		var this5 = this.elements[5];
		var this6 = this.elements[6];
		var this7 = this.elements[7];
		var this8 = this.elements[8];
		var this9 = this.elements[9];
		var this10 = this.elements[10];
		var this11 = this.elements[11];
		var this12 = this.elements[12];
		var this13 = this.elements[13];
		var this14 = this.elements[14];
		var this15 = this.elements[15];
		var m0 = m.elements[0];
		var m1 = m.elements[1];
		var m2 = m.elements[2];
		var m3 = m.elements[3];
		var m4 = m.elements[4];
		var m5 = m.elements[5];
		var m6 = m.elements[6];
		var m7 = m.elements[7];
		var m8 = m.elements[8];
		var m9 = m.elements[9];
		var m10 = m.elements[10];
		var m11 = m.elements[11];
		var m12 = m.elements[12];
		var m13 = m.elements[13];
		var m14 = m.elements[14];
		var m15 = m.elements[15];

		var result = [];
		result[0] = this0 * m0 + this1 * m4 + this2 * m8 + this3 * m12;
		result[1] = this0 * m1 + this1 * m5 + this2 * m9 + this3 * m13;
		result[2] = this0 * m2 + this1 * m6 + this2 * m10 + this3 * m14;
		result[3] = this0 * m3 + this1 * m7 + this2 * m11 + this3 * m15;
		result[4] = this4 * m0 + this5 * m4 + this6 * m8 + this7 * m12;
		result[5] = this4 * m1 + this5 * m5 + this6 * m9 + this7 * m13;
		result[6] = this4 * m2 + this5 * m6 + this6 * m10 + this7 * m14;
		result[7] = this4 * m3 + this5 * m7 + this6 * m11 + this7 * m15;
		result[8] = this8 * m0 + this9 * m4 + this10 * m8 + this11 * m12;
		result[9] = this8 * m1 + this9 * m5 + this10 * m9 + this11 * m13;
		result[10] = this8 * m2 + this9 * m6 + this10 * m10 + this11 * m14;
		result[11] = this8 * m3 + this9 * m7 + this10 * m11 + this11 * m15;
		result[12] = this12 * m0 + this13 * m4 + this14 * m8 + this15 * m12;
		result[13] = this12 * m1 + this13 * m5 + this14 * m9 + this15 * m13;
		result[14] = this12 * m2 + this13 * m6 + this14 * m10 + this15 * m14;
		result[15] = this12 * m3 + this13 * m7 + this14 * m11 + this15 * m15;
		return new CSG.Matrix4x4(result);
	},

	clone: function() {
		var elements = this.elements.map(function(p) {
			return p;
		});
		return new CSG.Matrix4x4(elements);
	},

	// Right multiply the matrix by a CSG.Vector3D (interpreted as 3 row, 1 column)
	// (result = M*v)
	// Fourth element is taken as 1
	rightMultiply1x3Vector: function(v) {
		var v0 = v._x;
		var v1 = v._y;
		var v2 = v._z;
		var v3 = 1;
		var x = v0 * this.elements[0] + v1 * this.elements[1] + v2 * this.elements[2] + v3 * this.elements[3];
		var y = v0 * this.elements[4] + v1 * this.elements[5] + v2 * this.elements[6] + v3 * this.elements[7];
		var z = v0 * this.elements[8] + v1 * this.elements[9] + v2 * this.elements[10] + v3 * this.elements[11];
		var w = v0 * this.elements[12] + v1 * this.elements[13] + v2 * this.elements[14] + v3 * this.elements[15];
		// scale such that fourth element becomes 1:
		if(w != 1) {
			var invw = 1.0 / w;
			x *= invw;
			y *= invw;
			z *= invw;
		}
		return new CSG.Vector3D(x, y, z);
	},

	// Multiply a CSG.Vector3D (interpreted as 3 column, 1 row) by this matrix
	// (result = v*M)
	// Fourth element is taken as 1
	leftMultiply1x3Vector: function(v) {
		var v0 = v._x;
		var v1 = v._y;
		var v2 = v._z;
		var v3 = 1;
		var x = v0 * this.elements[0] + v1 * this.elements[4] + v2 * this.elements[8] + v3 * this.elements[12];
		var y = v0 * this.elements[1] + v1 * this.elements[5] + v2 * this.elements[9] + v3 * this.elements[13];
		var z = v0 * this.elements[2] + v1 * this.elements[6] + v2 * this.elements[10] + v3 * this.elements[14];
		var w = v0 * this.elements[3] + v1 * this.elements[7] + v2 * this.elements[11] + v3 * this.elements[15];
		// scale such that fourth element becomes 1:
		if(w != 1) {
			var invw = 1.0 / w;
			x *= invw;
			y *= invw;
			z *= invw;
		}
		return new CSG.Vector3D(x, y, z);
	},

	// Right multiply the matrix by a CSG.Vector2D (interpreted as 2 row, 1 column)
	// (result = M*v)
	// Fourth element is taken as 1
	rightMultiply1x2Vector: function(v) {
		var v0 = v.x;
		var v1 = v.y;
		var v2 = 0;
		var v3 = 1;
		var x = v0 * this.elements[0] + v1 * this.elements[1] + v2 * this.elements[2] + v3 * this.elements[3];
		var y = v0 * this.elements[4] + v1 * this.elements[5] + v2 * this.elements[6] + v3 * this.elements[7];
		var z = v0 * this.elements[8] + v1 * this.elements[9] + v2 * this.elements[10] + v3 * this.elements[11];
		var w = v0 * this.elements[12] + v1 * this.elements[13] + v2 * this.elements[14] + v3 * this.elements[15];
		// scale such that fourth element becomes 1:
		if(w != 1) {
			var invw = 1.0 / w;
			x *= invw;
			y *= invw;
			z *= invw;
		}
		return new CSG.Vector2D(x, y);
	},

	// Multiply a CSG.Vector2D (interpreted as 2 column, 1 row) by this matrix
	// (result = v*M)
	// Fourth element is taken as 1
	leftMultiply1x2Vector: function(v) {
		var v0 = v.x;
		var v1 = v.y;
		var v2 = 0;
		var v3 = 1;
		var x = v0 * this.elements[0] + v1 * this.elements[4] + v2 * this.elements[8] + v3 * this.elements[12];
		var y = v0 * this.elements[1] + v1 * this.elements[5] + v2 * this.elements[9] + v3 * this.elements[13];
		var z = v0 * this.elements[2] + v1 * this.elements[6] + v2 * this.elements[10] + v3 * this.elements[14];
		var w = v0 * this.elements[3] + v1 * this.elements[7] + v2 * this.elements[11] + v3 * this.elements[15];
		// scale such that fourth element becomes 1:
		if(w != 1) {
			var invw = 1.0 / w;
			x *= invw;
			y *= invw;
			z *= invw;
		}
		return new CSG.Vector2D(x, y);
	},

	// determine whether this matrix is a mirroring transformation
	isMirroring: function() {
		var u = new CSG.Vector3D(this.elements[0], this.elements[4], this.elements[8]);
		var v = new CSG.Vector3D(this.elements[1], this.elements[5], this.elements[9]);
		var w = new CSG.Vector3D(this.elements[2], this.elements[6], this.elements[10]);

		// for a true orthogonal, non-mirrored base, u.cross(v) == w
		// If they have an opposite direction then we are mirroring
		var mirrorvalue = u.cross(v).dot(w);
		var ismirror = (mirrorvalue < 0);
		return ismirror;
	}
};

// return the unity matrix
CSG.Matrix4x4.unity = function() {
	return new CSG.Matrix4x4();
};

// Create a rotation matrix for rotating around the x axis
CSG.Matrix4x4.rotationX = function(degrees) {
	var radians = degrees * Math.PI * (1.0 / 180.0);
	var cos = Math.cos(radians);
	var sin = Math.sin(radians);
	var els = [
	1, 0, 0, 0, 0, cos, sin, 0, 0, -sin, cos, 0, 0, 0, 0, 1];
	return new CSG.Matrix4x4(els);
};

// Create a rotation matrix for rotating around the y axis
CSG.Matrix4x4.rotationY = function(degrees) {
	var radians = degrees * Math.PI * (1.0 / 180.0);
	var cos = Math.cos(radians);
	var sin = Math.sin(radians);
	var els = [
	cos, 0, -sin, 0, 0, 1, 0, 0, sin, 0, cos, 0, 0, 0, 0, 1];
	return new CSG.Matrix4x4(els);
};

// Create a rotation matrix for rotating around the z axis
CSG.Matrix4x4.rotationZ = function(degrees) {
	var radians = degrees * Math.PI * (1.0 / 180.0);
	var cos = Math.cos(radians);
	var sin = Math.sin(radians);
	var els = [
	cos, sin, 0, 0, -sin, cos, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1];
	return new CSG.Matrix4x4(els);
};

// Matrix for rotation about arbitrary point and axis
CSG.Matrix4x4.rotation = function(rotationCenter, rotationAxis, degrees) {
	rotationCenter = new CSG.Vector3D(rotationCenter);
	rotationAxis = new CSG.Vector3D(rotationAxis);
	var rotationPlane = CSG.Plane.fromNormalAndPoint(rotationAxis, rotationCenter);
	var orthobasis = new CSG.OrthoNormalBasis(rotationPlane);
	var transformation = CSG.Matrix4x4.translation(rotationCenter.negated());
	transformation = transformation.multiply(orthobasis.getProjectionMatrix());
	transformation = transformation.multiply(CSG.Matrix4x4.rotationZ(degrees));
	transformation = transformation.multiply(orthobasis.getInverseProjectionMatrix());
	transformation = transformation.multiply(CSG.Matrix4x4.translation(rotationCenter));
	return transformation;
};

// Create an affine matrix for translation:
CSG.Matrix4x4.translation = function(v) {
	// parse as CSG.Vector3D, so we can pass an array or a CSG.Vector3D
	var vec = new CSG.Vector3D(v);
	var els = [1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, vec.x, vec.y, vec.z, 1];
	return new CSG.Matrix4x4(els);
};

// Create an affine matrix for mirroring into an arbitrary plane:
CSG.Matrix4x4.mirroring = function(plane) {
	var nx = plane.normal.x;
	var ny = plane.normal.y;
	var nz = plane.normal.z;
	var w = plane.w;
	var els = [
		(1.0 - 2.0 * nx * nx), (-2.0 * ny * nx), (-2.0 * nz * nx), 0,
		(-2.0 * nx * ny), (1.0 - 2.0 * ny * ny), (-2.0 * nz * ny), 0,
		(-2.0 * nx * nz), (-2.0 * ny * nz), (1.0 - 2.0 * nz * nz), 0,
		(-2.0 * nx * w), (-2.0 * ny * w), (-2.0 * nz * w), 1
	];
	return new CSG.Matrix4x4(els);
};

// Create an affine matrix for scaling:
CSG.Matrix4x4.scaling = function(v) {
	// parse as CSG.Vector3D, so we can pass an array or a CSG.Vector3D
	var vec = new CSG.Vector3D(v);
	var els = [
	vec.x, 0, 0, 0, 0, vec.y, 0, 0, 0, 0, vec.z, 0, 0, 0, 0, 1];
	return new CSG.Matrix4x4(els);
};

///////////////////////////////////////////////////
// # class Vector2D:
// Represents a 2 element vector
CSG.Vector2D = function(x, y) {
	if(arguments.length == 2) {
		this._x = parseFloat(x);
		this._y = parseFloat(y);
	} else {
		var ok = true;
		if(arguments.length == 1) {
			if(typeof(x) == "object") {
				if(x instanceof CSG.Vector2D) {
					this._x = x._x;
					this._y = x._y;
				} else if(x instanceof Array) {
					this._x = parseFloat(x[0]);
					this._y = parseFloat(x[1]);
				} else if(('x' in x) && ('y' in x)) {
					this._x = parseFloat(x.x);
					this._y = parseFloat(x.y);
				} else ok = false;
			} else {
				var v = parseFloat(x);
				this._x = v;
				this._y = v;
			}
		} else ok = false;
		if(ok) {
			if((!CSG.IsFloat(this._x)) || (!CSG.IsFloat(this._y))) ok = false;
		}
		if(!ok) {
			throw new Error("wrong arguments");
		}
	}
};

CSG.Vector2D.fromAngle = function(radians) {
	return CSG.Vector2D.fromAngleRadians(radians);
};

CSG.Vector2D.fromAngleDegrees = function(degrees) {
	var radians = Math.PI * degrees / 180;
	return CSG.Vector2D.fromAngleRadians(radians);
};

CSG.Vector2D.fromAngleRadians = function(radians) {
	return new CSG.Vector2D(Math.cos(radians), Math.sin(radians));
};

CSG.Vector2D.prototype = {
	get x() {
		return this._x;
	}, get y() {
		return this._y;
	},

	set x(v) {
		throw new Error("Vector2D is immutable");
	}, set y(v) {
		throw new Error("Vector2D is immutable");
	},

	// extend to a 3D vector by adding a z coordinate:
	toVector3D: function(z) {
		return new CSG.Vector3D(this._x, this._y, z);
	},

	equals: function(a) {
		return(this._x == a._x) && (this._y == a._y);
	},

	clone: function() {
		return new CSG.Vector2D(this._x, this._y);
	},

	negated: function() {
		return new CSG.Vector2D(-this._x, -this._y);
	},

	plus: function(a) {
		return new CSG.Vector2D(this._x + a._x, this._y + a._y);
	},

	minus: function(a) {
		return new CSG.Vector2D(this._x - a._x, this._y - a._y);
	},

	times: function(a) {
		return new CSG.Vector2D(this._x * a, this._y * a);
	},

	dividedBy: function(a) {
		return new CSG.Vector2D(this._x / a, this._y / a);
	},

	dot: function(a) {
		return this._x * a._x + this._y * a._y;
	},

	lerp: function(a, t) {
		return this.plus(a.minus(this).times(t));
	},

	length: function() {
		return Math.sqrt(this.dot(this));
	},

	distanceTo: function(a) {
		return this.minus(a).length();
	},

	distanceToSquared: function(a) {
		return this.minus(a).lengthSquared();
	},

	lengthSquared: function() {
		return this.dot(this);
	},

	unit: function() {
		return this.dividedBy(this.length());
	},

	cross: function(a) {
		return this._x * a._y - this._y * a._x;
	},

	// returns the vector rotated by 90 degrees clockwise
	normal: function() {
		return new CSG.Vector2D(this._y, -this._x);
	},

	// Right multiply by a 4x4 matrix (the vector is interpreted as a row vector)
	// Returns a new CSG.Vector2D
	multiply4x4: function(matrix4x4) {
		return matrix4x4.leftMultiply1x2Vector(this);
	},

	transform: function(matrix4x4) {
		return matrix4x4.leftMultiply1x2Vector(this);
	},

	angle: function() {
		return this.angleRadians();
	},

	angleDegrees: function() {
		var radians = this.angleRadians();
		return 180 * radians / Math.PI;
	},

	angleRadians: function() {
		// y=sin, x=cos
		return Math.atan2(this._y, this._x);
	},

	min: function(p) {
		return new CSG.Vector2D(
		Math.min(this._x, p._x), Math.min(this._y, p._y));
	},

	max: function(p) {
		return new CSG.Vector2D(
		Math.max(this._x, p._x), Math.max(this._y, p._y));
	},

	toString: function() {
		return "("+this._x.toFixed(2)+", "+this._y.toFixed(2)+")";
	}
};


// # class Line2D
// Represents a directional line in 2D space
// A line is parametrized by its normal vector (perpendicular to the line, rotated 90 degrees counter clockwise)
// and w. The line passes through the point <normal>.times(w).
// normal must be a unit vector!
// Equation: p is on line if normal.dot(p)==w
CSG.Line2D = function(normal, w) {
	normal = new CSG.Vector2D(normal);
	w = parseFloat(w);
	var l = normal.length();
	// normalize:
	w *= l;
	normal = normal.times(1.0 / l);
	this.normal = normal;
	this.w = w;
};

CSG.Line2D.fromPoints = function(p1, p2) {
	p1 = new CSG.Vector2D(p1);
	p2 = new CSG.Vector2D(p2);
	var direction = p2.minus(p1);
	var normal = direction.normal().negated().unit();
	var w = p1.dot(normal);
	return new CSG.Line2D(normal, w);
};

CSG.Line2D.prototype = {
	// same line but opposite direction:
	reverse: function() {
		return new CSG.Line2D(this.normal.negated(), -this.w);
	},

	equals: function(l) {
		return(l.normal.equals(this.normal) && (l.w == this.w));
	},

	origin: function() {
		return this.normal.times(this.w);
	},

	direction: function() {
		return this.normal.normal();
	},

	xAtY: function(y) {
		// (py == y) && (normal * p == w)
		// -> px = (w - normal._y * y) / normal.x
		var x = (this.w - this.normal._y * y) / this.normal.x;
		return x;
	},

	absDistanceToPoint: function(point) {
		point = new CSG.Vector2D(point);
		var point_projected = point.dot(this.normal);
		var distance = Math.abs(point_projected - this.w);
		return distance;
	},
	/*FIXME: has error - origin is not defined, the method is never used
	closestPoint: function(point) {
		point = new CSG.Vector2D(point);
		var vector = point.dot(this.direction());
		return origin.plus(vector);
	},
	*/

	// intersection between two lines, returns point as Vector2D
	intersectWithLine: function(line2d) {
		var point = CSG.solve2Linear(this.normal.x, this.normal.y, line2d.normal.x, line2d.normal.y, this.w, line2d.w);
		point = new CSG.Vector2D(point); // make  vector2d
		return point;
	},

	transform: function(matrix4x4) {
		var origin = new CSG.Vector2D(0, 0);
		var pointOnPlane = this.normal.times(this.w);
		var neworigin = origin.multiply4x4(matrix4x4);
		var neworiginPlusNormal = this.normal.multiply4x4(matrix4x4);
		var newnormal = neworiginPlusNormal.minus(neworigin);
		var newpointOnPlane = pointOnPlane.multiply4x4(matrix4x4);
		var neww = newnormal.dot(newpointOnPlane);
		return new CSG.Line2D(newnormal, neww);
	}
};

// # class Line3D
// Represents a line in 3D space
// direction must be a unit vector
// point is a random point on the line
CSG.Line3D = function(point, direction) {
	point = new CSG.Vector3D(point);
	direction = new CSG.Vector3D(direction);
	this.point = point;
	this.direction = direction.unit();
};

CSG.Line3D.fromPoints = function(p1, p2) {
	p1 = new CSG.Vector3D(p1);
	p2 = new CSG.Vector3D(p2);
	var direction = p2.minus(p1).unit();
	return new CSG.Line3D(p1, direction);
};

CSG.Line3D.fromPlanes = function(p1, p2) {
	var direction = p1.normal.cross(p2.normal);
	var l = direction.length();
	if(l < 1e-10) {
		throw new Error("Parallel planes");
	}
	direction = direction.times(1.0 / l);

	var mabsx = Math.abs(direction.x);
	var mabsy = Math.abs(direction.y);
	var mabsz = Math.abs(direction.z);
	var origin;
	if((mabsx >= mabsy) && (mabsx >= mabsz)) {
		// direction vector is mostly pointing towards x
		// find a point p for which x is zero:
		var r = CSG.solve2Linear(p1.normal.y, p1.normal.z, p2.normal.y, p2.normal.z, p1.w, p2.w);
		origin = new CSG.Vector3D(0, r[0], r[1]);
	} else if((mabsy >= mabsx) && (mabsy >= mabsz)) {
		// find a point p for which y is zero:
		var r = CSG.solve2Linear(p1.normal.x, p1.normal.z, p2.normal.x, p2.normal.z, p1.w, p2.w);
		origin = new CSG.Vector3D(r[0], 0, r[1]);
	} else {
		// find a point p for which z is zero:
		var r = CSG.solve2Linear(p1.normal.x, p1.normal.y, p2.normal.x, p2.normal.y, p1.w, p2.w);
		origin = new CSG.Vector3D(r[0], r[1], 0);
	}
	return new CSG.Line3D(origin, direction);
};


CSG.Line3D.prototype = {
	intersectWithPlane: function(plane) {
		// plane: plane.normal * p = plane.w
		// line: p=line.point + labda * line.direction
		var labda = (plane.w - plane.normal.dot(this.point)) / plane.normal.dot(this.direction);
		var point = this.point.plus(this.direction.times(labda));
		return point;
	},

	clone: function(line) {
		return new CSG.Line3D(this.point.clone(), this.direction.clone());
	},

	reverse: function() {
		return new CSG.Line3D(this.point.clone(), this.direction.negated());
	},

	transform: function(matrix4x4) {
		var newpoint = this.point.multiply4x4(matrix4x4);
		var pointPlusDirection = this.point.plus(this.direction);
		var newPointPlusDirection = pointPlusDirection.multiply4x4(matrix4x4);
		var newdirection = newPointPlusDirection.minus(newpoint);
		return new CSG.Line3D(newpoint, newdirection);
	},

	closestPointOnLine: function(point) {
		point = new CSG.Vector3D(point);
		var t = point.minus(this.point).dot(this.direction) / this.direction.dot(this.direction);
		var closestpoint = this.point.plus(this.direction.times(t));
		return closestpoint;
	},

	distanceToPoint: function(point) {
		point = new CSG.Vector3D(point);
		var closestpoint = this.closestPointOnLine(point);
		var distancevector = point.minus(closestpoint);
		var distance = distancevector.length();
		return distance;
	},

	equals: function(line3d) {
		if(!this.direction.equals(line3d.direction)) return false;
		var distance = this.distanceToPoint(line3d.point);
		if(distance > 1e-8) return false;
		return true;
	}
};


// # class OrthoNormalBasis
// Reprojects points on a 3D plane onto a 2D plane
// or from a 2D plane back onto the 3D plane
CSG.OrthoNormalBasis = function(plane, rightvector) {
	if(arguments.length < 2) {
		// choose an arbitrary right hand vector, making sure it is somewhat orthogonal to the plane normal:
		rightvector = plane.normal.randomNonParallelVector();
	} else {
		rightvector = new CSG.Vector3D(rightvector);
	}
	this.v = plane.normal.cross(rightvector).unit();
	this.u = this.v.cross(plane.normal);
	this.plane = plane;
	this.planeorigin = plane.normal.times(plane.w);
};

// The z=0 plane, with the 3D x and y vectors mapped to the 2D x and y vector
CSG.OrthoNormalBasis.Z0Plane = function() {
	var plane = new CSG.Plane(new CSG.Vector3D([0, 0, 1]), 0);
	return new CSG.OrthoNormalBasis(plane, new CSG.Vector3D([1, 0, 0]));
};

CSG.OrthoNormalBasis.prototype = {
	getProjectionMatrix: function() {
		return new CSG.Matrix4x4([
			this.u.x, this.v.x, this.plane.normal.x, 0,
			this.u.y, this.v.y, this.plane.normal.y, 0,
			this.u.z, this.v.z, this.plane.normal.z, 0,
			0, 0, -this.plane.w, 1]);
	},

	getInverseProjectionMatrix: function() {
		var p = this.plane.normal.times(this.plane.w);
		return new CSG.Matrix4x4([
			this.u.x, this.u.y, this.u.z, 0,
			this.v.x, this.v.y, this.v.z, 0,
			this.plane.normal.x, this.plane.normal.y, this.plane.normal.z, 0,
			p.x, p.y, p.z, 1]);
	},

	to2D: function(vec3) {
		return new CSG.Vector2D(vec3.dot(this.u), vec3.dot(this.v));
	},

	to3D: function(vec2) {
		return this.planeorigin.plus(this.u.times(vec2.x)).plus(this.v.times(vec2.y));
	},

	line3Dto2D: function(line3d) {
		var a = line3d.point;
		var b = line3d.direction.plus(a);
		var a2d = this.to2D(a);
		var b2d = this.to2D(b);
		return CSG.Line2D.fromPoints(a2d, b2d);
	},

	line2Dto3D: function(line2d) {
		var a = line2d.origin();
		var b = line2d.direction().plus(a);
		var a3d = this.to3D(a);
		var b3d = this.to3D(b);
		return CSG.Line3D.fromPoints(a3d, b3d);
	},

	transform: function(matrix4x4) {
		// todo: this may not work properly in case of mirroring
		var newplane = this.plane.transform(matrix4x4);
		var rightpoint_transformed = this.u.transform(matrix4x4);
		var origin_transformed = new CSG.Vector3D(0, 0, 0).transform(matrix4x4);
		var newrighthandvector = rightpoint_transformed.minus(origin_transformed);
		var newbasis = new CSG.OrthoNormalBasis(newplane, newrighthandvector);
		return newbasis;
	}
};

function insertSorted(array, element, comparefunc) {
	var leftbound = 0;
	var rightbound = array.length;
	while(rightbound > leftbound) {
		var testindex = Math.floor((leftbound + rightbound) / 2);
		var testelement = array[testindex];
		var compareresult = comparefunc(element, testelement);
		if(compareresult > 0) // element > testelement
		{
			leftbound = testindex + 1;
		} else {
			rightbound = testindex;
		}
	}
	array.splice(leftbound, 0, element);
}

// Get the x coordinate of a point with a certain y coordinate, interpolated between two
// points (CSG.Vector2D).
// Interpolation is robust even if the points have the same y coordinate
CSG.interpolateBetween2DPointsForY = function(point1, point2, y) {
	var f1 = y - point1.y;
	var f2 = point2.y - point1.y;
	if(f2 < 0) {
		f1 = -f1;
		f2 = -f2;
	}
	var t;
	if(f1 <= 0) {
		t = 0.0;
	} else if(f1 >= f2) {
		t = 1.0;
	} else if(f2 < 1e-10) {
		t = 0.5;
	} else {
		t = f1 / f2;
	}
	var result = point1.x + t * (point2.x - point1.x);
	return result;
};

// Retesselation function for a set of coplanar polygons. See the introduction at the top of
// this file.
CSG.reTesselateCoplanarPolygons = function(sourcepolygons, destpolygons) {
	var EPS = 1e-5;

	var numpolygons = sourcepolygons.length;
	if(numpolygons > 0) {
		var plane = sourcepolygons[0].plane;
		var shared = sourcepolygons[0].shared;
		var orthobasis = new CSG.OrthoNormalBasis(plane);
		var polygonvertices2d = []; // array of array of CSG.Vector2D
		var polygontopvertexindexes = []; // array of indexes of topmost vertex per polygon
		var topy2polygonindexes = {};
		var ycoordinatetopolygonindexes = {};

		var xcoordinatebins = {};
		var ycoordinatebins = {};

		// convert all polygon vertices to 2D
		// Make a list of all encountered y coordinates
		// And build a map of all polygons that have a vertex at a certain y coordinate:
		var ycoordinateBinningFactor = 1.0 / EPS * 10;
		for(var polygonindex = 0; polygonindex < numpolygons; polygonindex++) {
			var poly3d = sourcepolygons[polygonindex];
			var vertices2d = [];
			var numvertices = poly3d.vertices.length;
			var minindex = -1;
			if(numvertices > 0) {
				var miny, maxy, maxindex;
				for(var i = 0; i < numvertices; i++) {
					var pos2d = orthobasis.to2D(poly3d.vertices[i].pos);
					// perform binning of y coordinates: If we have multiple vertices very
					// close to each other, give them the same y coordinate:
					var ycoordinatebin = Math.floor(pos2d.y * ycoordinateBinningFactor);
					var newy;
					if(ycoordinatebin in ycoordinatebins) {
						newy = ycoordinatebins[ycoordinatebin];
					} else if(ycoordinatebin + 1 in ycoordinatebins) {
						newy = ycoordinatebins[ycoordinatebin + 1];
					} else if(ycoordinatebin - 1 in ycoordinatebins) {
						newy = ycoordinatebins[ycoordinatebin - 1];
					} else {
						newy = pos2d.y;
						ycoordinatebins[ycoordinatebin] = pos2d.y;
					}
					pos2d = new CSG.Vector2D(pos2d.x, newy);
					vertices2d.push(pos2d);
					var y = pos2d.y;
					if((i === 0) || (y < miny)) {
						miny = y;
						minindex = i;
					}
					if((i === 0) || (y > maxy)) {
						maxy = y;
						maxindex = i;
					}
					if(!(y in ycoordinatetopolygonindexes)) {
						ycoordinatetopolygonindexes[y] = {};
					}
					ycoordinatetopolygonindexes[y][polygonindex] = true;
				}
				if(miny >= maxy) {
					// degenerate polygon, all vertices have same y coordinate. Just ignore it from now:
					vertices2d = [];
				} else {
					if(!(miny in topy2polygonindexes)) {
						topy2polygonindexes[miny] = [];
					}
					topy2polygonindexes[miny].push(polygonindex);
				}
			} // if(numvertices > 0)
			// reverse the vertex order:
			vertices2d.reverse();
			minindex = numvertices - minindex - 1;
			polygonvertices2d.push(vertices2d);
			polygontopvertexindexes.push(minindex);
		}
		var ycoordinates = [];
		for(var ycoordinate in ycoordinatetopolygonindexes) ycoordinates.push(ycoordinate);
		ycoordinates.sort(fnNumberSort);

		// Now we will iterate over all y coordinates, from lowest to highest y coordinate
		// activepolygons: source polygons that are 'active', i.e. intersect with our y coordinate
		//   Is sorted so the polygons are in left to right order
		// Each element in activepolygons has these properties:
		//        polygonindex: the index of the source polygon (i.e. an index into the sourcepolygons
		//                      and polygonvertices2d arrays)
		//        leftvertexindex: the index of the vertex at the left side of the polygon (lowest x)
		//                         that is at or just above the current y coordinate
		//        rightvertexindex: dito at right hand side of polygon
		//        topleft, bottomleft: coordinates of the left side of the polygon crossing the current y coordinate
		//        topright, bottomright: coordinates of the right hand side of the polygon crossing the current y coordinate
		var activepolygons = [];
		var prevoutpolygonrow = [];
		for(var yindex = 0; yindex < ycoordinates.length; yindex++) {
			var newoutpolygonrow = [];
			var ycoordinate_as_string = ycoordinates[yindex];
			var ycoordinate = Number(ycoordinate_as_string);

			// update activepolygons for this y coordinate:
			// - Remove any polygons that end at this y coordinate
			// - update leftvertexindex and rightvertexindex (which point to the current vertex index
			//   at the the left and right side of the polygon
			// Iterate over all polygons that have a corner at this y coordinate:
			var polygonindexeswithcorner = ycoordinatetopolygonindexes[ycoordinate_as_string];
			for(var activepolygonindex = 0; activepolygonindex < activepolygons.length; ++activepolygonindex) {
				var activepolygon = activepolygons[activepolygonindex];
				var polygonindex = activepolygon.polygonindex;
				if(polygonindexeswithcorner[polygonindex]) {
					// this active polygon has a corner at this y coordinate:
					var vertices2d = polygonvertices2d[polygonindex];
					var numvertices = vertices2d.length;
					var newleftvertexindex = activepolygon.leftvertexindex;
					var newrightvertexindex = activepolygon.rightvertexindex;
					// See if we need to increase leftvertexindex or decrease rightvertexindex:
					while(true) {
						var nextleftvertexindex = newleftvertexindex + 1;
						if(nextleftvertexindex >= numvertices) nextleftvertexindex = 0;
						if(vertices2d[nextleftvertexindex].y != ycoordinate) break;
						newleftvertexindex = nextleftvertexindex;
					}
					var nextrightvertexindex = newrightvertexindex - 1;
					if(nextrightvertexindex < 0) nextrightvertexindex = numvertices - 1;
					if(vertices2d[nextrightvertexindex].y == ycoordinate) {
						newrightvertexindex = nextrightvertexindex;
					}
					if((newleftvertexindex != activepolygon.leftvertexindex) && (newleftvertexindex == newrightvertexindex)) {
						// We have increased leftvertexindex or decreased rightvertexindex, and now they point to the same vertex
						// This means that this is the bottom point of the polygon. We'll remove it:
						activepolygons.splice(activepolygonindex, 1);
						--activepolygonindex;
					} else {
						activepolygon.leftvertexindex = newleftvertexindex;
						activepolygon.rightvertexindex = newrightvertexindex;
						activepolygon.topleft = vertices2d[newleftvertexindex];
						activepolygon.topright = vertices2d[newrightvertexindex];
						var nextleftvertexindex = newleftvertexindex + 1;
						if(nextleftvertexindex >= numvertices) nextleftvertexindex = 0;
						activepolygon.bottomleft = vertices2d[nextleftvertexindex];
						var nextrightvertexindex = newrightvertexindex - 1;
						if(nextrightvertexindex < 0) nextrightvertexindex = numvertices - 1;
						activepolygon.bottomright = vertices2d[nextrightvertexindex];
					}
				} // if polygon has corner here
			} // for activepolygonindex
			var nextycoordinate;
			if(yindex >= ycoordinates.length - 1) {
				// last row, all polygons must be finished here:
				activepolygons = [];
				nextycoordinate = null;
			} else // yindex < ycoordinates.length-1
			{
				nextycoordinate = Number(ycoordinates[yindex + 1]);
				var middleycoordinate = 0.5 * (ycoordinate + nextycoordinate);
				// update activepolygons by adding any polygons that start here:
				var startingpolygonindexes = topy2polygonindexes[ycoordinate_as_string];
				for(var polygonindex_key in startingpolygonindexes) {
					var polygonindex = startingpolygonindexes[polygonindex_key];
					var vertices2d = polygonvertices2d[polygonindex];
					var numvertices = vertices2d.length;
					var topvertexindex = polygontopvertexindexes[polygonindex];
					// the top of the polygon may be a horizontal line. In that case topvertexindex can point to any point on this line.
					// Find the left and right topmost vertices which have the current y coordinate:
					var topleftvertexindex = topvertexindex;
					while(true) {
						var i = topleftvertexindex + 1;
						if(i >= numvertices) i = 0;
						if(vertices2d[i].y != ycoordinate) break;
						if(i == topvertexindex) break; // should not happen, but just to prevent endless loops
						topleftvertexindex = i;
					}
					var toprightvertexindex = topvertexindex;
					while(true) {
						var i = toprightvertexindex - 1;
						if(i < 0) i = numvertices - 1;
						if(vertices2d[i].y != ycoordinate) break;
						if(i == topleftvertexindex) break; // should not happen, but just to prevent endless loops
						toprightvertexindex = i;
					}
					var nextleftvertexindex = topleftvertexindex + 1;
					if(nextleftvertexindex >= numvertices) nextleftvertexindex = 0;
					var nextrightvertexindex = toprightvertexindex - 1;
					if(nextrightvertexindex < 0) nextrightvertexindex = numvertices - 1;
					var newactivepolygon = {
						polygonindex: polygonindex,
						leftvertexindex: topleftvertexindex,
						rightvertexindex: toprightvertexindex,
						topleft: vertices2d[topleftvertexindex],
						topright: vertices2d[toprightvertexindex],
						bottomleft: vertices2d[nextleftvertexindex],
						bottomright: vertices2d[nextrightvertexindex],
					};
					insertSorted(activepolygons, newactivepolygon, function(el1, el2) {
						var x1 = CSG.interpolateBetween2DPointsForY(
									el1.topleft, el1.bottomleft, middleycoordinate);
						var x2 = CSG.interpolateBetween2DPointsForY(
									el2.topleft, el2.bottomleft, middleycoordinate);
						if(x1 > x2) return 1;
						if(x1 < x2) return -1;
						return 0;
					});
				} // for(var polygonindex in startingpolygonindexes)
			} //  yindex < ycoordinates.length-1
			//if( (yindex == ycoordinates.length-1) || (nextycoordinate - ycoordinate > EPS) )
			if(true) {
				// Now activepolygons is up to date
				// Build the output polygons for the next row in newoutpolygonrow:
				for(var activepolygon_key in activepolygons) {
					var activepolygon = activepolygons[activepolygon_key];
					var polygonindex = activepolygon.polygonindex;
					var vertices2d = polygonvertices2d[polygonindex];
					var numvertices = vertices2d.length;

					var x = CSG.interpolateBetween2DPointsForY(activepolygon.topleft, activepolygon.bottomleft, ycoordinate);
					var topleft = new CSG.Vector2D(x, ycoordinate);
					x = CSG.interpolateBetween2DPointsForY(activepolygon.topright, activepolygon.bottomright, ycoordinate);
					var topright = new CSG.Vector2D(x, ycoordinate);
					x = CSG.interpolateBetween2DPointsForY(activepolygon.topleft, activepolygon.bottomleft, nextycoordinate);
					var bottomleft = new CSG.Vector2D(x, nextycoordinate);
					x = CSG.interpolateBetween2DPointsForY(activepolygon.topright, activepolygon.bottomright, nextycoordinate);
					var bottomright = new CSG.Vector2D(x, nextycoordinate);
					var outpolygon = {
						topleft: topleft,
						topright: topright,
						bottomleft: bottomleft,
						bottomright: bottomright,
						leftline: CSG.Line2D.fromPoints(topleft, bottomleft),
						rightline: CSG.Line2D.fromPoints(bottomright, topright)
					};
					if(newoutpolygonrow.length > 0) {
						var prevoutpolygon = newoutpolygonrow[newoutpolygonrow.length - 1];
						var d1 = outpolygon.topleft.distanceTo(prevoutpolygon.topright);
						var d2 = outpolygon.bottomleft.distanceTo(prevoutpolygon.bottomright);
						if((d1 < EPS) && (d2 < EPS)) {
							// we can join this polygon with the one to the left:
							outpolygon.topleft = prevoutpolygon.topleft;
							outpolygon.leftline = prevoutpolygon.leftline;
							outpolygon.bottomleft = prevoutpolygon.bottomleft;
							newoutpolygonrow.splice(newoutpolygonrow.length - 1, 1);
						}
					}
					newoutpolygonrow.push(outpolygon);
				} // for(activepolygon in activepolygons)
				if(yindex > 0) {
					// try to match the new polygons against the previous row:
					var prevcontinuedindexes = {};
					var matchedindexes = {};
					for(var i = 0; i < newoutpolygonrow.length; i++) {
						var thispolygon = newoutpolygonrow[i];
						for(var ii = 0; ii < prevoutpolygonrow.length; ii++) {
							if(!matchedindexes[ii]) // not already processed?
							{
								// We have a match if the sidelines are equal or if the top coordinates
								// are on the sidelines of the previous polygon
								var prevpolygon = prevoutpolygonrow[ii];
								if(prevpolygon.bottomleft.distanceTo(thispolygon.topleft) < EPS) {
									if(prevpolygon.bottomright.distanceTo(thispolygon.topright) < EPS) {
										// Yes, the top of this polygon matches the bottom of the previous:
										matchedindexes[ii] = true;
										// Now check if the joined polygon would remain convex:
										var d1 = thispolygon.leftline.direction().x - prevpolygon.leftline.direction().x;
										var d2 = thispolygon.rightline.direction().x - prevpolygon.rightline.direction().x;
										var leftlinecontinues = Math.abs(d1) < EPS;
										var rightlinecontinues = Math.abs(d2) < EPS;
										var leftlineisconvex = leftlinecontinues || (d1 >= 0);
										var rightlineisconvex = rightlinecontinues || (d2 >= 0);
										if(leftlineisconvex && rightlineisconvex) {
											// yes, both sides have convex corners:
											// This polygon will continue the previous polygon
											thispolygon.outpolygon = prevpolygon.outpolygon;
											thispolygon.leftlinecontinues = leftlinecontinues;
											thispolygon.rightlinecontinues = rightlinecontinues;
											prevcontinuedindexes[ii] = true;
										}
										break;
									}
								}
							} // if(!prevcontinuedindexes[ii])
						} // for ii
					} // for i
					for(var ii = 0; ii < prevoutpolygonrow.length; ii++) {
						if(!prevcontinuedindexes[ii]) {
							// polygon ends here
							// Finish the polygon with the last point(s):
							var prevpolygon = prevoutpolygonrow[ii];
							prevpolygon.outpolygon.rightpoints.push(prevpolygon.bottomright);
							if(prevpolygon.bottomright.distanceTo(prevpolygon.bottomleft) > EPS) {
								// polygon ends with a horizontal line:
								prevpolygon.outpolygon.leftpoints.push(prevpolygon.bottomleft);
							}
							// reverse the left half so we get a counterclockwise circle:
							prevpolygon.outpolygon.leftpoints.reverse();
							var points2d = prevpolygon.outpolygon.rightpoints.concat(prevpolygon.outpolygon.leftpoints);
							var vertices3d = [];
							points2d.map(function(point2d) {
								var point3d = orthobasis.to3D(point2d);
								var vertex3d = new CSG.Vertex(point3d);
								vertices3d.push(vertex3d);
							});
							var polygon = new CSG.Polygon(vertices3d, shared, plane);
							destpolygons.push(polygon);
						}
					}
				} // if(yindex > 0)
				for(var i = 0; i < newoutpolygonrow.length; i++) {
					var thispolygon = newoutpolygonrow[i];
					if(!thispolygon.outpolygon) {
						// polygon starts here:
						thispolygon.outpolygon = {
							leftpoints: [],
							rightpoints: []
						};
						thispolygon.outpolygon.leftpoints.push(thispolygon.topleft);
						if(thispolygon.topleft.distanceTo(thispolygon.topright) > EPS) {
							// we have a horizontal line at the top:
							thispolygon.outpolygon.rightpoints.push(thispolygon.topright);
						}
					} else {
						// continuation of a previous row
						if(!thispolygon.leftlinecontinues) {
							thispolygon.outpolygon.leftpoints.push(thispolygon.topleft);
						}
						if(!thispolygon.rightlinecontinues) {
							thispolygon.outpolygon.rightpoints.push(thispolygon.topright);
						}
					}
				}
				prevoutpolygonrow = newoutpolygonrow;
			}
		} // for yindex
	} // if(numpolygons > 0)
};

////////////////////////////////
// ## class fuzzyFactory
// This class acts as a factory for objects. We can search for an object with approximately
// the desired properties (say a rectangle with width 2 and height 1)
// The lookupOrCreate() method looks for an existing object (for example it may find an existing rectangle
// with width 2.0001 and height 0.999. If no object is found, the user supplied callback is
// called, which should generate a new object. The new object is inserted into the database
// so it can be found by future lookupOrCreate() calls.
// Constructor:
//   numdimensions: the number of parameters for each object
//     for example for a 2D rectangle this would be 2
//   tolerance: The maximum difference for each parameter allowed to be considered a match
CSG.fuzzyFactory = function(numdimensions, tolerance) {
	var lookuptable = [];
	for(var i = 0; i < numdimensions; i++) {
		lookuptable.push({});
	}
	this.lookuptable = lookuptable;
	this.nextElementId = 1;
	this.multiplier = 1.0 / tolerance;
	this.objectTable = {};
};

CSG.fuzzyFactory.prototype = {
	// var obj = f.lookupOrCreate([el1, el2, el3], function(elements) {/* create the new object */});
	// Performs a fuzzy lookup of the object with the specified elements.
	// If found, returns the existing object
	// If not found, calls the supplied callback function which should create a new object with
	// the specified properties. This object is inserted in the lookup database.
	lookupOrCreate: function(els, creatorCallback) {
		var object;
		var key = this.lookupKey(els);
		if(key === null) {
			object = creatorCallback(els);
			key = this.nextElementId++;
			this.objectTable[key] = object;
			for(var dimension = 0; dimension < els.length; dimension++) {
				var elementLookupTable = this.lookuptable[dimension];
				var value = els[dimension];
				var valueMultiplied = value * this.multiplier;
				var valueQuantized1 = Math.floor(valueMultiplied);
				var valueQuantized2 = Math.ceil(valueMultiplied);
				CSG.fuzzyFactory.insertKey(key, elementLookupTable, valueQuantized1);
				CSG.fuzzyFactory.insertKey(key, elementLookupTable, valueQuantized2);
			}
		} else {
			object = this.objectTable[key];
		}
		return object;
	},

	// ----------- PRIVATE METHODS:
	lookupKey: function(els) {
		var keyset = {};
		for(var dimension = 0; dimension < els.length; dimension++) {
			var elementLookupTable = this.lookuptable[dimension];
			var value = els[dimension];
			var valueQuantized = Math.round(value * this.multiplier);
			valueQuantized += "";
			if(valueQuantized in elementLookupTable) {
				if(dimension === 0) {
					keyset = elementLookupTable[valueQuantized];
				} else {
					keyset = CSG.fuzzyFactory.intersectSets(keyset, elementLookupTable[valueQuantized]);
				}
			} else {
				return null;
			}
			if(CSG.fuzzyFactory.isEmptySet(keyset)) return null;
		}
		// return first matching key:
		for(var key in keyset) return key;
		return null;
	},

	lookupKeySetForDimension: function(dimension, value) {
		var result;
		var elementLookupTable = this.lookuptable[dimension];
		var valueMultiplied = value * this.multiplier;
		var valueQuantized = Math.floor(value * this.multiplier);
		if(valueQuantized in elementLookupTable) {
			result = elementLookupTable[valueQuantized];
		} else {
			result = {};
		}
		return result;
	}
};

CSG.fuzzyFactory.insertKey = function(key, lookuptable, quantizedvalue) {
	if(quantizedvalue in lookuptable) {
		lookuptable[quantizedvalue][key] = true;
	} else {
		var newset = {};
		newset[key] = true;
		lookuptable[quantizedvalue] = newset;
	}
};

CSG.fuzzyFactory.isEmptySet = function(obj) {
	for(var key in obj) return false;
	return true;
};

CSG.fuzzyFactory.intersectSets = function(set1, set2) {
	var result = {};
	for(var key in set1) {
		if(key in set2) {
			result[key] = true;
		}
	}
	return result;
};

CSG.fuzzyFactory.joinSets = function(set1, set2) {
	var result = {}, key;
	for(key in set1) {
		result[key] = true;
	}
	for(key in set2) {
		result[key] = true;
	}
	return result;
};

//////////////////////////////////////
CSG.fuzzyCSGFactory = function() {
	this.vertexfactory = new CSG.fuzzyFactory(3, 1e-5);
	this.planefactory = new CSG.fuzzyFactory(4, 1e-5);
	this.polygonsharedfactory = {};
};

CSG.fuzzyCSGFactory.prototype = {
	getPolygonShared: function(sourceshared) {
		var hash = sourceshared.getHash();
		if(hash in this.polygonsharedfactory) {
			return this.polygonsharedfactory[hash];
		} else {
			this.polygonsharedfactory[hash] = sourceshared;
			return sourceshared;
		}
	},

	getVertex: function(sourcevertex) {
		var elements = [sourcevertex.pos._x, sourcevertex.pos._y, sourcevertex.pos._z];
		var result = this.vertexfactory.lookupOrCreate(elements, function(els) {
			return sourcevertex;
		});
		return result;
	},

	getPlane: function(sourceplane) {
		var elements = [sourceplane.normal._x, sourceplane.normal._y, sourceplane.normal._z, sourceplane.w];
		var result = this.planefactory.lookupOrCreate(elements, function(els) {
			return sourceplane;
		});
		return result;
	},

	getPolygon: function(sourcepolygon) {
		var newplane = this.getPlane(sourcepolygon.plane);
		var newshared = this.getPolygonShared(sourcepolygon.shared);
		var _this = this;
		var newvertices = sourcepolygon.vertices.map(function(vertex) {
			return _this.getVertex(vertex);
		});
		return new CSG.Polygon(newvertices, newshared, newplane);
	},

	getCSG: function(sourcecsg) {
		var _this = this;
		var newpolygons = sourcecsg.polygons.map(function(polygon) {
			return _this.getPolygon(polygon);
		});
		return CSG.fromPolygons(newpolygons);
	}
};

//////////////////////////////////////
// Tag factory: we can request a unique tag through CSG.getTag()
CSG.staticTag = 1;

CSG.getTag = function() {
	return CSG.staticTag++;
};

//////////////////////////////////////
// # Class Properties
// This class is used to store properties of a solid
// A property can for example be a CSG.Vertex, a CSG.Plane or a CSG.Line3D
// Whenever an affine transform is applied to the CSG solid, all its properties are
// transformed as well.
// The properties can be stored in a complex nested structure (using arrays and objects)
CSG.Properties = function() {};

CSG.Properties.prototype = {
	_transform: function(matrix4x4) {
		var result = new CSG.Properties();
		CSG.Properties.transformObj(this, result, matrix4x4);
		return result;
	},
	_merge: function(otherproperties) {
		var result = new CSG.Properties();
		CSG.Properties.cloneObj(this, result);
		CSG.Properties.addFrom(result, otherproperties);
		return result;
	}
};

CSG.Properties.transformObj = function(source, result, matrix4x4) {
	for(var propertyname in source) {
		if(propertyname == "_transform") continue;
		if(propertyname == "_merge") continue;
		var propertyvalue = source[propertyname];
		var transformed = propertyvalue;
		if(typeof(propertyvalue) == "object") {
			if(('transform' in propertyvalue) && (typeof(propertyvalue.transform) == "function")) {
				transformed = propertyvalue.transform(matrix4x4);
			} else if(propertyvalue instanceof Array) {
				transformed = [];
				CSG.Properties.transformObj(propertyvalue, transformed, matrix4x4);
			} else if(propertyvalue instanceof CSG.Properties) {
				transformed = new CSG.Properties();
				CSG.Properties.transformObj(propertyvalue, transformed, matrix4x4);
			}
		}
		result[propertyname] = transformed;
	}
};

CSG.Properties.cloneObj = function(source, result) {
	for(var propertyname in source) {
		if(propertyname == "_transform") continue;
		if(propertyname == "_merge") continue;
		var propertyvalue = source[propertyname];
		var cloned = propertyvalue;
		if(typeof(propertyvalue) == "object") {
			if(propertyvalue instanceof Array) {
				cloned = [];
				for(var i = 0; i < propertyvalue.length; i++) {
					cloned.push(propertyvalue[i]);
				}
			} else if(propertyvalue instanceof CSG.Properties) {
				cloned = new CSG.Properties();
				CSG.Properties.cloneObj(propertyvalue, cloned);
			}
		}
		result[propertyname] = cloned;
	}
};

CSG.Properties.addFrom = function(result, otherproperties) {
	for(var propertyname in otherproperties) {
		if(propertyname == "_transform") continue;
		if(propertyname == "_merge") continue;
		if((propertyname in result) &&
				(typeof(result[propertyname]) == "object") &&
				(result[propertyname] instanceof CSG.Properties) &&
				(typeof(otherproperties[propertyname]) == "object") &&
				(otherproperties[propertyname] instanceof CSG.Properties)) {
			CSG.Properties.addFrom(result[propertyname], otherproperties[propertyname]);
		} else if(!(propertyname in result)) {
			result[propertyname] = otherproperties[propertyname];
		}
	}
};

//////////////////////////////////////
// # class Connector
// A connector allows to attach two objects at predefined positions
// For example a servo motor and a servo horn:
// Both can have a Connector called 'shaft'
// The horn can be moved and rotated such that the two connectors match
// and the horn is attached to the servo motor at the proper position.
// Connectors are stored in the properties of a CSG solid so they are
// ge the same transformations applied as the solid
CSG.Connector = function(point, axisvector, normalvector) {
	this.point = new CSG.Vector3D(point);
	this.axisvector = new CSG.Vector3D(axisvector).unit();
	this.normalvector = new CSG.Vector3D(normalvector).unit();
};

CSG.Connector.prototype = {
	normalized: function() {
		var axisvector = this.axisvector.unit();
		// make the normal vector truly normal:
		var n = this.normalvector.cross(axisvector).unit();
		var normalvector = axisvector.cross(n);
		return new CSG.Connector(this.point, axisvector, normalvector);
	},

	transform: function(matrix4x4) {
		var point = this.point.multiply4x4(matrix4x4);
		var axisvector = this.point.plus(this.axisvector).multiply4x4(matrix4x4).minus(point);
		var normalvector = this.point.plus(this.normalvector).multiply4x4(matrix4x4).minus(point);
		return new CSG.Connector(point, axisvector, normalvector);
	},

	// Get the transformation matrix to connect this Connector to another connector
	//   other: a CSG.Connector to which this connector should be connected
	//   mirror: false: the 'axis' vectors of the connectors should point in the same direction
	//           true: the 'axis' vectors of the connectors should point in opposite direction
	//   normalrotation: degrees of rotation between the 'normal' vectors of the two
	//                   connectors
	getTransformationTo: function(other, mirror, normalrotation) {
		mirror = mirror ? true : false;
		normalrotation = normalrotation ? Number(normalrotation) : 0;
		var us = this.normalized();
		other = other.normalized();
		// shift to the origin:
		var transformation = CSG.Matrix4x4.translation(this.point.negated());
		// construct the plane crossing through the origin and the two axes:
		var axesplane = CSG.Plane.anyPlaneFromVector3Ds(
		new CSG.Vector3D(0, 0, 0), us.axisvector, other.axisvector);
		var axesbasis = new CSG.OrthoNormalBasis(axesplane);
		var angle1 = axesbasis.to2D(us.axisvector).angle();
		var angle2 = axesbasis.to2D(other.axisvector).angle();
		var rotation = 180.0 * (angle2 - angle1) / Math.PI;
		if(mirror) rotation += 180.0;
		transformation = transformation.multiply(axesbasis.getProjectionMatrix());
		transformation = transformation.multiply(CSG.Matrix4x4.rotationZ(rotation));
		transformation = transformation.multiply(axesbasis.getInverseProjectionMatrix());
		var usAxesAligned = us.transform(transformation);
		// Now we have done the transformation for aligning the axes.
		// We still need to align the normals:
		var normalsplane = CSG.Plane.fromNormalAndPoint(other.axisvector, new CSG.Vector3D(0, 0, 0));
		var normalsbasis = new CSG.OrthoNormalBasis(normalsplane);
		angle1 = normalsbasis.to2D(usAxesAligned.normalvector).angle();
		angle2 = normalsbasis.to2D(other.normalvector).angle();
		rotation = 180.0 * (angle2 - angle1) / Math.PI;
		rotation += normalrotation;
		transformation = transformation.multiply(normalsbasis.getProjectionMatrix());
		transformation = transformation.multiply(CSG.Matrix4x4.rotationZ(rotation));
		transformation = transformation.multiply(normalsbasis.getInverseProjectionMatrix());
		// and translate to the destination point:
		transformation = transformation.multiply(CSG.Matrix4x4.translation(other.point));
		var usAligned = us.transform(transformation);
		return transformation;
	},

	axisLine: function() {
		return new CSG.Line3D(this.point, this.axisvector);
	},

	// creates a new Connector, with the connection point moved in the direction of the axisvector
	extend: function(distance) {
		var newpoint = this.point.plus(this.axisvector.unit().times(distance));
		return new CSG.Connector(newpoint, this.axisvector, this.normalvector);
	}
};


//////////////////////////////////////
// # Class Path2D
CSG.Path2D = function(points, closed) {
	closed = !! closed;
	points = points || [];
	// re-parse the points into CSG.Vector2D
	// and remove any duplicate points
	var prevpoint = null;
	if(closed && (points.length > 0)) {
		prevpoint = new CSG.Vector2D(points[points.length - 1]);
	}
	var newpoints = [];
	points.map(function(point) {
		point = new CSG.Vector2D(point);
		var skip = false;
		if(prevpoint !== null) {
			var distance = point.distanceTo(prevpoint);
			skip = distance < 1e-5;
		}
		if(!skip) newpoints.push(point);
		prevpoint = point;
	});
	this.points = newpoints;
	this.closed = closed;
};

/*
Construct a (part of a) circle. Parameters:
  options.center: the center point of the arc (CSG.Vector2D or array [x,y])
  options.radius: the circle radius (float)
  options.startangle: the starting angle of the arc, in degrees
	0 degrees corresponds to [1,0]
	90 degrees to [0,1]
	and so on
  options.endangle: the ending angle of the arc, in degrees
  options.resolution: number of points per 360 degree of rotation
  options.maketangent: adds two extra tiny line segments at both ends of the circle
	this ensures that the gradients at the edges are tangent to the circle
Returns a CSG.Path2D. The path is not closed (even if it is a 360 degree arc).
close() the resultin path if you want to create a true circle.
*/
CSG.Path2D.arc = function(options) {
	var center = CSG.parseOptionAs2DVector(options, "center", 0);
	var radius = CSG.parseOptionAsFloat(options, "radius", 1);
	var startangle = CSG.parseOptionAsFloat(options, "startangle", 0);
	var endangle = CSG.parseOptionAsFloat(options, "endangle", 360);
	var resolution = CSG.parseOptionAsInt(options, "resolution", CSG.defaultResolution2D);
	var maketangent = CSG.parseOptionAsBool(options, "maketangent", false);
	// no need to make multiple turns:
	while(endangle - startangle >= 720) {
		endangle -= 360;
	}
	while(endangle - startangle <= -720) {
		endangle += 360;
	}
	var points = [], point;
	var absangledif = Math.abs(endangle - startangle);
	if(absangledif < 1e-5) {
		point = CSG.Vector2D.fromAngle(startangle / 180.0 * Math.PI).times(radius);
		points.push(point.plus(center));
	} else {
		var numsteps = Math.floor(resolution * absangledif / 360) + 1;
		var edgestepsize = numsteps * 0.5 / absangledif; // step size for half a degree
		if(edgestepsize > 0.25) edgestepsize = 0.25;
		var numsteps_mod = maketangent ? (numsteps + 2) : numsteps;
		for(var i = 0; i <= numsteps_mod; i++) {
			var step = i;
			if(maketangent) {
				step = (i - 1) * (numsteps - 2 * edgestepsize) / numsteps + edgestepsize;
				if(step < 0) step = 0;
				if(step > numsteps) step = numsteps;
			}
			var angle = startangle + step * (endangle - startangle) / numsteps;
			point = CSG.Vector2D.fromAngle(angle / 180.0 * Math.PI).times(radius);
			points.push(point.plus(center));
		}
	}
	return new CSG.Path2D(points, false);
};

CSG.Path2D.prototype = {
	concat: function(otherpath) {
		if(this.closed || otherpath.closed) {
			throw new Error("Paths must not be closed");
		}
		var newpoints = this.points.concat(otherpath.points);
		return new CSG.Path2D(newpoints);
	},

	appendPoint: function(point) {
		if(this.closed) {
			throw new Error("Paths must not be closed");
		}
		var newpoints = this.points.concat([point]);
		return new CSG.Path2D(newpoints);
	},

	close: function() {
		return new CSG.Path2D(this.points, true);
	},

	// Extrude the path by following it with a rectangle (upright, perpendicular to the path direction)
	// Returns a CSG solid
	//   width: width of the extrusion, in the z=0 plane
	//   height: height of the extrusion in the z direction
	//   resolution: number of segments per 360 degrees for the curve in a corner
	rectangularExtrude: function(width, height, resolution) {
		var cag = this.expandToCAG(width / 2, resolution);
		var result = cag.extrude({
			offset: [0, 0, height]
		});
		return result;
	},

	// Expand the path to a CAG
	// This traces the path with a circle with radius pathradius
	expandToCAG: function(pathradius, resolution) {
		var sides = [];
		var numpoints = this.points.length;
		var startindex = 0;
		if(this.closed && (numpoints > 2)) startindex = -1;
		var prevvertex;
		for(var i = startindex; i < numpoints; i++) {
			var pointindex = i;
			if(pointindex < 0) pointindex = numpoints - 1;
			var point = this.points[pointindex];
			var vertex = new CAG.Vertex(point);
			if(i > startindex) {
				var side = new CAG.Side(prevvertex, vertex);
				sides.push(side);
			}
			prevvertex = vertex;
		}
		var shellcag = CAG.fromSides(sides);
		var expanded = shellcag.expandedShell(pathradius, resolution);
		return expanded;
	},

	innerToCAG: function() {
		if(!this.closed) throw new Error("The path should be closed!");
		return CAG.fromPoints(this.points);
	},

	transform: function(matrix4x4) {
		var newpoints = this.points.map(function(point) {
			return point.multiply4x4(matrix4x4);
		});
		return new CSG.Path2D(newpoints, this.closed);
	}
};

// Add several convenience methods to the classes that support a transform() method:
CSG.addTransformationMethodsToPrototype = function(prot) {
	prot.mirrored = function(plane) {
		return this.transform(CSG.Matrix4x4.mirroring(plane));
	};

	prot.mirroredX = function() {
		var plane = new CSG.Plane(new CSG.Vector3D(1, 0, 0), 0);
		return this.mirrored(plane);
	};

	prot.mirroredY = function() {
		var plane = new CSG.Plane(new CSG.Vector3D(0, 1, 0), 0);
		return this.mirrored(plane);
	};

	prot.mirroredZ = function() {
		var plane = new CSG.Plane(new CSG.Vector3D(0, 0, 1), 0);
		return this.mirrored(plane);
	};

	prot.translate = function(v) {
		return this.transform(CSG.Matrix4x4.translation(v));
	};

	prot.scale = function(f) {
		return this.transform(CSG.Matrix4x4.scaling(f));
	};

	prot.rotateX = function(deg) {
		return this.transform(CSG.Matrix4x4.rotationX(deg));
	};

	prot.rotateY = function(deg) {
		return this.transform(CSG.Matrix4x4.rotationY(deg));
	};

	prot.rotateZ = function(deg) {
		return this.transform(CSG.Matrix4x4.rotationZ(deg));
	};

	prot.rotate = function(rotationCenter, rotationAxis, degrees) {
		return this.transform(CSG.Matrix4x4.rotation(rotationCenter, rotationAxis, degrees));
	};
};

//////////////////
// CAG: solid area geometry: like CSG but 2D
// Each area consists of a number of sides
// Each side is a line between 2 points
var CAG = function() {
	this.sides = [];
};

// Construct a CAG from a list of `CAG.Side` instances.
CAG.fromSides = function(sides) {
	var cag = new CAG();
	cag.sides = sides;
	return cag;
};

// Construct a CAG from a list of points (a polygon)
// Rotation direction of the points is not relevant. Points can be a convex or concave polygon.
// Polygon must not self intersect
CAG.fromPoints = function(points) {
	var numpoints = points.length;
	if(numpoints < 3) throw new Error("CAG shape needs at least 3 points");
	var sides = [];
	var prevpoint = new CSG.Vector2D(points[numpoints - 1]);
	var prevvertex = new CAG.Vertex(prevpoint);
	points.map(function(p) {
		var point = new CSG.Vector2D(p);
		var vertex = new CAG.Vertex(point);
		var side = new CAG.Side(prevvertex, vertex);
		sides.push(side);
		prevvertex = vertex;
	});
	var result = CAG.fromSides(sides);
	if(result.isSelfIntersecting()) {
		throw new Error("Polygon is self intersecting!");
	}
	var area = result.area();
	if(Math.abs(area) < 1e-5) {
		throw new Error("Degenerate polygon!");
	}
	if(area < 0) {
		result = result.flipped();
	}
	result = result.canonicalized();
	return result;
};

// Like CAG.fromPoints but does not check if it's a valid polygon.
// Points should rotate counter clockwise
CAG.fromPointsNoCheck = function(points) {
	var sides = [];
	var prevpoint = new CSG.Vector2D(points[points.length - 1]);
	var prevvertex = new CAG.Vertex(prevpoint);
	points.map(function(p) {
		var point = new CSG.Vector2D(p);
		var vertex = new CAG.Vertex(point);
		var side = new CAG.Side(prevvertex, vertex);
		sides.push(side);
		prevvertex = vertex;
	});
	return CAG.fromSides(sides);
};

// Converts a CSG to a CAG. The CSG must consist of polygons with only z coordinates +1 and -1
// as constructed by CAG.toCSG(-1, 1). This is so we can use the 3D union(), intersect() etc
CAG.fromFakeCSG = function(csg) {
	var sides = csg.polygons.map(function(p) {
		return CAG.Side.fromFakePolygon(p);
	});
	return CAG.fromSides(sides);
};

// see if the line between p0start and p0end intersects with the line between p1start and p1end
// returns true if the lines strictly intersect, the end points are not counted!
CAG.linesIntersect = function(p0start, p0end, p1start, p1end) {
	if(p0end.equals(p1start) || p1end.equals(p0start)) {
		var d = p1end.minus(p1start).unit().plus(p0end.minus(p0start).unit()).length();
		if(d < 1e-5) {
			return true;
		}
	} else {
		var d0 = p0end.minus(p0start);
		var d1 = p1end.minus(p1start);
		if(Math.abs(d0.cross(d1)) < 1e-9) return false; // lines are parallel
		var alphas = CSG.solve2Linear(-d0.x, d1.x, -d0.y, d1.y, p0start.x - p1start.x, p0start.y - p1start.y);
		if((alphas[0] > 1e-6) && (alphas[0] < 0.999999) && (alphas[1] > 1e-5) && (alphas[1] < 0.999999)) return true;
		//    if( (alphas[0] >= 0) && (alphas[0] <= 1) && (alphas[1] >= 0) && (alphas[1] <= 1) ) return true;
	}
	return false;
};

/* Construct a circle
   options:
	 center: a 2D center point
	 radius: a scalar
	 resolution: number of sides per 360 degree rotation
   returns a CAG object
*/
CAG.circle = function(options) {
	options = options || {};
	var center = CSG.parseOptionAs2DVector(options, "center", [0, 0]);
	var radius = CSG.parseOptionAsFloat(options, "radius", 1);
	var resolution = CSG.parseOptionAsInt(options, "resolution", CSG.defaultResolution2D);
	var sides = [];
	var prevvertex;
	for(var i = 0; i <= resolution; i++) {
		var radians = 2 * Math.PI * i / resolution;
		var point = CSG.Vector2D.fromAngleRadians(radians).times(radius).plus(center);
		var vertex = new CAG.Vertex(point);
		if(i > 0) {
			sides.push(new CAG.Side(prevvertex, vertex));
		}
		prevvertex = vertex;
	}
	return CAG.fromSides(sides);
};

/* Construct a rectangle
   options:
	 center: a 2D center point
	 radius: a 2D vector with width and height
   returns a CAG object
*/
CAG.rectangle = function(options) {
	options = options || {};
	var c = CSG.parseOptionAs2DVector(options, "center", [0, 0]);
	var r = CSG.parseOptionAs2DVector(options, "radius", [1, 1]);
	var rswap = new CSG.Vector2D(r.x, -r.y);
	var points = [
	c.plus(r), c.plus(rswap), c.minus(r), c.minus(rswap)];
	return CAG.fromPoints(points);
};

//     var r = CSG.roundedRectangle({
//       center: [0, 0],
//       radius: [2, 1],
//       roundradius: 0.2,
//       resolution: 8,
//     });
CAG.roundedRectangle = function(options) {
	options = options || {};
	var center = CSG.parseOptionAs2DVector(options, "center", [0, 0]);
	var radius = CSG.parseOptionAs2DVector(options, "radius", [1, 1]);
	var roundradius = CSG.parseOptionAsFloat(options, "roundradius", 0.2);
	var resolution = CSG.parseOptionAsInt(options, "resolution", CSG.defaultResolution2D);
	var maxroundradius = Math.min(radius.x, radius.y);
	maxroundradius -= 0.1;
	roundradius = Math.min(roundradius, maxroundradius);
	roundradius = Math.max(0, roundradius);
	radius = new CSG.Vector2D(radius.x - roundradius, radius.y - roundradius);
	var rect = CAG.rectangle({
		center: center,
		radius: radius
	});
	if(roundradius > 0) {
		rect = rect.expand(roundradius, resolution);
	}
	return rect;
};

// Reconstruct a CAG from the output of toCompactBinary()
CAG.fromCompactBinary = function(bin) {
	if(bin['class'] != "CAG") throw new Error("Not a CAG");
	var vertices = [];
	var vertexData = bin.vertexData;
	var numvertices = vertexData.length / 2;
	var arrayindex = 0;
	for(var vertexindex = 0; vertexindex < numvertices; vertexindex++) {
		var x = vertexData[arrayindex++];
		var y = vertexData[arrayindex++];
		var pos = new CSG.Vector2D(x, y);
		var vertex = new CAG.Vertex(pos);
		vertices.push(vertex);
	}

	var sides = [];
	var numsides = bin.sideVertexIndices.length / 2;
	arrayindex = 0;
	for(var sideindex = 0; sideindex < numsides; sideindex++) {
		var vertexindex0 = bin.sideVertexIndices[arrayindex++];
		var vertexindex1 = bin.sideVertexIndices[arrayindex++];
		var side = new CAG.Side(vertices[vertexindex0], vertices[vertexindex1]);
		sides.push(side);
	}
	var cag = CAG.fromSides(sides);
	cag.isCanonicalized = true;
	return cag;
};

function fnSortByIndex(a, b) {
	return a.index - b.index;
}

CAG.prototype = {
	toString: function() {
		var result = "CAG (" + this.sides.length + " sides):\n";
		this.sides.map(function(side) {
			result += "  " + side.toString() + "\n";
		});
		return result;
	},

	toCSG: function(z0, z1) {
		var polygons = this.sides.map(function(side) {
			return side.toPolygon3D(z0, z1);
		});
		return CSG.fromPolygons(polygons);
	},

	toDebugString1: function() {
		this.sides.sort(function(a, b) {
			return a.vertex0.pos.x - b.vertex0.pos.x;
		});
		var str = "";
		this.sides.map(function(side) {
			str += "(" + side.vertex0.pos.x + "," + side.vertex0.pos.y + ") - (" + side.vertex1.pos.x + "," + side.vertex1.pos.y + ")\n";
		});
		return str;
	},

	toDebugString: function() {
		//    this.sides.sort(function(a,b){
		//      return a.vertex0.pos.x - b.vertex0.pos.x;
		//    });
		var str = "CAG.fromSides([\n";
		this.sides.map(function(side) {
			str += "  new CAG.Side(new CAG.Vertex(new CSG.Vector2D(" +
					side.vertex0.pos.x + "," + side.vertex0.pos.y +
				")), new CAG.Vertex(new CSG.Vector2D(" +
					side.vertex1.pos.x + "," + side.vertex1.pos.y + "))),\n";
		});
		str += "]);\n";
		return str;
	},

	union: function(cag) {
		var cags;
		if(cag instanceof Array) {
			cags = cag;
		} else {
			cags = [cag];
		}
		var r = this.toCSG(-1, 1);
		cags.map(function(cag) {
			r = r.unionSub(cag.toCSG(-1, 1), false, false);
		});
		r = r.reTesselated();
		r = r.canonicalized();
		cag = CAG.fromFakeCSG(r);
		var cag_canonicalized = cag.canonicalized();
		return cag_canonicalized;
	},

	subtract: function(cag) {
		var cags;
		if(cag instanceof Array) {
			cags = cag;
		} else {
			cags = [cag];
		}
		var r = this.toCSG(-1, 1);
		cags.map(function(cag) {
			r = r.subtractSub(cag.toCSG(-1, 1), false, false);
		});
		r = r.reTesselated();
		r = r.canonicalized();
		r = CAG.fromFakeCSG(r);
		r = r.canonicalized();
		return r;
	},

	intersect: function(cag) {
		var cags;
		if(cag instanceof Array) {
			cags = cag;
		} else {
			cags = [cag];
		}
		var r = this.toCSG(-1, 1);
		cags.map(function(cag) {
			r = r.intersectSub(cag.toCSG(-1, 1), false, false);
		});
		r = r.reTesselated();
		r = r.canonicalized();
		r = CAG.fromFakeCSG(r);
		r = r.canonicalized();
		return r;
	},

	transform: function(matrix4x4) {
		var ismirror = matrix4x4.isMirroring();
		var newsides = this.sides.map(function(side) {
			return side.transform(matrix4x4);
		});
		var result = CAG.fromSides(newsides);
		if(ismirror) {
			result = result.flipped();
		}
		return result;
	},

	// see http://local.wasp.uwa.edu.au/~pbourke/geometry/polyarea/ :
	// Area of the polygon. For a counter clockwise rotating polygon the area is positive, otherwise negative
	area: function() {
		var polygonArea = 0;
		this.sides.map(function(side) {
			polygonArea += side.vertex0.pos.cross(side.vertex1.pos);
		});
		polygonArea *= 0.5;
		return polygonArea;
	},

	flipped: function() {
		var newsides = this.sides.map(function(side) {
			return side.flipped();
		});
		newsides.reverse();
		return CAG.fromSides(newsides);
	},

	getBounds: function() {
		var minpoint;
		if(this.sides.length === 0) {
			minpoint = new CSG.Vector2D(0, 0);
		} else {
			minpoint = this.sides[0].vertex0.pos;
		}
		var maxpoint = minpoint;
		this.sides.map(function(side) {
			minpoint = minpoint.min(side.vertex0.pos);
			minpoint = minpoint.min(side.vertex1.pos);
			maxpoint = maxpoint.max(side.vertex0.pos);
			maxpoint = maxpoint.max(side.vertex1.pos);
		});
		return [minpoint, maxpoint];
	},

   center: function(c) {
      if(!c.length) c = [c,c];
      var b = this.getBounds();
      return this.translate([
         c[0]?-(b[1].x-b[0].x)/2-b[0].x:0,
         c[1]?-(b[1].y-b[0].y)/2-b[0].y:0]);
   },

	isSelfIntersecting: function() {
		var numsides = this.sides.length;
		for(var i = 0; i < numsides; i++) {
			var side0 = this.sides[i];
			for(var ii = i + 1; ii < numsides; ii++) {
				var side1 = this.sides[ii];
				if(CAG.linesIntersect(side0.vertex0.pos, side0.vertex1.pos, side1.vertex0.pos, side1.vertex1.pos)) {
					return true;
				}
			}
		}
		return false;
	},

	expandedShell: function(radius, resolution) {
		resolution = resolution || 8;
		if(resolution < 4) resolution = 4;
		var cags = [];
		var pointmap = {};
		var cag = this.canonicalized();
		cag.sides.map(function(side) {
			var d = side.vertex1.pos.minus(side.vertex0.pos);
			var dl = d.length();
			if(dl > 1e-5) {
				d = d.times(1.0 / dl);
				var normal = d.normal().times(radius);
				var shellpoints = [
					side.vertex1.pos.plus(normal),
					side.vertex1.pos.minus(normal),
					side.vertex0.pos.minus(normal),
					side.vertex0.pos.plus(normal)
				];
				//      var newcag = CAG.fromPointsNoCheck(shellpoints);
				var newcag = CAG.fromPoints(shellpoints);
				cags.push(newcag);
				for(var step = 0; step < 2; step++) {
					var p1 = (step === 0) ? side.vertex0.pos : side.vertex1.pos;
					var p2 = (step === 0) ? side.vertex1.pos : side.vertex0.pos;
					var tag = p1.x + " " + p1.y;
					if(!(tag in pointmap)) {
						pointmap[tag] = [];
					}
					pointmap[tag].push({
						"p1": p1,
						"p2": p2
					});
				}
			}
		});
		for(var tag in pointmap) {
			var m = pointmap[tag];
			var angle1, angle2;
			var pcenter = m[0].p1;
			if(m.length == 2) {
				var end1 = m[0].p2;
				var end2 = m[1].p2;
				angle1 = end1.minus(pcenter).angleDegrees();
				angle2 = end2.minus(pcenter).angleDegrees();
				if(angle2 < angle1) angle2 += 360;
				if(angle2 >= (angle1 + 360)) angle2 -= 360;
				if(angle2 < angle1 + 180) {
					var t = angle2;
					angle2 = angle1 + 360;
					angle1 = t;
				}
				angle1 += 90;
				angle2 -= 90;
			} else {
				angle1 = 0;
				angle2 = 360;
			}
			var fullcircle = (angle2 > angle1 + 359.999);
			if(fullcircle) {
				angle1 = 0;
				angle2 = 360;
			}
			if(angle2 > (angle1 + 1e-5)) {
				var points = [];
				if(!fullcircle) {
					points.push(pcenter);
				}
				var numsteps = Math.round(resolution * (angle2 - angle1) / 360);
				if(numsteps < 1) numsteps = 1;
				for(var step = 0; step <= numsteps; step++) {
					var angle = angle1 + step / numsteps * (angle2 - angle1);
					if(step == numsteps) angle = angle2; // prevent rounding errors
					var point = pcenter.plus(CSG.Vector2D.fromAngleDegrees(angle).times(radius));
					if((!fullcircle) || (step > 0)) {
						points.push(point);
					}
				}
				var newcag = CAG.fromPointsNoCheck(points);
				cags.push(newcag);
			}
		}
		var result = new CAG();
		result = result.union(cags);
		return result;
	},

	expand: function(radius, resolution) {
		var result = this.union(this.expandedShell(radius, resolution));
		return result;
	},

	contract: function(radius, resolution) {
		var result = this.subtract(this.expandedShell(radius, resolution));
		return result;
	},

	// extruded=cag.extrude({offset: [0,0,10], twistangle: 360, twiststeps: 100});
	// linear extrusion of 2D shape, with optional twist
	// The 2d shape is placed in in z=0 plane and extruded into direction <offset> (a CSG.Vector3D)
	// The final face is rotated <twistangle> degrees. Rotation is done around the origin of the 2d shape (i.e. x=0, y=0)
	// twiststeps determines the resolution of the twist (should be >= 1)
	// returns a CSG object
	extrude: function(options) {
		if(this.sides.length == 0) {
		// empty!
		return new CSG();
	}
	var offsetvector = CSG.parseOptionAs3DVector(options, "offset", [0,0,1]);
	var twistangle = CSG.parseOptionAsFloat(options, "twistangle", 0);
	var twiststeps = CSG.parseOptionAsInt(options, "twiststeps", 10);

	if(twistangle == 0) twiststeps = 1;
	if(twiststeps < 1) twiststeps = 1;

	var newpolygons = [];
	var prevtransformedcag;
	var prevstepz;
	for(var step=0; step <= twiststeps; step++) {
		var stepfraction = step / twiststeps;
		var transformedcag = this;
		var angle = twistangle * stepfraction;
		if(angle != 0) {
			transformedcag = transformedcag.rotateZ(angle);
		}
		var translatevector = new CSG.Vector2D(offsetvector.x, offsetvector.y).times(stepfraction);
		transformedcag = transformedcag.translate(translatevector);
		var bounds = transformedcag.getBounds();
		bounds[0] = bounds[0].minus(new CSG.Vector2D(1,1));
		bounds[1] = bounds[1].plus(new CSG.Vector2D(1,1));
		var stepz = offsetvector.z * stepfraction;
		if( (step == 0) || (step == twiststeps) ) {
			// bottom or top face:
			var csgshell = transformedcag.toCSG(stepz-1, stepz+1);
			var csgplane = CSG.fromPolygons([new CSG.Polygon([
				new CSG.Vertex(new CSG.Vector3D(bounds[0].x, bounds[0].y, stepz)),
				new CSG.Vertex(new CSG.Vector3D(bounds[1].x, bounds[0].y, stepz)),
				new CSG.Vertex(new CSG.Vector3D(bounds[1].x, bounds[1].y, stepz)),
				new CSG.Vertex(new CSG.Vector3D(bounds[0].x, bounds[1].y, stepz))
			])]);
			var flip = (step == 0);
			if(offsetvector.z < 0) flip = !flip;
			if(flip) {
				csgplane = csgplane.inverse();
			}
			csgplane = csgplane.intersect(csgshell);
			// only keep the polygons in the z plane:
			csgplane.polygons.map(function(polygon){
				if(Math.abs(polygon.plane.normal.z) > 0.99) {
					newpolygons.push(polygon);
				}
			});
		}
		if(step > 0) {
			var numsides = transformedcag.sides.length;
			for(var sideindex = 0; sideindex < numsides; sideindex++) {
				var thisside = transformedcag.sides[sideindex];
				var prevside = prevtransformedcag.sides[sideindex];
				var p1 = new CSG.Polygon([
					new CSG.Vertex(thisside.vertex1.pos.toVector3D(stepz)),
					new CSG.Vertex(thisside.vertex0.pos.toVector3D(stepz)),
					new CSG.Vertex(prevside.vertex0.pos.toVector3D(prevstepz))
				]);
				var p2 = new CSG.Polygon([
					new CSG.Vertex(thisside.vertex1.pos.toVector3D(stepz)),
					new CSG.Vertex(prevside.vertex0.pos.toVector3D(prevstepz)),
					new CSG.Vertex(prevside.vertex1.pos.toVector3D(prevstepz))
				]);
				if(offsetvector.z < 0) {
					p1 = p1.flipped();
					p2 = p2.flipped();
				}
				newpolygons.push(p1);
				newpolygons.push(p2);
			}
		}
		prevtransformedcag = transformedcag;
		prevstepz = stepz;
	} // for step
	return CSG.fromPolygons(newpolygons);
	},

	// check if we are a valid CAG (for debugging)
	check: function() {
		var errors = [];
		if(this.isSelfIntersecting()) {
			errors.push("Self intersects");
		}
		var pointcount = {};
		this.sides.map(function(side) {
			function mappoint(p) {
				var tag = p.x + " " + p.y;
				if(!(tag in pointcount)) pointcount[tag] = 0;
				pointcount[tag]++;
			}
			mappoint(side.vertex0.pos);
			mappoint(side.vertex1.pos);
		});
		for(var tag in pointcount) {
			var count = pointcount[tag];
			if(count & 1) {
				errors.push("Uneven number of sides (" + count + ") for point " + tag);
			}
		}
		var area = this.area();
		if(area < 1e-5) {
			errors.push("Area is " + area);
		}
		if(errors.length > 0) {
			var ertxt = "";
			errors.map(function(err) {
				ertxt += err + "\n";
			});
			throw new Error(ertxt);
		}
	},

	canonicalized: function() {
		if(this.isCanonicalized) {
			return this;
		} else {
			var factory = new CAG.fuzzyCAGFactory();
			var result = factory.getCAG(this);
			result.isCanonicalized = true;
			return result;
		}
	},

	toCompactBinary: function() {
		var cag = this.canonicalized();
		var numsides = cag.sides.length;
		var vertexmap = {};
		var vertices = [];
		var numvertices = 0;
		var sideVertexIndices = new Uint32Array(2 * numsides);
		var sidevertexindicesindex = 0;
		cag.sides.map(function(side) {
			[side.vertex0, side.vertex1].map(function(v) {
				var vertextag = v.getTag();
				var vertexindex;
				if(!(vertextag in vertexmap)) {
					vertexindex = numvertices++;
					vertexmap[vertextag] = vertexindex;
					vertices.push(v);
				} else {
					vertexindex = vertexmap[vertextag];
				}
				sideVertexIndices[sidevertexindicesindex++] = vertexindex;
			});
		});
		var vertexData = new Float64Array(numvertices * 2);
		var verticesArrayIndex = 0;
		vertices.map(function(v) {
			var pos = v.pos;
			vertexData[verticesArrayIndex++] = pos._x;
			vertexData[verticesArrayIndex++] = pos._y;
		});
		var result = {
			'class': "CAG",
			sideVertexIndices: sideVertexIndices,
			vertexData: vertexData
		};
		return result;
	},

	getOutlinePaths: function() {
		var cag = this.canonicalized();
		var sideTagToSideMap = {};
		var startVertexTagToSideTagMap = {};
		cag.sides.map(function(side) {
			var sidetag = side.getTag();
			sideTagToSideMap[sidetag] = side;
			var startvertextag = side.vertex0.getTag();
			if(!(startvertextag in startVertexTagToSideTagMap)) {
				startVertexTagToSideTagMap[startvertextag] = [];
			}
			startVertexTagToSideTagMap[startvertextag].push(sidetag);
		});
		var paths = [];
		while(true) {
			var startsidetag = null;
			for(var aVertexTag in startVertexTagToSideTagMap) {
				var sidesForThisVertex = startVertexTagToSideTagMap[aVertexTag];
				startsidetag = sidesForThisVertex[0];
				sidesForThisVertex.splice(0, 1);
				if(sidesForThisVertex.length === 0) {
					delete startVertexTagToSideTagMap[aVertexTag];
				}
				break;
			}
			if(startsidetag === null) break; // we've had all sides
			var connectedVertexPoints = [];
			var sidetag = startsidetag;
			var thisside = sideTagToSideMap[sidetag];
			var startvertextag = thisside.vertex0.getTag();
			while(true) {
				connectedVertexPoints.push(thisside.vertex0.pos);
				var nextvertextag = thisside.vertex1.getTag();
				if(nextvertextag == startvertextag) break; // we've closed the polygon
				if(!(nextvertextag in startVertexTagToSideTagMap)) {
					throw new Error("Area is not closed!");
				}
				var nextpossiblesidetags = startVertexTagToSideTagMap[nextvertextag];
				var nextsideindex = -1;
				if(nextpossiblesidetags.length == 1) {
					nextsideindex = 0;
				} else {
					// more than one side starting at the same vertex. This means we have
					// two shapes touching at the same corner
					var bestangle = null;
					var thisangle = thisside.direction().angleDegrees();
					for(var sideindex = 0; sideindex < nextpossiblesidetags.length; sideindex++) {
						var nextpossiblesidetag = nextpossiblesidetags[sideindex];
						var possibleside = sideTagToSideMap[nextpossiblesidetag];
						var angle = possibleside.direction().angleDegrees();
						var angledif = angle - thisangle;
						if(angledif < -180) angledif += 360;
						if(angledif >= 180) angledif -= 360;
						if((nextsideindex < 0) || (angledif > bestangle)) {
							nextsideindex = sideindex;
							bestangle = angledif;
						}
					}
				}
				var nextsidetag = nextpossiblesidetags[nextsideindex];
				nextpossiblesidetags.splice(nextsideindex, 1);
				if(nextpossiblesidetags.length === 0) {
					delete startVertexTagToSideTagMap[nextvertextag];
				}
				thisside = sideTagToSideMap[nextsidetag];
			} // inner loop
			var path = new CSG.Path2D(connectedVertexPoints, true);
			paths.push(path);
		} // outer loop
		return paths;
	},

	toDxfString: function() {
		var paths = this.getOutlinePaths();
		return CAG.PathsToDxf(paths);
	},

	toDxf: function() {
		var str = this.toDxfString();
		return new Blob([str], {
			type: "application/dxf"
		});
	}
};

CAG.PathsToDxf = function(paths) {
	var str = "999\nDXF generated by OpenJsCad\n";
	str += "  0\nSECTION\n  2\nHEADER\n";
	str += "  0\nENDSEC\n";
	str += "  0\nSECTION\n  2\nTABLES\n";
	str += "  0\nTABLE\n  2\nLTYPE\n  70\n1\n";
	str += "  0\nLTYPE\n  2\nCONTINUOUS\n  3\nSolid Line\n  72\n65\n  73\n0\n  40\n0.0\n";
	str += "  0\nENDTAB\n";
	str += "  0\nTABLE\n  2\nLAYER\n  70\n1\n";
	str += "  0\nLAYER\n  2\nOpenJsCad\n  62\n7\n  6\ncontinuous\n";
	str += "  0\nENDTAB\n";
	str += "  0\nTABLE\n  2\nSTYLE\n  70\n0\n  0\nENDTAB\n";
	str += "  0\nTABLE\n  2\nVIEW\n  70\n0\n  0\nENDTAB\n";
	str += "  0\nENDSEC\n";
	str += "  0\nSECTION\n  2\nBLOCKS\n";
	str += "  0\nENDSEC\n";
	str += "  0\nSECTION\n  2\nENTITIES\n";
	paths.map(function(path) {
		var numpoints_closed = path.points.length + (path.closed ? 1 : 0);
		str += "  0\nLWPOLYLINE\n  8\nOpenJsCad\n  90\n" + numpoints_closed + "\n  70\n" + (path.closed ? 1 : 0) + "\n";
		for(var pointindex = 0; pointindex < numpoints_closed; pointindex++) {
			var pointindexwrapped = pointindex;
			if(pointindexwrapped >= path.points.length) pointindexwrapped -= path.points.length;
			var point = path.points[pointindexwrapped];
			str += " 10\n" + point.x + "\n 20\n" + point.y + "\n 30\n0.0\n";
		}
	});
	str += "  0\nENDSEC\n  0\nEOF\n";
	return str;
};

CAG.Vertex = function(pos) {
	this.pos = pos;
};

CAG.Vertex.prototype = {
	toString: function() {
		return "("+this.pos.x.toFixed(2)+","+this.pos.y.toFixed(2)+")";
	},
	getTag: function() {
		var result = this.tag;
		if(!result) {
			result = CSG.getTag();
			this.tag = result;
		}
		return result;
	}
};

CAG.Side = function(vertex0, vertex1) {
	if(!(vertex0 instanceof CAG.Vertex)) throw new Error("Assertion failed");
	if(!(vertex1 instanceof CAG.Vertex)) throw new Error("Assertion failed");
	this.vertex0 = vertex0;
	this.vertex1 = vertex1;
};

CAG.Side.fromFakePolygon = function(polygon) {
	if(polygon.vertices.length != 4) {
		throw new Error("Assertion failed - 1");
	}
	var pointsZeroZ = [];
	var indicesZeroZ = [];
	for(var i = 0; i < 4; i++) {
		var pos = polygon.vertices[i].pos;
		if((pos.z >= -1.001) && (pos.z < -0.999)) {
		} else if((pos.z >= 0.999) && (pos.z < 1.001)) {
		} else {
			throw new Error("Assertion failed - 2");
		}
		if(pos.z > 0) {
			pointsZeroZ.push(new CSG.Vector2D(pos.x, pos.y));
			indicesZeroZ.push(i);
		}
	}
	if(pointsZeroZ.length != 2) {
		throw new Error("Assertion failed - 3");
	}
	var d = indicesZeroZ[1] - indicesZeroZ[0];
	var p1, p2;
	if(d == 1) {
		p1 = pointsZeroZ[1];
		p2 = pointsZeroZ[0];
	} else if(d == 3) {
		p1 = pointsZeroZ[0];
		p2 = pointsZeroZ[1];
	} else {
		throw new Error("Assertion failed - 4");
	}
	var result = new CAG.Side(new CAG.Vertex(p1), new CAG.Vertex(p2));
	return result;
};

CAG.Side.prototype = {
	toString: function() {
		return this.vertex0 + " -> "+ this.vertex1;
	},

	toPolygon3D: function(z0, z1) {
		var vertices = [
			new CSG.Vertex(this.vertex0.pos.toVector3D(z0)),
			new CSG.Vertex(this.vertex1.pos.toVector3D(z0)),
			new CSG.Vertex(this.vertex1.pos.toVector3D(z1)),
			new CSG.Vertex(this.vertex0.pos.toVector3D(z1))
		];
		return new CSG.Polygon(vertices);
	},

	transform: function(matrix4x4) {
		var newp1 = this.vertex0.pos.transform(matrix4x4);
		var newp2 = this.vertex1.pos.transform(matrix4x4);
		return new CAG.Side(new CAG.Vertex(newp1), new CAG.Vertex(newp2));
	},

	flipped: function() {
		return new CAG.Side(this.vertex1, this.vertex0);
	},

	direction: function() {
		return this.vertex1.pos.minus(this.vertex0.pos);
	},

	getTag: function() {
		var result = this.tag;
		if(!result) {
			result = CSG.getTag();
			this.tag = result;
		}
		return result;
	},

	lengthSquared: function() {
		var x = this.vertex1.pos.x - this.vertex0.pos.x,
		y = this.vertex1.pos.y - this.vertex0.pos.y;
		return x*x + y*y;
	},

	length: function() {
		return Math.sqrt(this.lengthSquared());
	}
};

//////////////////////////////////////
CAG.fuzzyCAGFactory = function() {
	this.vertexfactory = new CSG.fuzzyFactory(2, 1e-5);
};

CAG.fuzzyCAGFactory.prototype = {
	getVertex: function(sourcevertex) {
		var elements = [sourcevertex.pos._x, sourcevertex.pos._y];
		var result = this.vertexfactory.lookupOrCreate(elements, function(els) {
			return sourcevertex;
		});
		return result;
	},

	getSide: function(sourceside) {
		var vertex0 = this.getVertex(sourceside.vertex0);
		var vertex1 = this.getVertex(sourceside.vertex1);
		return new CAG.Side(vertex0, vertex1);
	},

	getCAG: function(sourcecag) {
		var _this = this;
		var newsides = sourcecag.sides.map(function(side) {
			return _this.getSide(side);
		});
		return CAG.fromSides(newsides);
	}
};

//////////////////////////////////////
CSG.addTransformationMethodsToPrototype(CSG.prototype);
CSG.addTransformationMethodsToPrototype(CSG.Vector2D.prototype);
CSG.addTransformationMethodsToPrototype(CSG.Vector3D.prototype);
CSG.addTransformationMethodsToPrototype(CSG.Vertex.prototype);
CSG.addTransformationMethodsToPrototype(CSG.Plane.prototype);
CSG.addTransformationMethodsToPrototype(CSG.Polygon.prototype);
CSG.addTransformationMethodsToPrototype(CSG.Line3D.prototype);
CSG.addTransformationMethodsToPrototype(CSG.Connector.prototype);
CSG.addTransformationMethodsToPrototype(CSG.Path2D.prototype);
CSG.addTransformationMethodsToPrototype(CSG.Line2D.prototype);
CSG.addTransformationMethodsToPrototype(CAG.prototype);
CSG.addTransformationMethodsToPrototype(CAG.Side.prototype);

/*
  2D polygons are now supported through the CAG class.
  With many improvements (see documentation):
	- shapes do no longer have to be convex
	- union/intersect/subtract is supported
	- expand / contract are supported

  But we'll keep CSG.Polygon2D as a stub for backwards compatibility
*/
CSG.Polygon2D = function(points) {
	var cag = CAG.fromPoints(points);
	this.sides = cag.sides;
};
CSG.Polygon2D.prototype = CAG.prototype;


module.CSG = CSG;
module.CAG = CAG;
})(this); //module to export to