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vnf.scad
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//////////////////////////////////////////////////////////////////////
// LibFile: vnf.scad
// The Vertices'N'Faces structure (VNF) holds the data used by polyhedron() to construct objects: a vertex
// list and a list of faces. This library makes it easier to construct polyhedra by providing
// functions to construct, merge, and modify VNF data, while avoiding common pitfalls such as
// reversed faces. It can find faults in your polyhedrons. Note that this file is for low level manipulation
// of lists of vertices and faces: it can perform some simple transformations on VNF structures
// but cannot perform boolean operations on the polyhedrons represented by VNFs.
// Includes:
// include <BOSL2/std.scad>
// FileGroup: Advanced Modeling
// FileSummary: Vertices 'n' Faces structure. Makes polyhedron() easier to use.
// FileFootnotes: STD=Included in std.scad
//////////////////////////////////////////////////////////////////////
// Section: Creating Polyhedrons with VNF Structures
// VNF stands for "Vertices'N'Faces". VNF structures are 2-item lists, `[VERTICES,FACES]` where the
// first item is a list of vertex points, and the second is a list of face indices into the vertex
// list. Each VNF is self contained, with face indices referring only to its own vertex list.
// You can construct a `polyhedron()` in parts by describing each part in a self-contained VNF, then
// merge the various VNFs to get the completed polyhedron vertex list and faces.
/// Constant: EMPTY_VNF
/// Description:
/// The empty VNF data structure. Equal to `[[],[]]`.
EMPTY_VNF = [[],[]]; // The standard empty VNF with no vertices or faces.
// Function: vnf_vertex_array()
// Synopsis: Returns a VNF structure from a rectangular vertex list.
// SynTags: VNF
// Topics: VNF Generators, Lists
// See Also: vnf_tri_array(), vnf_join(), vnf_from_polygons(), vnf_from_region()
// Usage:
// vnf = vnf_vertex_array(points, [caps=], [cap1=], [cap2=], [style=], [reverse=], [col_wrap=], [row_wrap=], [triangulate=]);
// Description:
// Creates a VNF structure from a rectangular vertex list, creating edges that connect the adjacent vertices in the vertex list
// and creating the faces defined by those edges. You can optionally create the edges and faces to wrap the last column
// back to the first column, or wrap the last row to the first. Endcaps can be added to either
// the first and/or last rows. The style parameter determines how the quadrilaterals are divided into
// triangles. The default style is an arbitrary, systematic subdivision in the same direction. The "alt" style
// is the uniform subdivision in the other (alternate) direction. The "flip1" style is an arbitrary division which alternates the
// direction for any adjacent pair of quadrilaterals. The "flip2" style is the alternating division that is the opposite of "flip1".
// The "min_edge" style picks the shorter edge to
// subdivide for each quadrilateral, so the division may not be uniform across the shape. The "quincunx" style
// adds a vertex in the center of each quadrilateral and creates four triangles, and the "convex" and "concave" styles
// choose the locally convex/concave subdivision. The "min_area" option creates the triangulation with the minimal area. Degenerate faces
// are not included in the output, but if this results in unused vertices they will still appear in the output.
// Arguments:
// points = A list of vertices to divide into columns and rows.
// ---
// caps = If true, add endcap faces to the first **and** last rows.
// cap1 = If true, add an endcap face to the first row.
// cap2 = If true, add an endcap face to the last row.
// col_wrap = If true, add faces to connect the last column to the first.
// row_wrap = If true, add faces to connect the last row to the first.
// reverse = If true, reverse all face normals.
// style = The style of subdividing the quads into faces. Valid options are "default", "alt", "flip1", "flip2", "min_edge", "min_area", "quincunx", "convex" and "concave".
// triangulate = If true, triangulates endcaps to resolve possible CGAL issues. This can be an expensive operation if the endcaps are complex. Default: false
// Example(3D):
// vnf = vnf_vertex_array(
// points=[
// for (h = [0:5:180-EPSILON]) [
// for (t = [0:5:360-EPSILON])
// cylindrical_to_xyz(100 + 12 * cos((h/2 + t)*6), t, h)
// ]
// ],
// col_wrap=true, caps=true, reverse=true, style="alt"
// );
// vnf_polyhedron(vnf);
// Example(3D,NoAxes,ThrownTogether,VPD=183): Open shape made from a three arcs.
// rows = [
// for(h=[-20:20:20])
// path3d(arc(r=40-abs(h), angle=280, 10), h)
// ];
// vnf = vnf_vertex_array(rows, reverse=true);
// vnf_polyhedron(vnf);
// color("green") vnf_wireframe(vnf);
// Example(3D,NoAxes,ThrownTogether,VPD=183): Open shape made from a three arcs, with `row_wrap=true`.
// rows = [
// for(h=[-20:20:20])
// path3d(arc(r=40-abs(h), angle=280, 10), h)
// ];
// vnf = vnf_vertex_array(rows, reverse=true, row_wrap=true);
// vnf_polyhedron(vnf);
// color("green") vnf_wireframe(vnf);
// Example(3D,NoAxes,ThrownTogether,VPD=183): Open shape made from a three arcs, with `col_wrap=true`.
// rows = [
// for(h=[-20:20:20])
// path3d(arc(r=40-abs(h), angle=280, 10), h)
// ];
// vnf = vnf_vertex_array(rows, reverse=true, col_wrap=true);
// vnf_polyhedron(vnf);
// color("green") vnf_wireframe(vnf);
// Example(3D,NoAxes,ThrownTogether,VPD=183): Open shape made from a three arcs, with `caps=true` and `col_wrap=true`.
// rows = [
// for(h=[-20:20:20])
// path3d(arc(r=40-abs(h), angle=280, 10), h)
// ];
// vnf = vnf_vertex_array(rows, reverse=true, caps=true, col_wrap=true);
// vnf_polyhedron(vnf);
// color("green") vnf_wireframe(vnf);
// Example(3D): Both `col_wrap` and `row_wrap` are true to make a torus.
// vnf = vnf_vertex_array(
// points=[
// for (a=[0:5:360-EPSILON])
// apply(
// zrot(a) * right(30) * xrot(90),
// path3d(circle(d=20))
// )
// ],
// col_wrap=true, row_wrap=true, reverse=true
// );
// vnf_polyhedron(vnf);
// Example(3D): Möbius Strip. Note that `row_wrap` is not used, and the first and last profile copies are the same.
// vnf = vnf_vertex_array(
// points=[
// for (a=[0:5:360]) apply(
// zrot(a) * right(30) * xrot(90) * zrot(a/2+60),
// path3d(square([1,10], center=true))
// )
// ],
// col_wrap=true, reverse=true
// );
// vnf_polyhedron(vnf);
// Example(3D): Assembling a Polyhedron from Multiple Parts
// wall_points = [
// for (a = [-90:2:90]) apply(
// up(a) * scale([1-0.1*cos(a*6),1-0.1*cos((a+90)*6),1]),
// path3d(circle(d=100))
// )
// ];
// cap = [
// for (a = [0:0.01:1+EPSILON]) apply(
// up(90-5*sin(a*360*2)) * scale([a,a,1]),
// wall_points[0]
// )
// ];
// cap1 = [for (p=cap) down(90, p=zscale(-1, p=p))];
// cap2 = [for (p=cap) up(90, p=p)];
// vnf1 = vnf_vertex_array(points=wall_points, col_wrap=true);
// vnf2 = vnf_vertex_array(points=cap1, col_wrap=true);
// vnf3 = vnf_vertex_array(points=cap2, col_wrap=true, reverse=true);
// vnf_polyhedron([vnf1, vnf2, vnf3]);
// Example(3D): Building a Multi-Stage Cylindrical Ramp
// include <BOSL2/rounding.scad>
// major_r = 50;
// groove_profile = [
// [-10,0], each arc(points=[[-7,0],[0,-3],[7,0]]), [10,0]
// ];
// ramp_profile = [ [-10,25], [90,25], [180,5], [190,5] ];
// rgroove = apply(right(major_r) * xrot(90), path3d(groove_profile));
// rprofile = round_corners(ramp_profile, radius=20, closed=false, $fn=72);
// vnf = vnf_vertex_array([
// for (a = [ramp_profile[0].x : 1 : last(ramp_profile).x]) let(
// z = lookup(a,rprofile),
// m = zrot(a) * up(z)
// )
// apply(m, [ [rgroove[0].x,0,-z], each rgroove, [last(rgroove).x,0,-z] ])
// ], caps=true, col_wrap=true, reverse=true);
// vnf_polyhedron(vnf, convexity=8);
function vnf_vertex_array(
points,
caps, cap1, cap2,
col_wrap=false,
row_wrap=false,
reverse=false,
style="default",
triangulate = false
) =
assert(!(any([caps,cap1,cap2]) && !col_wrap), "col_wrap must be true if caps are requested")
assert(!(any([caps,cap1,cap2]) && row_wrap), "Cannot combine caps with row_wrap")
assert(in_list(style,["default","alt","quincunx", "convex","concave", "min_edge","min_area","flip1","flip2"]))
assert(is_matrix(points[0], n=3),"Point array has the wrong shape or points are not 3d")
assert(is_consistent(points), "Non-rectangular or invalid point array")
assert(is_bool(triangulate))
let(
pts = flatten(points),
pcnt = len(pts),
rows = len(points),
cols = len(points[0])
)
rows<=1 || cols<=1 ? EMPTY_VNF :
let(
cap1 = first_defined([cap1,caps,false]),
cap2 = first_defined([cap2,caps,false]),
colcnt = cols - (col_wrap?0:1),
rowcnt = rows - (row_wrap?0:1),
verts = [
each pts,
if (style=="quincunx")
for (r = [0:1:rowcnt-1], c = [0:1:colcnt-1])
let(
i1 = ((r+0)%rows)*cols + ((c+0)%cols),
i2 = ((r+1)%rows)*cols + ((c+0)%cols),
i3 = ((r+1)%rows)*cols + ((c+1)%cols),
i4 = ((r+0)%rows)*cols + ((c+1)%cols)
)
mean([pts[i1], pts[i2], pts[i3], pts[i4]])
],
allfaces = [
if (cap1) count(cols,reverse=!reverse),
if (cap2) count(cols,(rows-1)*cols, reverse=reverse),
for (r = [0:1:rowcnt-1], c=[0:1:colcnt-1])
each
let(
i1 = ((r+0)%rows)*cols + ((c+0)%cols),
i2 = ((r+1)%rows)*cols + ((c+0)%cols),
i3 = ((r+1)%rows)*cols + ((c+1)%cols),
i4 = ((r+0)%rows)*cols + ((c+1)%cols),
faces =
style=="quincunx"?
let(i5 = pcnt + r*colcnt + c)
[[i1,i5,i2],[i2,i5,i3],[i3,i5,i4],[i4,i5,i1]]
: style=="alt" || (style=="flip1" && ((r+c)%2==0)) || (style=="flip2" && ((r+c)%2==1)) || (style=="random" && rands(0,1,1)[0]<.5)?
[[i1,i4,i2],[i2,i4,i3]]
: style=="min_area"?
let(
area42 = norm(cross(pts[i2]-pts[i1], pts[i4]-pts[i1]))+norm(cross(pts[i4]-pts[i3], pts[i2]-pts[i3])),
area13 = norm(cross(pts[i1]-pts[i4], pts[i3]-pts[i4]))+norm(cross(pts[i3]-pts[i2], pts[i1]-pts[i2])),
minarea_edge = area42 < area13 + EPSILON
? [[i1,i4,i2],[i2,i4,i3]]
: [[i1,i3,i2],[i1,i4,i3]]
)
minarea_edge
: style=="min_edge"?
let(
d42=norm(pts[i4]-pts[i2]),
d13=norm(pts[i1]-pts[i3]),
shortedge = d42<d13+EPSILON
? [[i1,i4,i2],[i2,i4,i3]]
: [[i1,i3,i2],[i1,i4,i3]]
)
shortedge
: style=="convex"?
let( // Find normal for 3 of the points. Is the other point above or below?
n = (reverse?-1:1)*cross(pts[i2]-pts[i1],pts[i3]-pts[i1]),
convexfaces = n==0
? [[i1,i4,i3]]
: n*pts[i4] > n*pts[i1]
? [[i1,i4,i2],[i2,i4,i3]]
: [[i1,i3,i2],[i1,i4,i3]]
)
convexfaces
: style=="concave"?
let( // Find normal for 3 of the points. Is the other point above or below?
n = (reverse?-1:1)*cross(pts[i2]-pts[i1],pts[i3]-pts[i1]),
concavefaces = n==0
? [[i1,i4,i3]]
: n*pts[i4] <= n*pts[i1]
? [[i1,i4,i2],[i2,i4,i3]]
: [[i1,i3,i2],[i1,i4,i3]]
)
concavefaces
: [[i1,i3,i2],[i1,i4,i3]],
// remove degenerate faces
culled_faces= [for(face=faces)
if (norm(cross(verts[face[1]]-verts[face[0]],
verts[face[2]]-verts[face[0]]))>EPSILON)
face
],
rfaces = reverse? [for (face=culled_faces) reverse(face)] : culled_faces
)
rfaces,
],
vnf = [verts, allfaces]
) triangulate? vnf_triangulate(vnf) : vnf;
// Function: vnf_tri_array()
// Synopsis: Returns a VNF from an array of points. The array need not be rectangular.
// SynTags: VNF
// Topics: VNF Generators, Lists
// See Also: vnf_vertex_array(), vnf_join(), vnf_from_polygons(), vnf_merge_points()
// Usage:
// vnf = vnf_tri_array(points, [caps=], [cap1=], [cap2=], [reverse=], [col_wrap=], [row_wrap=])
// Description:
// Produces a VNF from an array of points where each row length can differ from the adjacent rows by any amount. This enables the construction of triangular or even irregular VNF patches. The resulting VNF can be wrapped along the rows by setting `row_wrap` to true, and wrapped along columns by setting `col_wrap` to true. It is possible to do both at once.
// If `row_wrap` is false or not provided, end caps can be generated across the top and/or bottom rows.
// .
// The algorithm starts with the first point on each row and recursively walks around finding the minimum-length edge to make each new triangle face. This may result in several triangles being connected to one vertex. When triangulating two rows that happen to be equal length, the result is equivalent to {{vnf_vertex_array()}} using the "min_edge" style. If you already have a rectangular vertex list (equal length rows), you should use `vnf_vertex_array()` if you need a different triangulation style.
// .
// If you need to merge two VNF arrays that share edges using `vnf_join()` you can remove the duplicated vertices using `vnf_merge_points()`.
// Arguments:
// points = List of point lists for each row.
// ---
// caps = If true, add endcap faces to the first **and** last rows.
// cap1 = If true, add an endcap face to the first row.
// cap2 = If true, add an endcap face to the last row.
// col_wrap = If true, add faces to connect the last column to the first.
// row_wrap = If true, add faces to connect the last row to the first.
// reverse = If true, reverse all face normals.
// Example(3D,NoAxes): Each row has one more point than the preceeding one.
// pts = [for(y=[1:1:10]) [for(x=[0:y-1]) [x,y,y]]];
// vnf = vnf_tri_array(pts);
// vnf_wireframe(vnf,width=0.1);
// color("red")move_copies(flatten(pts)) sphere(r=.15,$fn=9);
// Example(3D,NoAxes): Each row has two more points than the preceeding one.
// pts = [for(y=[0:2:10]) [for(x=[-y/2:y/2]) [x,y,y]]];
// vnf = vnf_tri_array(pts);
// vnf_wireframe(vnf,width=0.1);
// color("red")move_copies(flatten(pts)) sphere(r=.15,$fn=9);
// Example(3D): Merging two VNFs to construct a cone with one point length change between rows.
// pts1 = [for(z=[0:10]) path3d(arc(3+z,r=z/2+1, angle=[0,180]),10-z)];
// pts2 = [for(z=[0:10]) path3d(arc(3+z,r=z/2+1, angle=[180,360]),10-z)];
// vnf = vnf_join([vnf_tri_array(pts1),
// vnf_tri_array(pts2)]);
// color("green")vnf_wireframe(vnf,width=0.1);
// vnf_polyhedron(vnf);
// Example(3D): Cone with length change two between rows
// pts1 = [for(z=[0:1:10]) path3d(arc(3+2*z,r=z/2+1, angle=[0,180]),10-z)];
// pts2 = [for(z=[0:1:10]) path3d(arc(3+2*z,r=z/2+1, angle=[180,360]),10-z)];
// vnf = vnf_join([vnf_tri_array(pts1),
// vnf_tri_array(pts2)]);
// color("green")vnf_wireframe(vnf,width=0.1);
// vnf_polyhedron(vnf);
// Example(3D,NoAxes): The number of points per row can change irregularly by any amount.
// lens = [10,9,8,6,4,8,2,5,3,10,4];
// pts = [for(y=idx(lens)) lerpn([-lens[y],y,y],[lens[y],y,y],lens[y])];
// vnf = vnf_tri_array(pts);
// vnf_wireframe(vnf,width=0.1);
// color("red")move_copies(flatten(pts)) sphere(r=.15,$fn=9);
// Example(3D,NoAxes,Edges,VPR=[65,0,25],VPD=380,Med): Model of a cymbal with roughly same-size facets, using a different number of points for each concentric ring of vertices.
// include <BOSL2/beziers.scad>
// bez = [
// [[0,26], [35,26], [29,0], [80,16], [102,0]], //top
// [[99,-1], [79,15], [28,-1], [34,25], [-1,25]] // bottom
// ];
// points = [
// for(b=bez)
// for(u=[0.01:0.04:1]) let(
// bzp = bezier_points(b,u),
// r = bzp[0],
// n = max(3, round(360 / (6/r * 180/PI)))
// ) path3d(regular_ngon(n, r=r), bzp[1])
// ];
// vnf = vnf_tri_array(points, reverse=true, col_wrap=true, caps=true);
// color("brown") difference() {
// vnf_polyhedron(vnf);
// cylinder(30, d=8);
// }
function vnf_tri_array(
points,
caps, cap1, cap2,
col_wrap=false,
row_wrap=false,
reverse=false
) =
assert(!(any([caps,cap1,cap2]) && row_wrap), "Cannot combine caps with row_wrap")
let(
plen = len(points),
// append first vertex of each polygon to its end if wrapping columns
st = col_wrap ? [
for(i=[0:plen-1])
points[i][0] != points[i][len(points[i])-1]
? concat(points[i], [points[i][0]])
: points[i]
] : points,
addcol = col_wrap ? len(st[0])-len(points[0]) : 0,
rowstarts = [ for(i=[0:plen-1]) len(st[i]) ],
capfirst = first_defined([cap1,caps,false]),
caplast = first_defined([cap2,caps,false]),
pcumlen = [0, each cumsum(rowstarts)],
faces = flatten([
// close first end
if (capfirst)
if (reverse) [[ for(i=[0:rowstarts[0]-1-addcol]) i ]]
else [[ for(i=[rowstarts[0]-1-addcol:-1:0]) i ]],
// triangulate between the two polygons
for(i = [0:plen-2+(row_wrap?1:0)]) let(j = (i+1)%plen)
_lofttri(st[i], st[j], pcumlen[i], pcumlen[j], rowstarts[i], rowstarts[j], reverse),
// close up the last end
if (caplast)
if (reverse) [[ for(i=[pcumlen[plen]-1-addcol:-1:pcumlen[plen-1]]) i ]]
else [[ for(i=[pcumlen[plen-1]:pcumlen[plen]-1-addcol]) i ]]
]),
vnf = [flatten(st), faces]
) col_wrap ? vnf_merge_points(vnf) : vnf;
/*
Recursively triangulate between two 3D paths (which can be different
in length by any amount), starting at index 0 and generate a list of
triangles with minimum new-side length.
The first side of the first triangle always connects the two first
vertices of each path.
To triangulate between two closed paths, the first and last vertices
must be the same.
Parameters:
p1 = first path, an array of [x,y,z] vertices
p2 = second path, an array of [x,y,z] vertices
i1offset = index offset of first vertex in the first path
(sum of any prior path lengths)
i2offset = index offset of first vertex in the second path
(sum of any prior path lengths)
n1 = number of vertices in first path
n2 = number of vertices in second path
reverse = if true, assume a polygon path goes around
counterclockwise with respect to the direction from
p1 to p2 (right hand rule), clockwise if false
Other parameters are for internal use:
trilist[] = array of triangles to return
i1 = vertex index on p1 of the next triangle
i2 = vertex index on p2 of the next triangle
(next triangle vertex found can be on either p1 or p2, depending
on which triangle is smaller.)
Returns an array of triangles using vertex indices offset by
i1offset and i2offset
*/
function _lofttri(p1, p2, i1offset, i2offset, n1, n2, reverse=false, trilist=[], i1=0, i2=0) = n1!=n2 ?
// unequal row lengths
let(
t1 = i1 < n1 ? i1+1 : n1, // test point 1
t2 = i2 < n2 ? i2+1 : n2, // test point 2
d12 = t2>=n2 ? 9e+9 : norm(p2[t2]-p1[i1]), // distance from i1 to t2
d21 = t1>=n1 ? 9e+9 : norm(p1[t1]-p2[i2]), // distance from i2 to t1
triangle = reverse ?
[i1offset+i1, i2offset+i2, d12<d21 ? i2offset+t2 : i1offset+t1] :
[i2offset+i2, i1offset+i1, d12<d21 ? i2offset+t2 : i1offset+t1]
) t1>=n1 && t2>=n2 ? trilist :
_lofttri(p1, p2, i1offset, i2offset, n1, n2, reverse, concat(trilist, [triangle]), d12<d21 ? i1 : t1, d12<d21 ? t2 : i2)
: // equal row lengths
let(n=n1, i=i1,
t = i < n ? i+1 : n, // test point
d12 = t>=n ? 9e+9 : norm(p2[t]-p1[i]), // distance from p1 to new p2
d21 = t>=n ? 9e+9 : norm(p1[t]-p2[i]), // distance from p2 to new p1
triangle1 = reverse ?
[i1offset+i, i2offset+i, d12<d21 ? i2offset+t : i1offset+t] :
[i2offset+i, i1offset+i, d12<d21 ? i2offset+t : i1offset+t],
triangle2 = reverse ?
[i2offset+t, i1offset+t, d12<d21 ? i1offset+i : i2offset+i] :
[i1offset+t, i2offset+t, d12<d21 ? i1offset+i : i2offset+i]
) t>=n ? trilist :
_lofttri(p1, p2, i1offset, i2offset, n, n, reverse, concat(trilist, [triangle1, triangle2]), t, t);
// Function: vnf_join()
// Synopsis: Returns a single VNF structure from a list of VNF structures.
// SynTags: VNF
// Topics: VNF Generators, Lists
// See Also: vnf_tri_array(), vnf_vertex_array(), vnf_from_polygons(), vnf_from_region()
// Usage:
// vnf = vnf_join([VNF, VNF, VNF, ...]);
// Description:
// Given a list of VNF structures, merges them all into a single VNF structure.
// Combines all the points of the input VNFs and labels the faces appropriately.
// All the points in the input VNFs will appear in the output, even if they are
// duplicates of each other. It is valid to repeat points in a VNF, but if you
// with to remove the duplicates that will occur along joined edges, use {{vnf_merge_points()}}.
// .
// Note that this is a tool for manipulating polyhedron data. It is for
// building up a full polyhedron from partial polyhedra.
// It is *not* a union operator for VNFs. The VNFs to be joined must not intersect each other,
// except at edges, or the result will be an invalid polyhedron. Similarly the
// result must not have any other illegal polyhedron characteristics, such as creating
// more than two faces sharing the same edge.
// If you want a valid result it is your responsibility to ensure that the polyhedron
// has no holes, no intersecting faces or edges, and obeys all the requirements
// that CGAL expects.
// .
// For example, if you combine two pyramids to try to make an octahedron, the result will
// be invalid because of the two internal faces created by the pyramid bases. A valid
// use would be to build a cube missing one face and a pyramid missing its base and
// then join them into a cube with a point.
// Arguments:
// vnfs = a list of the VNFs to joint into one VNF.
// Example(3D,VPR=[60,0,26],VPD=55,VPT=[5.6,-5.3,9.8]): Here is a VNF where the top face is missing. It is not a valid polyhedron like this, but we can use it as a building block to make a polyhedron.
// bottom = vnf_vertex_array([path3d(rect(8)), path3d(rect(5),4)],col_wrap=true,cap1=true);
// vnf_polyhedron(bottom);
// Example(3D,VPR=[60,0,26],VPD=55,VPT=[5.6,-5.3,9.8]): Here is a VNF that also has a missing face.
// triangle = yrot(-90,path3d(regular_ngon(n=3,side=5,anchor=LEFT)));
// top = up(4,vnf_vertex_array([list_set(right(2.5,triangle),0,[0,0,7]),
// right(6,triangle)
// ], col_wrap=true, cap2=true));
// vnf_polyhedron(zrot(90,top));
// Example(3D,VPR=[60,0,26],VPD=55,VPT=[5.6,-5.3,9.8]): Using vnf_join combines the two VNFs into a single VNF. Note that they share an edge. But the result still isn't closed, so it is not yet a valid polyhedron.
// bottom = vnf_vertex_array([path3d(rect(8)), path3d(rect(5),4)],col_wrap=true,cap1=true);
// triangle = yrot(-90,path3d(regular_ngon(n=3,side=5,anchor=LEFT)));
// top = up(4,vnf_vertex_array([list_set(right(2.5,triangle),0,[0,0,7]),
// right(6,triangle)
// ], col_wrap=true, cap2=true));
// full = vnf_join([bottom,zrot(90,top)]);
// vnf_polyhedron(full);
// Example(3D,VPR=[60,0,26],VPD=55,VPT=[5.6,-5.3,9.8]): If we add enough pieces, and the pieces are all consistent with each other, then we can arrive at a valid polyhedron like this one. To be valid you need to meet all the CGAL requirements: every edge has exactly two faces, all faces are in clockwise order, no intersections of edges.
// bottom = vnf_vertex_array([path3d(rect(8)), path3d(rect(5),4)],col_wrap=true,cap1=true);
// triangle = yrot(-90,path3d(regular_ngon(n=3,side=5,anchor=LEFT)));
// top = up(4,vnf_vertex_array([list_set(right(2.5,triangle),0,[0,0,7]),
// right(6,triangle)
// ], col_wrap=true, cap2=true));
// full = vnf_join([bottom,
// for(theta=[0:90:359]) zrot(theta,top)
// ]);
// vnf_polyhedron(full);
// Example(3D): The vnf_join function is not a union operator for polyhedra. If any faces intersect, like they do in this example where we combine the faces of two cubes, the result is invalid and will give rise to CGAL errors when you add more objects into the model.
// cube1 = cube(5);
// cube2 = move([2,2,2],cube1);
// badvnf = vnf_join([cube1,cube2]);
// vnf_polyhedron(badvnf);
// right(2.5)up(3)color("red")
// text3d("Invalid",size=1,anchor=CENTER,
// orient=FRONT,h=.1);
function vnf_join(vnfs) =
assert(is_vnf_list(vnfs) , "Input must be a list of VNFs")
len(vnfs)==1 ? vnfs[0]
:
let (
offs = cumsum([ 0, for (vnf = vnfs) len(vnf[0]) ]),
verts = [for (vnf=vnfs) each vnf[0]],
faces =
[ for (i = idx(vnfs))
let( faces = vnfs[i][1] )
for (face = faces)
if ( len(face) >= 3 )
[ for (j = face)
assert( j>=0 && j<len(vnfs[i][0]),
str("VNF number ", i, " has a face indexing an nonexistent vertex") )
offs[i] + j ]
]
)
[verts,faces];
// Function: vnf_from_polygons()
// Synopsis: Returns a VNF from a list of 3D polygons.
// SynTags: VNF
// Topics: VNF Generators, Lists
// See Also: vnf_tri_array(), vnf_join(), vnf_vertex_array(), vnf_from_region()
// Usage:
// vnf = vnf_from_polygons(polygons, [eps]);
// Description:
// Given a list of 3D polygons, produces a VNF containing those polygons.
// It is up to the caller to make sure that the points are in the correct order to make the face
// normals point outwards. No checking for duplicate vertices is done. If you want to
// remove duplicate vertices use {{vnf_merge_points()}}. Polygons with zero area are discarded from the face list by default.
// If you give non-coplanar faces an error is displayed. These checks increase run time by about 2x for triangular polygons, but
// about 10x for pentagons; the checks can be disabled by setting fast=true.
// Arguments:
// polygons = The list of 3D polygons to turn into a VNF
// fast = Set to true to skip area and coplanarity checks for increased speed. Default: false
// eps = Polygons with area smaller than this are discarded. Default: EPSILON
// Example(3D,VPR=[60,0,40]): Construction of a dodecahedron from pentagon faces.
// dihedral = 2*atan(PHI); // dodecahedron face dihedral
// rpenta = 10; // pentagon face radius
// edge = 2*rpenta*sin(36); // edge length
// inrad = 0.5*edge * PHI*PHI/sqrt(3-PHI); // inner radius
// face3d = path3d(pentagon(rpenta), inrad); // single face
// facepoints = [
// face3d,
// for(a=[36:72:360]) zrot(a, yrot(180-dihedral, face3d)),
// for(a=[36:72:360]) zrot(a, yrot(360-dihedral, face3d)),
// yrot(180, face3d)
// ];
// vnf = vnf_from_polygons(facepoints, fast=true);
// vnf_polyhedron(vnf);
function vnf_from_polygons(polygons,fast=false,eps=EPSILON) =
assert(is_list(polygons) && is_path(polygons[0]),"Input should be a list of polygons")
let(
offs = cumsum([0, for(p=polygons) len(p)]),
faces = [for(i=idx(polygons))
let(
area=fast ? 1 : polygon_area(polygons[i]),
dummy=assert(is_def(area) || is_collinear(polygons[i],eps=eps),str("Polygon ", i, " is not coplanar"))
)
if (is_def(area) && area > eps)
[for (j=idx(polygons[i])) offs[i]+j]
]
)
[flatten(polygons), faces];
function _path_path_closest_vertices(path1,path2) =
let(
dists = [for (i=idx(path1)) let(j=closest_point(path1[i],path2)) [j,norm(path2[j]-path1[i])]],
i1 = min_index(column(dists,1)),
i2 = dists[i1][0]
) [dists[i1][1], i1, i2];
function _join_paths_at_vertices(path1,path2,v1,v2) =
let(
repeat_start = !approx(path1[v1],path2[v2]),
path1 = clockwise_polygon(list_rotate(path1,v1)),
path2 = ccw_polygon(list_rotate(path2,v2))
)
[
each path1,
if (repeat_start) path1[0],
each path2,
if (repeat_start) path2[0],
];
/// Internal Function: _cleave_connected_region(region, eps)
/// Description:
/// Given a region that is connected and has its outer border in region[0],
/// produces a overlapping connected path to join internal holes to
/// the outer border without adding points. Output is a single non-simple polygon.
/// Requirements:
/// It expects that all region paths be simple closed paths, with region[0] CW and
/// the other paths CCW and encircled by region[0]. The input region paths are also
/// supposed to be disjoint except for common vertices and common edges but with
/// no crossings. It may return `undef` if these conditions are not met.
/// This function implements an extension of the algorithm discussed in:
/// https://www.geometrictools.com/Documentation/TriangulationByEarClipping.pdf
function _cleave_connected_region(region, eps=EPSILON) =
len(region)==1 ? region[0] :
let(
outer = deduplicate(region[0]), //
holes = [for(i=[1:1:len(region)-1]) // deduplication possibly unneeded
deduplicate( region[i] ) ], //
extridx = [for(li=holes) max_index(column(li,0)) ],
// the right extreme vertex for each hole sorted by decreasing x values
extremes = sort( [for(i=idx(holes)) [ i, extridx[i], -holes[i][extridx[i]].x] ], idx=2 )
)
_polyHoles(outer, holes, extremes, eps, 0);
// connect the hole paths one at a time to the outer path.
// 'extremes' is the list of the right extreme vertex of each hole sorted by decreasing abscissas
// see: _cleave_connected_region(region, eps)
function _polyHoles(outer, holes, extremes, eps=EPSILON, n=0) =
let(
extr = extremes[n], //
hole = holes[extr[0]], // hole path to bridge to the outer path
ipt = extr[1], // index of the hole point with maximum abscissa
brdg = _bridge(hole[ipt], outer, eps) // the index of a point in outer to bridge hole[ipt] to
)
brdg == undef ? undef :
let(
l = len(outer),
lh = len(hole),
// the new outer polygon bridging the hole to the old outer
npoly =
approx(outer[brdg], hole[ipt], eps)
? [ for(i=[brdg: 1: brdg+l]) outer[i%l] ,
for(i=[ipt+1: 1: ipt+lh-1]) hole[i%lh] ]
: [ for(i=[brdg: 1: brdg+l]) outer[i%l] ,
for(i=[ipt: 1: ipt+lh]) hole[i%lh] ]
)
n==len(holes)-1 ? npoly :
_polyHoles(npoly, holes, extremes, eps, n+1);
// find a point in outer to be connected to pt in the interior of outer
// by a segment that not cross or touch any non adjacente edge of outer.
// return the index of a vertex in the outer path where the bridge should end
// see _polyHoles(outer, holes, extremes, eps)
function _bridge(pt, outer,eps) =
// find the intersection of a ray from pt to the right
// with the boundary of the outer cycle
let(
l = len(outer),
crxs =
let( edges = pair(outer,wrap=true) )
[for( i = idx(edges) )
let( edge = edges[i] )
// consider just descending outer edges at right of pt crossing ordinate pt.y
if( (edge[0].y > pt.y) //+eps)
&& (edge[1].y <= pt.y)
&& _is_at_left(pt, [edge[1], edge[0]], eps) )
[ i,
// the point of edge with ordinate pt.y
abs(pt.y-edge[1].y)<eps ? edge[1] :
let( u = (pt-edge[1]).y / (edge[0]-edge[1]).y )
(1-u)*edge[1] + u*edge[0]
]
]
)
crxs == [] ? undef :
let(
// the intersection point of the nearest edge to pt with minimum slope
minX = min([for(p=crxs) p[1].x]),
crxcand = [for(crx=crxs) if(crx[1].x < minX+eps) crx ], // nearest edges
nearest = min_index([for(crx=crxcand)
(outer[crx[0]].x - pt.x) / (outer[crx[0]].y - pt.y) ]), // minimum slope
proj = crxcand[nearest],
vert0 = outer[proj[0]], // the two vertices of the nearest crossing edge
vert1 = outer[(proj[0]+1)%l],
isect = proj[1] // the intersection point
)
norm(pt-vert1) < eps ? (proj[0]+1)%l : // if pt touches an outer vertex, return its index
// as vert0.y > pt.y then pt!=vert0
norm(pt-isect) < eps ? undef : // if pt touches the middle of an outer edge -> error
let(
// the edge [vert0, vert1] necessarily satisfies vert0.y > vert1.y
// indices of candidates to an outer bridge point
cand =
(vert0.x > pt.x)
? [ proj[0],
// select reflex vertices inside of the triangle [pt, vert0, isect]
for(i=idx(outer))
if( _tri_class(select(outer,i-1,i+1),eps) <= 0
&& _pt_in_tri(outer[i], [pt, vert0, isect], eps)>=0 )
i
]
: [ (proj[0]+1)%l,
// select reflex vertices inside of the triangle [pt, isect, vert1]
for(i=idx(outer))
if( _tri_class(select(outer,i-1,i+1),eps) <= 0
&& _pt_in_tri(outer[i], [pt, isect, vert1], eps)>=0 )
i
],
// choose the candidate outer[i] such that the line [pt, outer[i]] has minimum slope
// among those with minimum slope choose the nearest to pt
slopes = [for(i=cand) 1-abs(outer[i].x-pt.x)/norm(outer[i]-pt) ],
min_slp = min(slopes),
cand2 = [for(i=idx(cand)) if(slopes[i]<=min_slp+eps) cand[i] ],
nearest = min_index([for(i=cand2) norm(pt-outer[i]) ])
)
cand2[nearest];
// Function: vnf_from_region()
// Synopsis: Returns a 3D VNF given a 2D region.
// SynTags: VNF
// Topics: VNF Generators, Lists
// See Also: vnf_vertex_array(), vnf_tri_array(), vnf_join(), vnf_from_polygons()
// Usage:
// vnf = vnf_from_region(region, [transform], [reverse]);
// Description:
// Given a (two-dimensional) region, applies the given transformation matrix to it and makes a (three-dimensional) triangulated VNF of
// faces for that region, reversed if desired.
// Arguments:
// region = The region to convert to a vnf.
// transform = If given, a transformation matrix to apply to the faces generated from the region. Default: No transformation applied.
// reverse = If true, reverse the normals of the faces generated from the region. An untransformed region will have face normals pointing `UP`. Default: false
// Example(3D):
// region = [square([20,10],center=true),
// right(5,square(4,center=true)),
// left(5,square(6,center=true))];
// vnf = vnf_from_region(region);
// color("gray")down(.125)
// linear_extrude(height=.125)region(region);
// vnf_wireframe(vnf,width=.25);
function vnf_from_region(region, transform, reverse=false, triangulate=true) =
let (
region = [for (path = region) deduplicate(path, closed=true)],
regions = region_parts(force_region(region)),
vnfs =
[
for (rgn = regions)
let(
cleaved = path3d(_cleave_connected_region(rgn))
)
assert( cleaved, "The region is invalid")
let(
face = is_undef(transform)? cleaved : apply(transform,cleaved),
faceidxs = reverse? [for (i=[len(face)-1:-1:0]) i] : [for (i=[0:1:len(face)-1]) i]
) [face, [faceidxs]]
],
outvnf = vnf_join(vnfs)
)
triangulate ? vnf_triangulate(outvnf) : outvnf;
// Section: VNF Testing and Access
// Function: is_vnf()
// Synopsis: Returns true given a VNF-like structure.
// Topics: VNF Manipulation
// See Also: is_vnf_list(), vnf_vertices(), vnf_faces()
// Usage:
// bool = is_vnf(x);
// Description:
// Returns true if the given value looks like a VNF structure.
function is_vnf(x) =
is_list(x) &&
len(x)==2 &&
is_list(x[0]) &&
is_list(x[1]) &&
(x[0]==[] || (len(x[0])>=3 && is_vector(x[0][0],3))) &&
(x[1]==[] || is_vector(x[1][0]));
// Function: is_vnf_list()
// Synopsis: Returns true given a list of VNF-like structures.
// Topics: VNF Manipulation
// See Also: is_vnf(), vnf_vertices(), vnf_faces()
//
// Description: Returns true if the given value looks passingly like a list of VNF structures.
function is_vnf_list(x) = is_list(x) && all([for (v=x) is_vnf(v)]);
// Function: vnf_vertices()
// Synopsis: Returns the list of vertex points from a VNF.
// Topics: VNF Manipulation
// See Also: is_vnf(), is_vnf_list(), vnf_faces()
// Description: Given a VNF structure, returns the list of vertex points.
function vnf_vertices(vnf) = vnf[0];
// Function: vnf_faces()
// Synopsis: Returns the list of faces from a VNF.
// Topics: VNF Manipulation
// See Also: is_vnf(), is_vnf_list(), vnf_vertices()
// Description: Given a VNF structure, returns the list of faces, where each face is a list of indices into the VNF vertex list.
function vnf_faces(vnf) = vnf[1];
// Section: Altering the VNF Internals
// Function: vnf_reverse_faces()
// Synopsis: Reverses the faces of a VNF.
// SynTags: VNF
// Topics: VNF Manipulation
// See Also: vnf_quantize(), vnf_merge_points(), vnf_drop_unused_points(), vnf_triangulate(), vnf_slice(), vnf_unify_faces()
// Usage:
// rvnf = vnf_reverse_faces(vnf);
// Description:
// Reverses the orientation of all the faces in the given VNF.
function vnf_reverse_faces(vnf) =
[vnf[0], [for (face=vnf[1]) reverse(face)]];
// Function: vnf_quantize()
// Synopsis: Quantizes the vertex coordinates of a VNF.
// SynTags: VNF
// Topics: VNF Manipulation
// See Also: vnf_reverse_faces(), vnf_merge_points(), vnf_drop_unused_points(), vnf_triangulate(), vnf_slice()
// Usage:
// vnf2 = vnf_quantize(vnf,[q]);
// Description:
// Quantizes the vertex coordinates of the VNF to the given quanta `q`.
// Arguments:
// vnf = The VNF to quantize.
// q = The quanta to quantize the VNF coordinates to.
function vnf_quantize(vnf,q=pow(2,-12)) =
[[for (pt = vnf[0]) quant(pt,q)], vnf[1]];
// Function: vnf_merge_points()
// Synopsis: Consolidates duplicate vertices of a VNF.
// SynTags: VNF
// Topics: VNF Manipulation
// See Also: vnf_reverse_faces(), vnf_quantize(), vnf_drop_unused_points(), vnf_triangulate(), vnf_slice(), vnf_unify_faces()
// Usage:
// new_vnf = vnf_merge_points(vnf, [eps]);
// Description:
// Given a VNF, consolidates all duplicate vertices with a tolerance `eps`, relabeling the faces as necessary,
// and eliminating any face with fewer than 3 vertices. Unreferenced vertices of the input VNF are not dropped.
// To remove such vertices uses {{vnf_drop_unused_points()}}.
// Arguments:
// vnf = a VNF to consolidate
// eps = the tolerance in finding duplicates. Default: EPSILON
function vnf_merge_points(vnf,eps=EPSILON) =
let(
verts = vnf[0],
dedup = vector_search(verts,eps,verts), // collect vertex duplicates
map = [for(i=idx(verts)) min(dedup[i]) ], // remap duplic vertices
offset = cumsum([for(i=idx(verts)) map[i]==i ? 0 : 1 ]), // remaping face vertex offsets
map2 = list(idx(verts))-offset, // map old vertex indices to new indices
nverts = [for(i=idx(verts)) if(map[i]==i) verts[i] ], // this doesn't eliminate unreferenced vertices
nfaces =
[ for(face=vnf[1])
let(
nface = [ for(vi=face) map2[map[vi]] ],
dface = [for (i=idx(nface))
if( nface[i]!=nface[(i+1)%len(nface)])
nface[i] ]
)
if(len(dface) >= 3) dface
]
)
[nverts, nfaces];
// Function: vnf_drop_unused_points()
// Synopsis: Removes unreferenced vertices from a VNF.
// SynTags: VNF
// Topics: VNF Manipulation
// See Also: vnf_reverse_faces(), vnf_quantize(), vnf_merge_points(), vnf_triangulate(), vnf_slice(), vnf_unify_faces()
// Usage:
// clean_vnf = vnf_drop_unused_points(vnf);
// Description:
// Remove all unreferenced vertices from a VNF. Note that in most
// cases unreferenced vertices cause no harm, and this function may
// be slow on large VNFs.
function vnf_drop_unused_points(vnf) =
let(
flat = flatten(vnf[1]),
ind = _link_indicator(flat,0,len(vnf[0])-1),
verts = [for(i=idx(vnf[0])) if(ind[i]==1) vnf[0][i] ],
map = cumsum(ind)
)
[ verts, [for(face=vnf[1]) [for(v=face) map[v]-1 ] ] ];
function _link_indicator(l,imin,imax) =
len(l) == 0 ? repeat(imax-imin+1,0) :
imax-imin<100 || len(l)<400 ? [for(si=search(list([imin:1:imax]),l,1)) si!=[] ? 1: 0 ] :
let(
pivot = floor((imax+imin)/2),
lesser = [ for(li=l) if( li< pivot) li ],
greater = [ for(li=l) if( li> pivot) li ]
)
concat( _link_indicator(lesser ,imin,pivot-1),
search(pivot,l,1) ? 1 : 0 ,
_link_indicator(greater,pivot+1,imax) ) ;
// Function: vnf_triangulate()
// Synopsis: Triangulates the faces of a VNF.
// SynTags: VNF
// Topics: VNF Manipulation
// See Also: vnf_reverse_faces(), vnf_quantize(), vnf_merge_points(), vnf_drop_unused_points(), vnf_slice(), vnf_unify_faces()
// Usage:
// vnf2 = vnf_triangulate(vnf);
// Description:
// Triangulates faces in the VNF that have more than 3 vertices.
// Arguments:
// vnf = VNF to triangulate
// Example(3D):
// include <BOSL2/polyhedra.scad>
// vnf = zrot(33,regular_polyhedron_info("vnf", "dodecahedron", side=12));
// vnf_polyhedron(vnf);
// triangulated = vnf_triangulate(vnf);
// color("red")vnf_wireframe(triangulated,width=.3);
function vnf_triangulate(vnf) =
let(
verts = vnf[0],
faces = [for (face=vnf[1])
each (len(face)==3 ? [face] :
let( tris = polygon_triangulate(verts, face) )
assert( tris!=undef, "Some `vnf` face cannot be triangulated.")
tris ) ]
)
[verts, faces];
// Function: vnf_unify_faces()
// Synopsis: Remove triangulation from VNF, returning a copy with full faces
// SynTags: VNF
// Topics: VNF Manipulation
// See Also: vnf_reverse_faces(), vnf_quantize(), vnf_merge_points(), vnf_triangulate(), vnf_slice()
// Usage:
// newvnf = vnf_unify_faces(vnf);
// Description:
// When a VNF has been triangulated, the polygons that form the true faces have been chopped up into
// triangles. This can create problems for algorithms that operate on the VNF itself, where you might
// want to be able to identify the true faces. This function merges together the triangles that
// form those true faces, turning a VNF where each true face is represented by a single entry
// in the faces list of the VNF. This function requires that the true faces have no internal vertices.
// This will always be true for a triangulated VNF, but might fail for a VNF with some other
// face partition. If internal vertices are present, the output will include backtracking paths from
// the boundary to all of those vertices.
// Arguments:
// vnf = vnf whose faces you want to unify
// Example(3D,Med,NoAxes): Original prism on the left is triangulated. On the right, the result of unifying the faces.
// $fn=16;
// poly = linear_sweep(hexagon(side=10),h=35);
// vnf = vnf_unify_faces(poly);
// vnf_wireframe(poly);
// color([0,1,1,.70])vnf_polyhedron(poly);
// right(25){
// vnf_wireframe(vnf);
// color([0,1,1,.70])vnf_polyhedron(vnf);
// }
function vnf_unify_faces(vnf) =
let(
faces = vnf[1],
edges = [for(i=idx(faces), edge=pair(faces[i],wrap=true))
[[min(edge),max(edge)],i]],
normals = [for(face=faces) polygon_normal(select(vnf[0],face))],
facelist = count(faces), //[for(i=[1:1:len(faces)-1]) i],
newfaces = _detri_combine_faces(edges,faces,normals,facelist,0)
)
[vnf[0],newfaces];
function _detri_combine_faces(edgelist,faces,normals,facelist,curface) =
curface==len(faces)? select(faces,facelist)
: !in_list(curface,facelist) ? _detri_combine_faces(edgelist,faces,normals,facelist,curface+1)
:
let(
thisface=faces[curface],
neighbors = [for(i=idx(thisface))
let(
edgepair = search([sort(select(thisface,i,i+1))],edgelist,0)[0],
choices = select(edgelist,edgepair),
good_choice=[for(choice=choices)
if (choice[1]!=curface && in_list(choice[1],facelist) && normals[choice[1]]*normals[curface]>1-EPSILON)
choice],
d=assert(len(good_choice)<=1)
)
len(good_choice)==1 ? good_choice[0][1] : -1
],
// Check for duplicates in the neighbor list so we don't add them twice
dups = search([for(n=neighbors) if (n>=0) n], neighbors,0),
goodind = column(dups,0),
newface = [for(i=idx(thisface))