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BezierCurve.cs

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+// Bezier Curve functions for Unity 3D
+// Copyright (C) 2017 Guney Ozsan
+// 
+// This program is free software: you can redistribute it and/or modify
+// it under the terms of the GNU General Public License as published by
+// the Free Software Foundation, either version 3 of the License, or
+// (at your option) any later version.
+// 
+// This program is distributed in the hope that it will be useful,
+// but WITHOUT ANY WARRANTY; without even the implied warranty of
+// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+// GNU General Public License for more details.
+// 
+// You should have received a copy of the GNU General Public License
+// along with this program.  If not, see <http://www.gnu.org/licenses/>.
+// ---------------------------------------------------------------------
+// 
+// Mathematical formulas: "Bézier curve" from https://en.wikipedia.org/wiki/B%C3%A9zier_curve
+
+using System;
+// .NET:
+// using System.Numerics;
+// Unity 3D:
+using UnityEngine;
+
+// Notation follows the Mathematical convention.
+public static class BezierCurve
+{
+
+    public static Vector2 Linear(Vector2 p0, Vector2 p1, float t)
+    {
+        // polynomial form (recursive)
+        return p0 + t*(p1 - p0);
+        // explicit form
+        // return (1 - t)*p0 + t*p1;
+    }
+
+
+    public static Vector2 Quadratic(Vector2 p0, Vector2 p1, Vector2 p2, float t)
+    {
+        // polynomial form (recursive)
+        return (1 - t)*(p0 + t*(p1 - p0)) + t*(p1 + t*(p2 - p1));
+        // Alternative recursive function:
+        // return (1 - t)*Linear(p0, p1, t) + t*Linear(p1, p2, t);
+        // explicit form
+        // return (1 - t)*(1 - t)*p0 + 2*(1 - t)*t*p1 + t*t*p2;
+    }
+
+
+    public static Vector2 Cubic(Vector2 p0, Vector2 p1, Vector2 p2, Vector2 p3, float t)
+    {
+        // polynomial form (recursive)
+        return (1 - t)*((1 - t)*(p0 + t*(p1 - p0)) + t*(p1 + t*(p2 - p1))) + t*((1 - t)*(p1 + t*(p2 - p1)) + t*(p2 + t*(p3 - p2)));
+        // Alternative recursive function:
+        // return (1 - t)*Quadratic(p0, p1, p2, t) + t*Quadratic(p1, p2, p3, t);
+        // explicit form
+        // return (1 - t)*(1 - t)*(1 - t)*p0 + 3*(1 - t)*(1 - t)*t*p1 + 3*(1 - t)*t*t*p2 + t*t*t*p3;
+    }
+
+
+    // Recursive definition
+    public static Vector2 Recursive(Vector2[] p, float t)
+    {
+        Vector2 bt = p[0];
+
+        if (p.Length > 1)
+        {
+            Vector2[] p1pn = new Vector2[p.Length - 1];
+            Array.Copy(p, 1, p1pn, 0, p.Length - 1);
+            // The following should be like this but skipped for optimization
+            // Vector2[] p0pnMinus1 = Array.Resize(ref p, p.Length - 1);
+            // bt = (1 - t)*Recursive(p0pnMinus1 , t) + t*Recursive(p1pn, t);
+            Array.Resize(ref p, p.Length - 1);
+            bt = (1 - t)*Recursive(p, t) + t*Recursive(p1pn, t);
+        }
+
+        return bt;
+    }
+
+
+    // Explicit definition
+    public static Vector2 General(Vector2[] p, float t)
+    {
+        Vector2 bt = Vector2.zero;
+        int n = p.Length - 1;
+
+        for (int i = 0; i <= n; i++)
+        {
+            bt += Combination(n, i)*Power(1 - t, n - i)*Power(t, i)*p[i];
+        }
+
+        return bt;
+    }
+
+
+    // Polynomial form (good for repetitive use).
+    // Prior to using this, calculate polynomial coefficients using PolynomialCoefficients() method,
+    // keep them in a Vector2 array c[] and pass the array here.
+    // If you really need an efficient algorithm, you can make a dictionary of PolynomialCoefficients.
+    public static Vector2 Polynomial(Vector2[] p, float t, Vector2[] c)
+    {
+        Vector2 bt = Vector2.zero;
+
+        for (int j = 0; j < p.Length; j++)
+        {
+            bt += Power(t, j)*c[j];
+        }
+
+        return bt;
+    }
+
+
+    public static class FirstDerivative
+    {
+
+        public static Vector2 Linear(Vector2 p0, Vector2 p1, float t)
+        {
+            return p1 - p0;
+        }
+
+
+        public static Vector2 Quadratic(Vector2 p0, Vector2 p1, Vector2 p2, float t)
+        {
+            return 2*(1 - t)*(p1 - p0) + 2*t*(p2 - p1);
+        }
+
+
+        public static Vector2 Cubic(Vector2 p0, Vector2 p1, Vector2 p2, Vector2 p3, float t)
+        {
+            return 3*(1 - t)*(1 - t)*(p1 - p0) + 6*(1 - t)*t*(p2 - p1) + 3*t*t*(p3 - p2);
+        }
+
+    }
+
+
+    public static class SecondDerivative
+    {
+
+        public static Vector2 Quadratic(Vector2 p0, Vector2 p1, Vector2 p2, float t)
+        {
+            return 2*(p2 - 2*p1 + p0);
+        }
+
+
+        public static Vector2 Cubic(Vector2 p0, Vector2 p1, Vector2 p2, Vector2 p3, float t)
+        {
+            return 6*(1 - t)*(p2 - 2*p1 + p0) + 6*t*(p3 - 2*p2 + p1);
+        }
+
+    }
+
+
+    // Parametric derivative order version for flexible use, less optimal.
+    public static class Derivative
+    {
+        public static Vector2 Linear(Vector2 p0, Vector2 p1, float t, int order)
+        {
+            switch (order)
+            {
+                case 1:
+                    return p1 - p0;
+                    break;
+
+                default:
+                    return Vector2.zero;
+            }
+        }
+
+
+        public static Vector2 Quadratic(Vector2 p0, Vector2 p1, Vector2 p2, float t, int order)
+        {
+            switch (order)
+            {
+                case 2:
+                    return 2*(p2 - 2*p1 + p0);
+                    break;
+
+                case 1:
+                    return 2*(1 - t)*(p1 - p0) + 2*t*(p2 - p1);
+                    break;
+
+                default:
+                    return Vector2.zero;
+            }
+        }
+
+
+        public static Vector2 Cubic(Vector2 p0, Vector2 p1, Vector2 p2, Vector2 p3, float t, int order)
+        {
+            switch (order)
+            {
+                case 3:
+                    return 6*(p3 - 3*p2 + 3*p1 - p0);
+                    break;
+
+                case 2:
+                    return 6*(1 - t)*(p2 - 2*p1 + p0) + 6*t*(p3 - 2*p2 + p1);
+                    break;
+
+                case 1:
+                    return 3*(1 - t)*(1 - t)*(p1 - p0) + 6*(1 - t)*t*(p2 - p1) + 3*t*t*(p3 - p2);
+                    break;
+
+                default:
+                    return Vector2.zero;
+            }
+        }
+    }
+
+
+    // If you really need an efficient algorithm, you can make a dictionary of PolynomialCoefficients.
+    public static Vector2[] PolynomialCoefficients(Vector2[] p)
+    {
+        Vector2[] c = new Vector2[p.Length];
+
+        int n = p.Length - 1;
+
+        for (int j = 0; j <= n; j++)
+        {
+            c[j] = Vector2.one;
+
+            // Pi part
+            int pi = 1;
+
+            for (int m = 0; m <= j - 1; m++)
+            {
+                pi *= n - m;
+            }
+
+            // Sigma part
+            Vector2 sigma = Vector2.zero;
+
+            for (int i = 0; i <= j; i++)
+            {
+                sigma += (Power(-1, i + j)*p[i]) / (Factorial(i)*Factorial(j - i));
+            }
+
+            c[j] = pi*sigma;
+        }
+
+        return c;
+    }
+
+
+    public static int Combination(int n, int i)
+    {
+        return Factorial(n) / (Factorial(i)*Factorial(n - i));
+    }
+
+
+    public static int Factorial(int n)
+    {
+        int y = 1;
+
+        for (int i = 1; i <= n; i++)
+        {
+            y *= i;
+        }
+
+        return y;
+    }
+
+
+    public static float Power(float b, int n)
+    {
+        if (n == 0)
+        {
+            return 1;
+        }
+        else
+        {
+            float y = b;
+
+            for (int i = 1; i <= n - 1; i++)
+            {
+                y *= y;
+            }
+
+            return y;
+        }
+    }
+}