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CubicBezierFitter.cs
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CubicBezierFitter.cs
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// Porting to Unity by PeDev 2020
// The origin is from: https://stackoverflow.com/questions/5525665/smoothing-a-hand-drawn-curve
/*
An Algorithm for Automatically Fitting Digitized Curves
by Philip J. Schneider
from "Graphics Gems", Academic Press, 1990
*/
using System.Collections.Generic;
using UnityEngine;
namespace EasingCurve {
/// <summary>
/// To make cubic bezier spline by data.
/// </summary>
public static class CubicBezierFitter {
private const int MAX_DATA_COUNT = 100000;
private const float MIN_ERROR = 1e-06f;
public static List<Vector2> FitCurve(Vector2[] d, float error) {
Vector2 tHat1, tHat2; /* Unit tangent vectors at endVector2s */
tHat1 = ComputeLeftTangent(d, 0);
tHat2 = ComputeRightTangent(d, d.Length - 1);
List<Vector2> result = new List<Vector2>() {
new Vector2(0, 0) // The first cp is always (0, 0)
};
FitCubic(d, 0, d.Length - 1, tHat1, tHat2, error, result);
return result;
}
private static void FitCubic(Vector2[] d, int first, int last, Vector2 tHat1, Vector2 tHat2, float error, List<Vector2> result) {
Vector2[] bezCurve; /*Control Vector2s of fitted Bezier curve*/
float[] u; /* Parameter values for Vector2 */
float[] uPrime; /* Improved parameter values */
float maxError; /* Maximum fitting error */
int splitVector2; /* Vector2 to split Vector2 set at */
int nPts; /* Number of Vector2s in subset */
float iterationError; /*Error below which you try iterating */
int maxIterations = 4; /* Max times to try iterating */
Vector2 tHatCenter; /* Unit tangent vector at splitVector2 */
int i;
error = Mathf.Max(error, MIN_ERROR);
iterationError = error * error;
nPts = last - first + 1;
/* Use heuristic if region only has two Vector2s in it */
if (nPts == 2) {
float dist = (d[first]-d[last]).magnitude / 3.0f;
bezCurve = new Vector2[4];
bezCurve[0] = d[first];
bezCurve[3] = d[last];
bezCurve[1] = (tHat1 * dist) + bezCurve[0];
bezCurve[2] = (tHat2 * dist) + bezCurve[3];
result.Add(bezCurve[1]);
result.Add(bezCurve[2]);
result.Add(bezCurve[3]);
return;
}
/* Parameterize Vector2s, and attempt to fit curve */
u = ChordLengthParameterize(d, first, last);
bezCurve = GenerateBezier(d, first, last, u, tHat1, tHat2);
/* Find max deviation of Vector2s to fitted curve */
maxError = ComputeMaxError(d, first, last, bezCurve, u, out splitVector2);
if (maxError < error) {
result.Add(bezCurve[1]);
result.Add(bezCurve[2]);
result.Add(bezCurve[3]);
return;
}
/* If error not too large, try some reparameterization */
/* and iteration */
if (maxError < iterationError) {
for (i = 0; i < maxIterations; i++) {
uPrime = Reparameterize(d, first, last, u, bezCurve);
bezCurve = GenerateBezier(d, first, last, uPrime, tHat1, tHat2);
maxError = ComputeMaxError(d, first, last,
bezCurve, uPrime, out splitVector2);
if (maxError < error) {
result.Add(bezCurve[1]);
result.Add(bezCurve[2]);
result.Add(bezCurve[3]);
return;
}
u = uPrime;
}
}
/* Fitting failed -- split at max error Vector2 and fit recursively */
tHatCenter = ComputeCenterTangent(d, splitVector2);
FitCubic(d, first, splitVector2, tHat1, tHatCenter, error, result);
tHatCenter = -tHatCenter;
FitCubic(d, splitVector2, last, tHatCenter, tHat2, error, result);
}
static Vector2[] GenerateBezier(Vector2[] d, int first, int last, float[] uPrime, Vector2 tHat1, Vector2 tHat2) {
int i;
Vector2[,] A = new Vector2[MAX_DATA_COUNT,2];/* Precomputed rhs for eqn */
int nPts; /* Number of pts in sub-curve */
float[,] C = new float[2,2]; /* Matrix C */
float[] X = new float[2]; /* Matrix X */
float det_C0_C1, /* Determinants of matrices */
det_C0_X,
det_X_C1;
float alpha_l, /* Alpha values, left and right */
alpha_r;
Vector2 tmp; /* Utility variable */
Vector2[] bezCurve = new Vector2[4]; /* RETURN bezier curve ctl pts */
nPts = last - first + 1;
/* Compute the A's */
for (i = 0; i < nPts; i++) {
Vector2 v1, v2;
v1 = tHat1;
v2 = tHat2;
v1 *= B1(uPrime[i]);
v2 *= B2(uPrime[i]);
A[i, 0] = v1;
A[i, 1] = v2;
}
/* Create the C and X matrices */
C[0, 0] = 0.0f;
C[0, 1] = 0.0f;
C[1, 0] = 0.0f;
C[1, 1] = 0.0f;
X[0] = 0.0f;
X[1] = 0.0f;
for (i = 0; i < nPts; i++) {
C[0, 0] += V2Dot(A[i, 0], A[i, 0]);
C[0, 1] += V2Dot(A[i, 0], A[i, 1]);
/* C[1][0] += V2Dot(&A[i][0], &A[i][9]);*/
C[1, 0] = C[0, 1];
C[1, 1] += V2Dot(A[i, 1], A[i, 1]);
tmp = ((Vector2)d[first + i] -
(
((Vector2)d[first] * B0(uPrime[i])) +
(
((Vector2)d[first] * B1(uPrime[i])) +
(
((Vector2)d[last] * B2(uPrime[i])) +
((Vector2)d[last] * B3(uPrime[i]))))));
X[0] += V2Dot(A[i, 0], tmp);
X[1] += V2Dot(A[i, 1], tmp);
}
/* Compute the determinants of C and X */
det_C0_C1 = C[0, 0] * C[1, 1] - C[1, 0] * C[0, 1];
det_C0_X = C[0, 0] * X[1] - C[1, 0] * X[0];
det_X_C1 = X[0] * C[1, 1] - X[1] * C[0, 1];
/* Finally, derive alpha values */
alpha_l = (det_C0_C1 == 0) ? 0.0f : det_X_C1 / det_C0_C1;
alpha_r = (det_C0_C1 == 0) ? 0.0f : det_C0_X / det_C0_C1;
/* If alpha negative, use the Wu/Barsky heuristic (see text) */
/* (if alpha is 0, you get coincident control Vector2s that lead to
* divide by zero in any subsequent NewtonRaphsonRootFind() call. */
float segLength = (d[first] - d[last]).magnitude;
float epsilon = 1.0e-6f * segLength;
if (alpha_l < epsilon || alpha_r < epsilon) {
/* fall back on standard (probably inaccurate) formula, and subdivide further if needed. */
float dist = segLength / 3.0f;
bezCurve[0] = d[first];
bezCurve[3] = d[last];
bezCurve[1] = (tHat1 * dist) + bezCurve[0];
bezCurve[2] = (tHat2 * dist) + bezCurve[3];
return (bezCurve);
}
/* First and last control Vector2s of the Bezier curve are */
/* positioned exactly at the first and last data Vector2s */
/* Control Vector2s 1 and 2 are positioned an alpha distance out */
/* on the tangent vectors, left and right, respectively */
bezCurve[0] = d[first];
bezCurve[3] = d[last];
bezCurve[1] = (tHat1 * alpha_l) + bezCurve[0];
bezCurve[2] = (tHat2 * alpha_r) + bezCurve[3];
return (bezCurve);
}
/*
* Reparameterize:
* Given set of Vector2s and their parameterization, try to find
* a better parameterization.
*
*/
static float[] Reparameterize(Vector2[] d, int first, int last, float[] u, Vector2[] bezCurve) {
int nPts = last-first+1;
int i;
float[] uPrime = new float[nPts]; /* New parameter values */
for (i = first; i <= last; i++) {
uPrime[i - first] = NewtonRaphsonRootFind(bezCurve, d[i], u[i - first]);
}
return uPrime;
}
/*
* NewtonRaphsonRootFind :
* Use Newton-Raphson iteration to find better root.
*/
static float NewtonRaphsonRootFind(Vector2[] Q, Vector2 P, float u) {
float numerator, denominator;
Vector2[] Q1 = new Vector2[3], Q2 = new Vector2[2]; /* Q' and Q'' */
Vector2 Q_u, Q1_u, Q2_u; /*u evaluated at Q, Q', & Q'' */
float uPrime; /* Improved u */
int i;
/* Compute Q(u) */
Q_u = BezierII(3, Q, u);
/* Generate control vertices for Q' */
for (i = 0; i <= 2; i++) {
Q1[i].x = (Q[i + 1].x - Q[i].x) * 3.0f;
Q1[i].y = (Q[i + 1].y - Q[i].y) * 3.0f;
}
/* Generate control vertices for Q'' */
for (i = 0; i <= 1; i++) {
Q2[i].x = (Q1[i + 1].x - Q1[i].x) * 2.0f;
Q2[i].y = (Q1[i + 1].y - Q1[i].y) * 2.0f;
}
/* Compute Q'(u) and Q''(u) */
Q1_u = BezierII(2, Q1, u);
Q2_u = BezierII(1, Q2, u);
/* Compute f(u)/f'(u) */
numerator = (Q_u.x - P.x) * (Q1_u.x) + (Q_u.y - P.y) * (Q1_u.y);
denominator = (Q1_u.x) * (Q1_u.x) + (Q1_u.y) * (Q1_u.y) +
(Q_u.x - P.x) * (Q2_u.x) + (Q_u.y - P.y) * (Q2_u.y);
if (denominator == 0.0f) return u;
/* u = u - f(u)/f'(u) */
uPrime = u - (numerator / denominator);
return (uPrime);
}
/*
* Bezier :
* Evaluate a Bezier curve at a particular parameter value
*
*/
static Vector2 BezierII(int degree, Vector2[] V, float t) {
int i, j;
Vector2 Q; /* Vector2 on curve at parameter t */
Vector2[] Vtemp; /* Local copy of control Vector2s */
/* Copy array */
Vtemp = new Vector2[degree + 1];
for (i = 0; i <= degree; i++) {
Vtemp[i] = V[i];
}
/* Triangle computation */
for (i = 1; i <= degree; i++) {
for (j = 0; j <= degree - i; j++) {
Vtemp[j].x = (1.0f - t) * Vtemp[j].x + t * Vtemp[j + 1].x;
Vtemp[j].y = (1.0f - t) * Vtemp[j].y + t * Vtemp[j + 1].y;
}
}
Q = Vtemp[0];
return Q;
}
/*
* B0, B1, B2, B3 :
* Bezier multipliers
*/
static float B0(float u) {
float tmp = 1.0f - u;
return (tmp * tmp * tmp);
}
static float B1(float u) {
float tmp = 1.0f - u;
return (3 * u * (tmp * tmp));
}
static float B2(float u) {
float tmp = 1.0f - u;
return (3 * u * u * tmp);
}
static float B3(float u) {
return (u * u * u);
}
/*
* ComputeLeftTangent, ComputeRightTangent, ComputeCenterTangent :
*Approximate unit tangents at endVector2s and "center" of digitized curve
*/
static Vector2 ComputeLeftTangent(Vector2[] d, int end) {
Vector2 tHat1;
tHat1 = d[end + 1] - d[end];
tHat1.Normalize();
return tHat1;
}
static Vector2 ComputeRightTangent(Vector2[] d, int end) {
Vector2 tHat2;
tHat2 = d[end - 1] - d[end];
tHat2.Normalize();
return tHat2;
}
static Vector2 ComputeCenterTangent(Vector2[] d, int center) {
Vector2 V1, V2, tHatCenter = new Vector2();
V1 = d[center - 1] - d[center];
V2 = d[center] - d[center + 1];
tHatCenter.x = (V1.x + V2.x) / 2.0f;
tHatCenter.y = (V1.y + V2.y) / 2.0f;
tHatCenter.Normalize();
return tHatCenter;
}
/*
* ChordLengthParameterize :
* Assign parameter values to digitized Vector2s
* using relative distances between Vector2s.
*/
static float[] ChordLengthParameterize(Vector2[] d, int first, int last) {
int i;
float[] u = new float[last-first+1]; /* Parameterization */
u[0] = 0.0f;
for (i = first + 1; i <= last; i++) {
u[i - first] = u[i - first - 1] + (d[i - 1] - d[i]).magnitude;
}
for (i = first + 1; i <= last; i++) {
u[i - first] = u[i - first] / u[last - first];
}
return u;
}
/*
* ComputeMaxError :
* Find the maximum squared distance of digitized Vector2s
* to fitted curve.
*/
static float ComputeMaxError(Vector2[] d, int first, int last, Vector2[] bezCurve, float[] u, out int splitVector2) {
int i;
float maxDist; /* Maximum error */
float dist; /* Current error */
Vector2 P; /* Vector2 on curve */
Vector2 v; /* Vector2 from Vector2 to curve */
splitVector2 = (last - first + 1) / 2;
maxDist = 0.0f;
for (i = first + 1; i < last; i++) {
P = BezierII(3, bezCurve, u[i - first]);
v = P - d[i];
dist = v.sqrMagnitude;
if (dist >= maxDist) {
maxDist = dist;
splitVector2 = i;
}
}
return maxDist;
}
private static float V2Dot(Vector2 a, Vector2 b) {
return ((a.x * b.x) + (a.y * b.y));
}
}
}