e955c94ed3
Listing the "Blender Foundation" as copyright holder implied the Blender Foundation holds copyright to files which may include work from many developers. While keeping copyright on headers makes sense for isolated libraries, Blender's own code may be refactored or moved between files in a way that makes the per file copyright holders less meaningful. Copyright references to the "Blender Foundation" have been replaced with "Blender Authors", with the exception of `./extern/` since these this contains libraries which are more isolated, any changed to license headers there can be handled on a case-by-case basis. Some directories in `./intern/` have also been excluded: - `./intern/cycles/` it's own `AUTHORS` file is planned. - `./intern/opensubdiv/`. An "AUTHORS" file has been added, using the chromium projects authors file as a template. Design task: #110784 Ref !110783.
570 lines
16 KiB
C++
570 lines
16 KiB
C++
/* SPDX-FileCopyrightText: 2008-2022 Blender Authors
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*
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* SPDX-License-Identifier: GPL-2.0-or-later */
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/** \file
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* \ingroup freestyle
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* \brief An Algorithm for Automatically Fitting Digitized Curves by Philip J. Schneider,
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* \brief from "Graphics Gems", Academic Press, 1990
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*/
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#include <cmath>
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#include <cstdio>
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#include <cstdlib> // for malloc and free
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#include "FitCurve.h"
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#include "BLI_sys_types.h"
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using namespace std;
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namespace Freestyle {
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using BezierCurve = Vector2 *;
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/* Forward declarations */
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static double *Reparameterize(Vector2 *d, int first, int last, double *u, BezierCurve bezCurve);
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static double NewtonRaphsonRootFind(BezierCurve Q, Vector2 P, double u);
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static Vector2 BezierII(int degree, Vector2 *V, double t);
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static double B0(double u);
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static double B1(double u);
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static double B2(double u);
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static double B3(double u);
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static Vector2 ComputeLeftTangent(Vector2 *d, int end);
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static double ComputeMaxError(
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Vector2 *d, int first, int last, BezierCurve bezCurve, double *u, int *splitPoint);
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static double *ChordLengthParameterize(Vector2 *d, int first, int last);
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static BezierCurve GenerateBezier(
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Vector2 *d, int first, int last, double *uPrime, Vector2 tHat1, Vector2 tHat2);
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static Vector2 V2AddII(Vector2 a, Vector2 b);
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static Vector2 V2ScaleIII(Vector2 v, double s);
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static Vector2 V2SubII(Vector2 a, Vector2 b);
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/* returns squared length of input vector */
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static double V2SquaredLength(Vector2 *a)
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{
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return (((*a)[0] * (*a)[0]) + ((*a)[1] * (*a)[1]));
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}
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/* returns length of input vector */
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static double V2Length(Vector2 *a)
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{
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return sqrt(V2SquaredLength(a));
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}
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static Vector2 *V2Scale(Vector2 *v, double newlen)
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{
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double len = V2Length(v);
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if (len != 0.0) {
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(*v)[0] *= newlen / len;
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(*v)[1] *= newlen / len;
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}
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return v;
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}
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/* return the dot product of vectors a and b */
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static double V2Dot(Vector2 *a, Vector2 *b)
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{
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return (((*a)[0] * (*b)[0]) + ((*a)[1] * (*b)[1]));
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}
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/* return the distance between two points */
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static double V2DistanceBetween2Points(Vector2 *a, Vector2 *b)
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{
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double dx = (*a)[0] - (*b)[0];
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double dy = (*a)[1] - (*b)[1];
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return sqrt((dx * dx) + (dy * dy));
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}
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/* return vector sum c = a+b */
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static Vector2 *V2Add(Vector2 *a, Vector2 *b, Vector2 *c)
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{
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(*c)[0] = (*a)[0] + (*b)[0];
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(*c)[1] = (*a)[1] + (*b)[1];
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return c;
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}
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/* normalizes the input vector and returns it */
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static Vector2 *V2Normalize(Vector2 *v)
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{
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double len = V2Length(v);
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if (len != 0.0) {
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(*v)[0] /= len;
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(*v)[1] /= len;
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}
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return v;
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}
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/* negates the input vector and returns it */
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static Vector2 *V2Negate(Vector2 *v)
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{
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(*v)[0] = -(*v)[0];
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(*v)[1] = -(*v)[1];
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return v;
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}
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/* GenerateBezier:
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* Use least-squares method to find Bezier control points for region.
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* Vector2 *d; Array of digitized points
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* int first, last; Indices defining region
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* double *uPrime; Parameter values for region
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* Vector2 tHat1, tHat2; Unit tangents at endpoints
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*/
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static BezierCurve GenerateBezier(
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Vector2 *d, int first, int last, double *uPrime, Vector2 tHat1, Vector2 tHat2)
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{
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int i;
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Vector2 A[2]; /* rhs for eqn */
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int nPts; /* Number of pts in sub-curve */
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double C[2][2]; /* Matrix C */
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double X[2]; /* Matrix X */
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double det_C0_C1; /* Determinants of matrices */
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double det_C0_X;
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double det_X_C1;
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double alpha_l; /* Alpha values, left and right */
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double alpha_r;
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Vector2 tmp; /* Utility variable */
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BezierCurve bezCurve; /* RETURN bezier curve control points. */
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bezCurve = (Vector2 *)malloc(4 * sizeof(Vector2));
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nPts = last - first + 1;
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/* Create the C and X matrices */
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C[0][0] = 0.0;
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C[0][1] = 0.0;
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C[1][0] = 0.0;
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C[1][1] = 0.0;
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X[0] = 0.0;
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X[1] = 0.0;
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for (i = 0; i < nPts; i++) {
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/* Compute the A's */
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A[0] = tHat1;
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A[1] = tHat2;
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V2Scale(&A[0], B1(uPrime[i]));
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V2Scale(&A[1], B2(uPrime[i]));
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C[0][0] += V2Dot(&A[0], &A[0]);
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C[0][1] += V2Dot(&A[0], &A[1]);
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// C[1][0] += V2Dot(&A[0], &A[1]);
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C[1][0] = C[0][1];
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C[1][1] += V2Dot(&A[1], &A[1]);
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tmp = V2SubII(d[first + i],
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V2AddII(V2ScaleIII(d[first], B0(uPrime[i])),
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V2AddII(V2ScaleIII(d[first], B1(uPrime[i])),
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V2AddII(V2ScaleIII(d[last], B2(uPrime[i])),
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V2ScaleIII(d[last], B3(uPrime[i]))))));
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X[0] += V2Dot(&A[0], &tmp);
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X[1] += V2Dot(&A[1], &tmp);
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}
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/* Compute the determinants of C and X */
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det_C0_C1 = C[0][0] * C[1][1] - C[1][0] * C[0][1];
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det_C0_X = C[0][0] * X[1] - C[0][1] * X[0];
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det_X_C1 = X[0] * C[1][1] - X[1] * C[0][1];
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/* Finally, derive alpha values */
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if (det_C0_C1 == 0.0) {
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det_C0_C1 = (C[0][0] * C[1][1]) * 10.0e-12;
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}
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alpha_l = det_X_C1 / det_C0_C1;
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alpha_r = det_C0_X / det_C0_C1;
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/* If alpha negative, use the Wu/Barsky heuristic (see text) (if alpha is 0, you get coincident
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* control points that lead to divide by zero in any subsequent NewtonRaphsonRootFind() call).
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*/
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if (alpha_l < 1.0e-6 || alpha_r < 1.0e-6) {
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double dist = V2DistanceBetween2Points(&d[last], &d[first]) / 3.0;
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bezCurve[0] = d[first];
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bezCurve[3] = d[last];
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V2Add(&(bezCurve[0]), V2Scale(&(tHat1), dist), &(bezCurve[1]));
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V2Add(&(bezCurve[3]), V2Scale(&(tHat2), dist), &(bezCurve[2]));
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return bezCurve;
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}
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/* First and last control points of the Bezier curve are positioned exactly at the first and last
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* data points Control points 1 and 2 are positioned an alpha distance out on the tangent
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* vectors, left and right, respectively
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*/
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bezCurve[0] = d[first];
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bezCurve[3] = d[last];
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V2Add(&bezCurve[0], V2Scale(&tHat1, alpha_l), &bezCurve[1]);
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V2Add(&bezCurve[3], V2Scale(&tHat2, alpha_r), &bezCurve[2]);
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return bezCurve;
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}
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/*
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* Reparameterize:
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* Given set of points and their parameterization, try to find a better parameterization.
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* Vector2 *d; Array of digitized points
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* int first, last; Indices defining region
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* double *u; Current parameter values
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* BezierCurve bezCurve; Current fitted curve
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*/
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static double *Reparameterize(Vector2 *d, int first, int last, double *u, BezierCurve bezCurve)
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{
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int nPts = last - first + 1;
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int i;
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double *uPrime; /* New parameter values */
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uPrime = (double *)malloc(nPts * sizeof(double));
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for (i = first; i <= last; i++) {
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uPrime[i - first] = NewtonRaphsonRootFind(bezCurve, d[i], u[i - first]);
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}
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return uPrime;
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}
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/*
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* NewtonRaphsonRootFind:
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* Use Newton-Raphson iteration to find better root.
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* BezierCurve Q; Current fitted curve
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* Vector2 P; Digitized point
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* double u; Parameter value for "P"
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*/
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static double NewtonRaphsonRootFind(BezierCurve Q, Vector2 P, double u)
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{
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double numerator, denominator;
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Vector2 Q1[3], Q2[2]; /* Q' and Q'' */
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Vector2 Q_u, Q1_u, Q2_u; /* u evaluated at Q, Q', & Q'' */
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double uPrime; /* Improved u */
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int i;
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/* Compute Q(u) */
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Q_u = BezierII(3, Q, u);
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/* Generate control vertices for Q' */
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for (i = 0; i <= 2; i++) {
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Q1[i][0] = (Q[i + 1][0] - Q[i][0]) * 3.0;
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Q1[i][1] = (Q[i + 1][1] - Q[i][1]) * 3.0;
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}
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/* Generate control vertices for Q'' */
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for (i = 0; i <= 1; i++) {
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Q2[i][0] = (Q1[i + 1][0] - Q1[i][0]) * 2.0;
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Q2[i][1] = (Q1[i + 1][1] - Q1[i][1]) * 2.0;
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}
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/* Compute Q'(u) and Q''(u) */
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Q1_u = BezierII(2, Q1, u);
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Q2_u = BezierII(1, Q2, u);
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/* Compute f(u)/f'(u) */
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numerator = (Q_u[0] - P[0]) * (Q1_u[0]) + (Q_u[1] - P[1]) * (Q1_u[1]);
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denominator = (Q1_u[0]) * (Q1_u[0]) + (Q1_u[1]) * (Q1_u[1]) + (Q_u[0] - P[0]) * (Q2_u[0]) +
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(Q_u[1] - P[1]) * (Q2_u[1]);
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/* u = u - f(u)/f'(u) */
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if (denominator == 0) { // FIXME
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return u;
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}
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uPrime = u - (numerator / denominator);
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return uPrime;
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}
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/*
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* Bezier:
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* Evaluate a Bezier curve at a particular parameter value
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* int degree; The degree of the bezier curve
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* Vector2 *V; Array of control points
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* double t; Parametric value to find point for
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*/
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static Vector2 BezierII(int degree, Vector2 *V, double t)
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{
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int i, j;
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Vector2 Q; /* Point on curve at parameter t */
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Vector2 *Vtemp; /* Local copy of control points */
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/* Copy array */
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Vtemp = (Vector2 *)malloc(uint((degree + 1) * sizeof(Vector2)));
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for (i = 0; i <= degree; i++) {
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Vtemp[i] = V[i];
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}
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/* Triangle computation */
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for (i = 1; i <= degree; i++) {
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for (j = 0; j <= degree - i; j++) {
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Vtemp[j][0] = (1.0 - t) * Vtemp[j][0] + t * Vtemp[j + 1][0];
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Vtemp[j][1] = (1.0 - t) * Vtemp[j][1] + t * Vtemp[j + 1][1];
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}
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}
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Q = Vtemp[0];
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free((void *)Vtemp);
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return Q;
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}
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/*
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* B0, B1, B2, B3:
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* Bezier multipliers
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*/
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static double B0(double u)
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{
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double tmp = 1.0 - u;
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return (tmp * tmp * tmp);
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}
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static double B1(double u)
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{
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double tmp = 1.0 - u;
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return (3 * u * (tmp * tmp));
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}
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static double B2(double u)
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{
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double tmp = 1.0 - u;
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return (3 * u * u * tmp);
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}
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static double B3(double u)
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{
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return (u * u * u);
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}
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/*
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* ComputeLeftTangent, ComputeRightTangent, ComputeCenterTangent:
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* Approximate unit tangents at endpoints and "center" of digitized curve
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*/
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/* Vector2 *d; Digitized points
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* int end; Index to "left" end of region
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*/
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static Vector2 ComputeLeftTangent(Vector2 *d, int end)
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{
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Vector2 tHat1;
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tHat1 = V2SubII(d[end + 1], d[end]);
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tHat1 = *V2Normalize(&tHat1);
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return tHat1;
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}
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/* Vector2 *d; Digitized points
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* int end; Index to "right" end of region
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*/
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static Vector2 ComputeRightTangent(Vector2 *d, int end)
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{
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Vector2 tHat2;
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tHat2 = V2SubII(d[end - 1], d[end]);
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tHat2 = *V2Normalize(&tHat2);
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return tHat2;
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}
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/* Vector2 *d; Digitized points
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* int end; Index to point inside region
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*/
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static Vector2 ComputeCenterTangent(Vector2 *d, int center)
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{
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Vector2 V1, V2, tHatCenter;
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V1 = V2SubII(d[center - 1], d[center]);
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V2 = V2SubII(d[center], d[center + 1]);
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tHatCenter[0] = (V1[0] + V2[0]) / 2.0;
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tHatCenter[1] = (V1[1] + V2[1]) / 2.0;
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tHatCenter = *V2Normalize(&tHatCenter);
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/* avoid numerical singularity in the special case when V1 == -V2 */
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if (V2Length(&tHatCenter) < M_EPSILON) {
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tHatCenter = *V2Normalize(&V1);
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}
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return tHatCenter;
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}
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/*
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* ChordLengthParameterize:
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* Assign parameter values to digitized points using relative distances between points.
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* Vector2 *d; Array of digitized points
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* int first, last; Indices defining region
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*/
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static double *ChordLengthParameterize(Vector2 *d, int first, int last)
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{
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int i;
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double *u; /* Parameterization */
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u = (double *)malloc(uint(last - first + 1) * sizeof(double));
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u[0] = 0.0;
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for (i = first + 1; i <= last; i++) {
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u[i - first] = u[i - first - 1] + V2DistanceBetween2Points(&d[i], &d[i - 1]);
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}
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for (i = first + 1; i <= last; i++) {
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u[i - first] = u[i - first] / u[last - first];
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}
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return u;
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}
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/*
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* ComputeMaxError :
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* Find the maximum squared distance of digitized points to fitted curve.
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* Vector2 *d; Array of digitized points
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* int first, last; Indices defining region
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* BezierCurve bezCurve; Fitted Bezier curve
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* double *u; Parameterization of points
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* int *splitPoint; Point of maximum error
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*/
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static double ComputeMaxError(
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Vector2 *d, int first, int last, BezierCurve bezCurve, double *u, int *splitPoint)
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{
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int i;
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double maxDist; /* Maximum error */
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double dist; /* Current error */
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Vector2 P; /* Point on curve */
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Vector2 v; /* Vector from point to curve */
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*splitPoint = (last - first + 1) / 2;
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maxDist = 0.0;
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for (i = first + 1; i < last; i++) {
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P = BezierII(3, bezCurve, u[i - first]);
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v = V2SubII(P, d[i]);
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dist = V2SquaredLength(&v);
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if (dist >= maxDist) {
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maxDist = dist;
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*splitPoint = i;
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}
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}
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return maxDist;
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}
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static Vector2 V2AddII(Vector2 a, Vector2 b)
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{
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Vector2 c;
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c[0] = a[0] + b[0];
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c[1] = a[1] + b[1];
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return c;
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}
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static Vector2 V2ScaleIII(Vector2 v, double s)
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{
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Vector2 result;
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result[0] = v[0] * s;
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result[1] = v[1] * s;
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return result;
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}
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static Vector2 V2SubII(Vector2 a, Vector2 b)
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{
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Vector2 c;
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c[0] = a[0] - b[0];
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c[1] = a[1] - b[1];
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return c;
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}
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//------------------------- WRAPPER -----------------------------//
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FitCurveWrapper::~FitCurveWrapper()
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{
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_vertices.clear();
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}
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void FitCurveWrapper::DrawBezierCurve(int n, Vector2 *curve)
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{
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for (int i = 0; i <= n; ++i) {
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_vertices.push_back(curve[i]);
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}
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}
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void FitCurveWrapper::FitCurve(vector<Vec2d> &data, vector<Vec2d> &oCurve, double error)
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{
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int size = data.size();
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Vector2 *d = new Vector2[size];
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for (int i = 0; i < size; ++i) {
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|
d[i][0] = data[i][0];
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|
d[i][1] = data[i][1];
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|
}
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|
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|
FitCurve(d, size, error);
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|
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|
delete[] d;
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|
|
|
// copy results
|
|
for (vector<Vector2>::iterator v = _vertices.begin(), vend = _vertices.end(); v != vend; ++v) {
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|
oCurve.emplace_back(v->x(), v->y());
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|
}
|
|
}
|
|
|
|
void FitCurveWrapper::FitCurve(Vector2 *d, int nPts, double error)
|
|
{
|
|
Vector2 tHat1, tHat2; /* Unit tangent vectors at endpoints */
|
|
|
|
tHat1 = ComputeLeftTangent(d, 0);
|
|
tHat2 = ComputeRightTangent(d, nPts - 1);
|
|
FitCubic(d, 0, nPts - 1, tHat1, tHat2, error);
|
|
}
|
|
|
|
void FitCurveWrapper::FitCubic(
|
|
Vector2 *d, int first, int last, Vector2 tHat1, Vector2 tHat2, double error)
|
|
{
|
|
BezierCurve bezCurve; /* Control points of fitted Bezier curve */
|
|
double *u; /* Parameter values for point */
|
|
double *uPrime; /* Improved parameter values */
|
|
double maxError; /* Maximum fitting error */
|
|
int splitPoint; /* Point to split point set at */
|
|
int nPts; /* Number of points in subset */
|
|
double iterationError; /* Error below which you try iterating */
|
|
int maxIterations = 4; /* Max times to try iterating */
|
|
Vector2 tHatCenter; /* Unit tangent vector at splitPoint */
|
|
int i;
|
|
|
|
iterationError = error * error;
|
|
nPts = last - first + 1;
|
|
|
|
/* Use heuristic if region only has two points in it */
|
|
if (nPts == 2) {
|
|
double dist = V2DistanceBetween2Points(&d[last], &d[first]) / 3.0;
|
|
|
|
bezCurve = (Vector2 *)malloc(4 * sizeof(Vector2));
|
|
bezCurve[0] = d[first];
|
|
bezCurve[3] = d[last];
|
|
V2Add(&bezCurve[0], V2Scale(&tHat1, dist), &bezCurve[1]);
|
|
V2Add(&bezCurve[3], V2Scale(&tHat2, dist), &bezCurve[2]);
|
|
DrawBezierCurve(3, bezCurve);
|
|
free((void *)bezCurve);
|
|
return;
|
|
}
|
|
|
|
/* Parameterize points, and attempt to fit curve */
|
|
u = ChordLengthParameterize(d, first, last);
|
|
bezCurve = GenerateBezier(d, first, last, u, tHat1, tHat2);
|
|
|
|
/* Find max deviation of points to fitted curve */
|
|
maxError = ComputeMaxError(d, first, last, bezCurve, u, &splitPoint);
|
|
if (maxError < error) {
|
|
DrawBezierCurve(3, bezCurve);
|
|
free((void *)u);
|
|
free((void *)bezCurve);
|
|
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);
|
|
|
|
free((void *)u);
|
|
free((void *)bezCurve);
|
|
u = uPrime;
|
|
|
|
bezCurve = GenerateBezier(d, first, last, u, tHat1, tHat2);
|
|
maxError = ComputeMaxError(d, first, last, bezCurve, u, &splitPoint);
|
|
|
|
if (maxError < error) {
|
|
DrawBezierCurve(3, bezCurve);
|
|
free((void *)u);
|
|
free((void *)bezCurve);
|
|
return;
|
|
}
|
|
}
|
|
}
|
|
|
|
/* Fitting failed -- split at max error point and fit recursively */
|
|
free((void *)u);
|
|
free((void *)bezCurve);
|
|
tHatCenter = ComputeCenterTangent(d, splitPoint);
|
|
FitCubic(d, first, splitPoint, tHat1, tHatCenter, error);
|
|
V2Negate(&tHatCenter);
|
|
FitCubic(d, splitPoint, last, tHatCenter, tHat2, error);
|
|
}
|
|
|
|
} /* namespace Freestyle */
|