19e49fb63d355446b91d20ff78ad78b297e89a50dcaryclark@google.com/* 29e49fb63d355446b91d20ff78ad78b297e89a50dcaryclark@google.com * Copyright 2012 Google Inc. 39e49fb63d355446b91d20ff78ad78b297e89a50dcaryclark@google.com * 49e49fb63d355446b91d20ff78ad78b297e89a50dcaryclark@google.com * Use of this source code is governed by a BSD-style license that can be 59e49fb63d355446b91d20ff78ad78b297e89a50dcaryclark@google.com * found in the LICENSE file. 69e49fb63d355446b91d20ff78ad78b297e89a50dcaryclark@google.com */ 7c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com#include "CubicUtilities.h" 845a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com#include "Extrema.h" 91304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com#include "LineUtilities.h" 10c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com#include "QuadraticUtilities.h" 11c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com 127ff5c841bf669826b4cbd708ae1a6b3527f15dcacaryclark@google.comconst int gPrecisionUnit = 256; // FIXME: arbitrary -- should try different values in test framework 1305c4bad470722bc4e5e6ae3d79aa8bcf9e732f06caryclark@google.com 1405c4bad470722bc4e5e6ae3d79aa8bcf9e732f06caryclark@google.com// FIXME: cache keep the bounds and/or precision with the caller? 1573ca6243b31e225e9fd5b75a96cbc82d62557de6caryclark@google.comdouble calcPrecision(const Cubic& cubic) { 1673ca6243b31e225e9fd5b75a96cbc82d62557de6caryclark@google.com _Rect dRect; 1705c4bad470722bc4e5e6ae3d79aa8bcf9e732f06caryclark@google.com dRect.setBounds(cubic); // OPTIMIZATION: just use setRawBounds ? 1873ca6243b31e225e9fd5b75a96cbc82d62557de6caryclark@google.com double width = dRect.right - dRect.left; 1973ca6243b31e225e9fd5b75a96cbc82d62557de6caryclark@google.com double height = dRect.bottom - dRect.top; 207ff5c841bf669826b4cbd708ae1a6b3527f15dcacaryclark@google.com return (width > height ? width : height) / gPrecisionUnit; 2173ca6243b31e225e9fd5b75a96cbc82d62557de6caryclark@google.com} 2273ca6243b31e225e9fd5b75a96cbc82d62557de6caryclark@google.com 23753b870c62bd22cee3d9a15efc634822724084acmtklein#ifdef SK_DEBUG 249d5f99bc309a7d733e33a149bef295ae3c8b3ac1caryclark@google.comdouble calcPrecision(const Cubic& cubic, double t, double scale) { 259d5f99bc309a7d733e33a149bef295ae3c8b3ac1caryclark@google.com Cubic part; 269f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com sub_divide(cubic, SkTMax(0., t - scale), SkTMin(1., t + scale), part); 279d5f99bc309a7d733e33a149bef295ae3c8b3ac1caryclark@google.com return calcPrecision(part); 289d5f99bc309a7d733e33a149bef295ae3c8b3ac1caryclark@google.com} 299d5f99bc309a7d733e33a149bef295ae3c8b3ac1caryclark@google.com#endif 309d5f99bc309a7d733e33a149bef295ae3c8b3ac1caryclark@google.com 311304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.combool clockwise(const Cubic& c) { 321304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com double sum = (c[0].x - c[3].x) * (c[0].y + c[3].y); 331304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com for (int idx = 0; idx < 3; ++idx){ 341304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com sum += (c[idx + 1].x - c[idx].x) * (c[idx + 1].y + c[idx].y); 351304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com } 361304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com return sum <= 0; 371304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com} 389d5f99bc309a7d733e33a149bef295ae3c8b3ac1caryclark@google.com 39c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.comvoid coefficients(const double* cubic, double& A, double& B, double& C, double& D) { 40c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com A = cubic[6]; // d 41c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com B = cubic[4] * 3; // 3*c 42c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com C = cubic[2] * 3; // 3*b 43c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com D = cubic[0]; // a 44c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com A -= D - C + B; // A = -a + 3*b - 3*c + d 45c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com B += 3 * D - 2 * C; // B = 3*a - 6*b + 3*c 46c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com C -= 3 * D; // C = -3*a + 3*b 47c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com} 48c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com 491304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.combool controls_contained_by_ends(const Cubic& c) { 501304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com _Vector startTan = c[1] - c[0]; 511304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com if (startTan.x == 0 && startTan.y == 0) { 521304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com startTan = c[2] - c[0]; 531304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com } 541304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com _Vector endTan = c[2] - c[3]; 551304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com if (endTan.x == 0 && endTan.y == 0) { 561304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com endTan = c[1] - c[3]; 571304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com } 581304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com if (startTan.dot(endTan) >= 0) { 591304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com return false; 601304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com } 611304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com _Line startEdge = {c[0], c[0]}; 621304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com startEdge[1].x -= startTan.y; 631304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com startEdge[1].y += startTan.x; 641304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com _Line endEdge = {c[3], c[3]}; 651304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com endEdge[1].x -= endTan.y; 661304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com endEdge[1].y += endTan.x; 671304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com double leftStart1 = is_left(startEdge, c[1]); 681304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com if (leftStart1 * is_left(startEdge, c[2]) < 0) { 691304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com return false; 701304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com } 711304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com double leftEnd1 = is_left(endEdge, c[1]); 721304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com if (leftEnd1 * is_left(endEdge, c[2]) < 0) { 731304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com return false; 741304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com } 751304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com return leftStart1 * leftEnd1 >= 0; 761304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com} 771304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com 781304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.combool ends_are_extrema_in_x_or_y(const Cubic& c) { 791304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com return (between(c[0].x, c[1].x, c[3].x) && between(c[0].x, c[2].x, c[3].x)) 801304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com || (between(c[0].y, c[1].y, c[3].y) && between(c[0].y, c[2].y, c[3].y)); 811304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com} 821304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com 831304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.combool monotonic_in_y(const Cubic& c) { 841304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com return between(c[0].y, c[1].y, c[3].y) && between(c[0].y, c[2].y, c[3].y); 851304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com} 861304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com 871304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.combool serpentine(const Cubic& c) { 881304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com if (!controls_contained_by_ends(c)) { 891304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com return false; 901304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com } 911304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com double wiggle = (c[0].x - c[2].x) * (c[0].y + c[2].y); 921304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com for (int idx = 0; idx < 2; ++idx){ 931304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com wiggle += (c[idx + 1].x - c[idx].x) * (c[idx + 1].y + c[idx].y); 941304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com } 951304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com double waggle = (c[1].x - c[3].x) * (c[1].y + c[3].y); 961304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com for (int idx = 1; idx < 3; ++idx){ 971304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com waggle += (c[idx + 1].x - c[idx].x) * (c[idx + 1].y + c[idx].y); 981304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com } 991304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com return wiggle * waggle < 0; 1001304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com} 1011304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com 102c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com// cubic roots 103c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com 104c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.comconst double PI = 4 * atan(1); 105c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com 106c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com// from SkGeometry.cpp (and Numeric Solutions, 5.6) 1079f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.comint cubicRootsValidT(double A, double B, double C, double D, double t[3]) { 1089f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com#if 0 109c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com if (approximately_zero(A)) { // we're just a quadratic 1109f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com return quadraticRootsValidT(B, C, D, t); 111c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com } 112c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com double a, b, c; 113c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com { 114c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com double invA = 1 / A; 115c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com a = B * invA; 116c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com b = C * invA; 117c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com c = D * invA; 118c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com } 119c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com double a2 = a * a; 120c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com double Q = (a2 - b * 3) / 9; 121c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com double R = (2 * a2 * a - 9 * a * b + 27 * c) / 54; 122c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com double Q3 = Q * Q * Q; 123c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com double R2MinusQ3 = R * R - Q3; 124c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com double adiv3 = a / 3; 125c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com double* roots = t; 126c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com double r; 127c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com 128c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com if (R2MinusQ3 < 0) // we have 3 real roots 129c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com { 130c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com double theta = acos(R / sqrt(Q3)); 131c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com double neg2RootQ = -2 * sqrt(Q); 132c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com 133c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com r = neg2RootQ * cos(theta / 3) - adiv3; 134c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com if (is_unit_interval(r)) 135c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com *roots++ = r; 136c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com 137c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com r = neg2RootQ * cos((theta + 2 * PI) / 3) - adiv3; 138c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com if (is_unit_interval(r)) 139c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com *roots++ = r; 140c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com 141c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com r = neg2RootQ * cos((theta - 2 * PI) / 3) - adiv3; 142c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com if (is_unit_interval(r)) 143c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com *roots++ = r; 144c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com } 145c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com else // we have 1 real root 146c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com { 147c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com double A = fabs(R) + sqrt(R2MinusQ3); 148c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com A = cube_root(A); 149c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com if (R > 0) { 150c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com A = -A; 151c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com } 152c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com if (A != 0) { 153c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com A += Q / A; 154c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com } 155c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com r = A - adiv3; 156c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com if (is_unit_interval(r)) 157c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com *roots++ = r; 158c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com } 159c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com return (int)(roots - t); 1609f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com#else 1619f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com double s[3]; 1629f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com int realRoots = cubicRootsReal(A, B, C, D, s); 1639f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com int foundRoots = add_valid_ts(s, realRoots, t); 1649f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com return foundRoots; 1659f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com#endif 1669f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com} 1679f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com 1689f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.comint cubicRootsReal(double A, double B, double C, double D, double s[3]) { 169753b870c62bd22cee3d9a15efc634822724084acmtklein#ifdef SK_DEBUG 1709f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com // create a string mathematica understands 1719f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com // GDB set print repe 15 # if repeated digits is a bother 1729f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com // set print elements 400 # if line doesn't fit 1739f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com char str[1024]; 1749f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com bzero(str, sizeof(str)); 1759f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com sprintf(str, "Solve[%1.19g x^3 + %1.19g x^2 + %1.19g x + %1.19g == 0, x]", A, B, C, D); 1765e0500fb5f17fe14db42fc3e0aad08e6b41ccc5fcaryclark@google.com mathematica_ize(str, sizeof(str)); 1774aaaaeace7e617ddc473645756fb7c20790bc270caryclark@google.com#if ONE_OFF_DEBUG && ONE_OFF_DEBUG_MATHEMATICA 1785e0500fb5f17fe14db42fc3e0aad08e6b41ccc5fcaryclark@google.com SkDebugf("%s\n", str); 1795e0500fb5f17fe14db42fc3e0aad08e6b41ccc5fcaryclark@google.com#endif 1809f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com#endif 18145a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com if (approximately_zero(A) 18245a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com && approximately_zero_when_compared_to(A, B) 183beda389e646d6be3cfef853584a78ca8ba39d0fccaryclark@google.com && approximately_zero_when_compared_to(A, C) 184beda389e646d6be3cfef853584a78ca8ba39d0fccaryclark@google.com && approximately_zero_when_compared_to(A, D)) { // we're just a quadratic 1859f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com return quadraticRootsReal(B, C, D, s); 1869f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com } 187f9502d7dfad5b361a3cdaa42eb75b593c95f79d8caryclark@google.com if (approximately_zero_when_compared_to(D, A) 188f9502d7dfad5b361a3cdaa42eb75b593c95f79d8caryclark@google.com && approximately_zero_when_compared_to(D, B) 189f9502d7dfad5b361a3cdaa42eb75b593c95f79d8caryclark@google.com && approximately_zero_when_compared_to(D, C)) { // 0 is one root 1909f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com int num = quadraticRootsReal(A, B, C, s); 1919f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com for (int i = 0; i < num; ++i) { 1929f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com if (approximately_zero(s[i])) { 1939f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com return num; 1949f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com } 1959f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com } 1969f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com s[num++] = 0; 1979f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com return num; 1989f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com } 1999f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com if (approximately_zero(A + B + C + D)) { // 1 is one root 2009f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com int num = quadraticRootsReal(A, A + B, -D, s); 2019f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com for (int i = 0; i < num; ++i) { 2029f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com if (AlmostEqualUlps(s[i], 1)) { 2039f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com return num; 2049f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com } 2059f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com } 2069f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com s[num++] = 1; 2079f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com return num; 2089f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com } 2099f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com double a, b, c; 2109f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com { 2119f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com double invA = 1 / A; 2129f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com a = B * invA; 2139f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com b = C * invA; 2149f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com c = D * invA; 2159f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com } 2169f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com double a2 = a * a; 2179f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com double Q = (a2 - b * 3) / 9; 2189f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com double R = (2 * a2 * a - 9 * a * b + 27 * c) / 54; 2199f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com double R2 = R * R; 2209f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com double Q3 = Q * Q * Q; 2219f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com double R2MinusQ3 = R2 - Q3; 2229f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com double adiv3 = a / 3; 2239f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com double r; 2249f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com double* roots = s; 2259f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com#if 0 2269f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com if (approximately_zero_squared(R2MinusQ3) && AlmostEqualUlps(R2, Q3)) { 2279f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com if (approximately_zero_squared(R)) {/* one triple solution */ 2289f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com *roots++ = -adiv3; 2299f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com } else { /* one single and one double solution */ 2309f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com 2319f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com double u = cube_root(-R); 2329f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com *roots++ = 2 * u - adiv3; 2339f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com *roots++ = -u - adiv3; 2349f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com } 2359f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com } 2364024f32d99b63a599c544a49f526e53c25135159skia.committer@gmail.com else 2379f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com#endif 2389f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com if (R2MinusQ3 < 0) // we have 3 real roots 2399f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com { 2409f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com double theta = acos(R / sqrt(Q3)); 2419f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com double neg2RootQ = -2 * sqrt(Q); 2429f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com 2439f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com r = neg2RootQ * cos(theta / 3) - adiv3; 2449f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com *roots++ = r; 2459f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com 2469f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com r = neg2RootQ * cos((theta + 2 * PI) / 3) - adiv3; 2479f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com if (!AlmostEqualUlps(s[0], r)) { 2489f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com *roots++ = r; 2499f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com } 2509f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com r = neg2RootQ * cos((theta - 2 * PI) / 3) - adiv3; 2519f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com if (!AlmostEqualUlps(s[0], r) && (roots - s == 1 || !AlmostEqualUlps(s[1], r))) { 2529f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com *roots++ = r; 2539f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com } 2549f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com } 2559f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com else // we have 1 real root 2569f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com { 2579f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com double sqrtR2MinusQ3 = sqrt(R2MinusQ3); 2589f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com double A = fabs(R) + sqrtR2MinusQ3; 2599f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com A = cube_root(A); 2609f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com if (R > 0) { 2619f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com A = -A; 2629f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com } 2639f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com if (A != 0) { 2649f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com A += Q / A; 2659f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com } 2669f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com r = A - adiv3; 2679f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com *roots++ = r; 2689f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com if (AlmostEqualUlps(R2, Q3)) { 2699f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com r = -A / 2 - adiv3; 2709f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com if (!AlmostEqualUlps(s[0], r)) { 2719f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com *roots++ = r; 2729f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com } 2739f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com } 2749f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com } 2759f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com return (int)(roots - s); 276c682590538a27d73489bc91c098e000fdfb07ccfcaryclark@google.com} 2778dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com 2788dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com// from http://www.cs.sunysb.edu/~qin/courses/geometry/4.pdf 2798dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com// c(t) = a(1-t)^3 + 3bt(1-t)^2 + 3c(1-t)t^2 + dt^3 2808dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com// c'(t) = -3a(1-t)^2 + 3b((1-t)^2 - 2t(1-t)) + 3c(2t(1-t) - t^2) + 3dt^2 2818dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com// = 3(b-a)(1-t)^2 + 6(c-b)t(1-t) + 3(d-c)t^2 28205c4bad470722bc4e5e6ae3d79aa8bcf9e732f06caryclark@google.comstatic double derivativeAtT(const double* cubic, double t) { 2838dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com double one_t = 1 - t; 2848dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com double a = cubic[0]; 2858dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com double b = cubic[2]; 2868dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com double c = cubic[4]; 2878dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com double d = cubic[6]; 28805c4bad470722bc4e5e6ae3d79aa8bcf9e732f06caryclark@google.com return 3 * ((b - a) * one_t * one_t + 2 * (c - b) * t * one_t + (d - c) * t * t); 2898dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com} 2908dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com 29105c4bad470722bc4e5e6ae3d79aa8bcf9e732f06caryclark@google.comdouble dx_at_t(const Cubic& cubic, double t) { 29205c4bad470722bc4e5e6ae3d79aa8bcf9e732f06caryclark@google.com return derivativeAtT(&cubic[0].x, t); 29305c4bad470722bc4e5e6ae3d79aa8bcf9e732f06caryclark@google.com} 29405c4bad470722bc4e5e6ae3d79aa8bcf9e732f06caryclark@google.com 29505c4bad470722bc4e5e6ae3d79aa8bcf9e732f06caryclark@google.comdouble dy_at_t(const Cubic& cubic, double t) { 29605c4bad470722bc4e5e6ae3d79aa8bcf9e732f06caryclark@google.com return derivativeAtT(&cubic[0].y, t); 29705c4bad470722bc4e5e6ae3d79aa8bcf9e732f06caryclark@google.com} 29805c4bad470722bc4e5e6ae3d79aa8bcf9e732f06caryclark@google.com 29905c4bad470722bc4e5e6ae3d79aa8bcf9e732f06caryclark@google.com// OPTIMIZE? compute t^2, t(1-t), and (1-t)^2 and pass them to another version of derivative at t? 3007ff5c841bf669826b4cbd708ae1a6b3527f15dcacaryclark@google.com_Vector dxdy_at_t(const Cubic& cubic, double t) { 3017ff5c841bf669826b4cbd708ae1a6b3527f15dcacaryclark@google.com _Vector result = { derivativeAtT(&cubic[0].x, t), derivativeAtT(&cubic[0].y, t) }; 30245a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com return result; 30345a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com} 30445a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com 3051304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com// OPTIMIZE? share code with formulate_F1DotF2 30673ca6243b31e225e9fd5b75a96cbc82d62557de6caryclark@google.comint find_cubic_inflections(const Cubic& src, double tValues[]) 30773ca6243b31e225e9fd5b75a96cbc82d62557de6caryclark@google.com{ 30873ca6243b31e225e9fd5b75a96cbc82d62557de6caryclark@google.com double Ax = src[1].x - src[0].x; 30973ca6243b31e225e9fd5b75a96cbc82d62557de6caryclark@google.com double Ay = src[1].y - src[0].y; 31073ca6243b31e225e9fd5b75a96cbc82d62557de6caryclark@google.com double Bx = src[2].x - 2 * src[1].x + src[0].x; 31173ca6243b31e225e9fd5b75a96cbc82d62557de6caryclark@google.com double By = src[2].y - 2 * src[1].y + src[0].y; 31273ca6243b31e225e9fd5b75a96cbc82d62557de6caryclark@google.com double Cx = src[3].x + 3 * (src[1].x - src[2].x) - src[0].x; 31373ca6243b31e225e9fd5b75a96cbc82d62557de6caryclark@google.com double Cy = src[3].y + 3 * (src[1].y - src[2].y) - src[0].y; 3141304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com return quadraticRootsValidT(Bx * Cy - By * Cx, Ax * Cy - Ay * Cx, Ax * By - Ay * Bx, tValues); 3151304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com} 3161304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com 3171304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.comstatic void formulate_F1DotF2(const double src[], double coeff[4]) 3181304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com{ 3191304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com double a = src[2] - src[0]; 3201304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com double b = src[4] - 2 * src[2] + src[0]; 3211304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com double c = src[6] + 3 * (src[2] - src[4]) - src[0]; 3221304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com coeff[0] = c * c; 3231304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com coeff[1] = 3 * b * c; 3241304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com coeff[2] = 2 * b * b + c * a; 3251304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com coeff[3] = a * b; 3261304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com} 3271304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com 3281304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com/* from SkGeometry.cpp 3291304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com Looking for F' dot F'' == 0 3301304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com 3311304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com A = b - a 3321304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com B = c - 2b + a 3331304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com C = d - 3c + 3b - a 3341304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com 3351304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com F' = 3Ct^2 + 6Bt + 3A 3361304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com F'' = 6Ct + 6B 3371304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com 3381304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB 3391304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com*/ 3401304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.comint find_cubic_max_curvature(const Cubic& src, double tValues[]) 3411304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com{ 3421304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com double coeffX[4], coeffY[4]; 3431304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com int i; 3441304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com formulate_F1DotF2(&src[0].x, coeffX); 3451304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com formulate_F1DotF2(&src[0].y, coeffY); 3461304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com for (i = 0; i < 4; i++) { 3471304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com coeffX[i] = coeffX[i] + coeffY[i]; 3481304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com } 3491304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com return cubicRootsValidT(coeffX[0], coeffX[1], coeffX[2], coeffX[3], tValues); 35073ca6243b31e225e9fd5b75a96cbc82d62557de6caryclark@google.com} 35173ca6243b31e225e9fd5b75a96cbc82d62557de6caryclark@google.com 3521304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com 3538dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.combool rotate(const Cubic& cubic, int zero, int index, Cubic& rotPath) { 3548dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com double dy = cubic[index].y - cubic[zero].y; 3558dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com double dx = cubic[index].x - cubic[zero].x; 3569f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com if (approximately_zero(dy)) { 3579f60291c5375457f8adf228dbe6e8ff1186b13e1caryclark@google.com if (approximately_zero(dx)) { 3588dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com return false; 3598dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com } 3608dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com memcpy(rotPath, cubic, sizeof(Cubic)); 3618dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com return true; 3628dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com } 3638dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com for (int index = 0; index < 4; ++index) { 3648dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com rotPath[index].x = cubic[index].x * dx + cubic[index].y * dy; 3658dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com rotPath[index].y = cubic[index].y * dx - cubic[index].x * dy; 3668dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com } 3678dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com return true; 3688dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com} 3698dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com 37005c4bad470722bc4e5e6ae3d79aa8bcf9e732f06caryclark@google.com#if 0 // unused for now 3718dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.comdouble secondDerivativeAtT(const double* cubic, double t) { 3728dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com double a = cubic[0]; 3738dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com double b = cubic[2]; 3748dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com double c = cubic[4]; 3758dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com double d = cubic[6]; 3768dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com return (c - 2 * b + a) * (1 - t) + (d - 2 * c + b) * t; 3778dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com} 37805c4bad470722bc4e5e6ae3d79aa8bcf9e732f06caryclark@google.com#endif 3798dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com 38045a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com_Point top(const Cubic& cubic, double startT, double endT) { 38145a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com Cubic sub; 38245a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com sub_divide(cubic, startT, endT, sub); 38345a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com _Point topPt = sub[0]; 38445a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com if (topPt.y > sub[3].y || (topPt.y == sub[3].y && topPt.x > sub[3].x)) { 38545a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com topPt = sub[3]; 38645a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com } 38745a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com double extremeTs[2]; 3881304bb25aa3b0baa61fc2e2900fabcef88801b59caryclark@google.com if (!monotonic_in_y(sub)) { 38945a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com int roots = findExtrema(sub[0].y, sub[1].y, sub[2].y, sub[3].y, extremeTs); 39045a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com for (int index = 0; index < roots; ++index) { 39145a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com _Point mid; 39245a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com double t = startT + (endT - startT) * extremeTs[index]; 39345a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com xy_at_t(cubic, t, mid.x, mid.y); 39445a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com if (topPt.y > mid.y || (topPt.y == mid.y && topPt.x > mid.x)) { 39545a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com topPt = mid; 39645a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com } 39745a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com } 39845a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com } 39945a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com return topPt; 40045a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com} 40145a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com 4027ff5c841bf669826b4cbd708ae1a6b3527f15dcacaryclark@google.com// OPTIMIZE: avoid computing the unused half 4037ff5c841bf669826b4cbd708ae1a6b3527f15dcacaryclark@google.comvoid xy_at_t(const Cubic& cubic, double t, double& x, double& y) { 4047ff5c841bf669826b4cbd708ae1a6b3527f15dcacaryclark@google.com _Point xy = xy_at_t(cubic, t); 4057ff5c841bf669826b4cbd708ae1a6b3527f15dcacaryclark@google.com if (&x) { 4067ff5c841bf669826b4cbd708ae1a6b3527f15dcacaryclark@google.com x = xy.x; 4077ff5c841bf669826b4cbd708ae1a6b3527f15dcacaryclark@google.com } 4087ff5c841bf669826b4cbd708ae1a6b3527f15dcacaryclark@google.com if (&y) { 4097ff5c841bf669826b4cbd708ae1a6b3527f15dcacaryclark@google.com y = xy.y; 4107ff5c841bf669826b4cbd708ae1a6b3527f15dcacaryclark@google.com } 4117ff5c841bf669826b4cbd708ae1a6b3527f15dcacaryclark@google.com} 4127ff5c841bf669826b4cbd708ae1a6b3527f15dcacaryclark@google.com 41345a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com_Point xy_at_t(const Cubic& cubic, double t) { 4148dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com double one_t = 1 - t; 4158dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com double one_t2 = one_t * one_t; 4168dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com double a = one_t2 * one_t; 4178dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com double b = 3 * one_t2 * t; 4188dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com double t2 = t * t; 4198dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com double c = 3 * one_t * t2; 4208dcf114db9762c02d217beba6e29dffa4e92d298caryclark@google.com double d = t2 * t; 42145a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com _Point result = {a * cubic[0].x + b * cubic[1].x + c * cubic[2].x + d * cubic[3].x, 42245a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com a * cubic[0].y + b * cubic[1].y + c * cubic[2].y + d * cubic[3].y}; 42345a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com return result; 42445a8fc6a8b00451f807783f2a6ec640e9bcc7256caryclark@google.com} 425