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 /* * Copyright 2012 Google Inc. * * Use of this source code is governed by a BSD-style license that can be * found in the LICENSE file. */ #include "CubicUtilities.h" #include "Extrema.h" #include "LineUtilities.h" #include "QuadraticUtilities.h" const int gPrecisionUnit = 256; // FIXME: arbitrary -- should try different values in test framework // FIXME: cache keep the bounds and/or precision with the caller? double calcPrecision(const Cubic& cubic) { _Rect dRect; dRect.setBounds(cubic); // OPTIMIZATION: just use setRawBounds ? double width = dRect.right - dRect.left; double height = dRect.bottom - dRect.top; return (width > height ? width : height) / gPrecisionUnit; } #if SK_DEBUG double calcPrecision(const Cubic& cubic, double t, double scale) { Cubic part; sub_divide(cubic, SkTMax(0., t - scale), SkTMin(1., t + scale), part); return calcPrecision(part); } #endif bool clockwise(const Cubic& c) { double sum = (c[0].x - c[3].x) * (c[0].y + c[3].y); for (int idx = 0; idx < 3; ++idx){ sum += (c[idx + 1].x - c[idx].x) * (c[idx + 1].y + c[idx].y); } return sum <= 0; } void coefficients(const double* cubic, double& A, double& B, double& C, double& D) { A = cubic[6]; // d B = cubic[4] * 3; // 3*c C = cubic[2] * 3; // 3*b D = cubic[0]; // a A -= D - C + B; // A = -a + 3*b - 3*c + d B += 3 * D - 2 * C; // B = 3*a - 6*b + 3*c C -= 3 * D; // C = -3*a + 3*b } bool controls_contained_by_ends(const Cubic& c) { _Vector startTan = c[1] - c[0]; if (startTan.x == 0 && startTan.y == 0) { startTan = c[2] - c[0]; } _Vector endTan = c[2] - c[3]; if (endTan.x == 0 && endTan.y == 0) { endTan = c[1] - c[3]; } if (startTan.dot(endTan) >= 0) { return false; } _Line startEdge = {c[0], c[0]}; startEdge[1].x -= startTan.y; startEdge[1].y += startTan.x; _Line endEdge = {c[3], c[3]}; endEdge[1].x -= endTan.y; endEdge[1].y += endTan.x; double leftStart1 = is_left(startEdge, c[1]); if (leftStart1 * is_left(startEdge, c[2]) < 0) { return false; } double leftEnd1 = is_left(endEdge, c[1]); if (leftEnd1 * is_left(endEdge, c[2]) < 0) { return false; } return leftStart1 * leftEnd1 >= 0; } bool ends_are_extrema_in_x_or_y(const Cubic& c) { return (between(c[0].x, c[1].x, c[3].x) && between(c[0].x, c[2].x, c[3].x)) || (between(c[0].y, c[1].y, c[3].y) && between(c[0].y, c[2].y, c[3].y)); } bool monotonic_in_y(const Cubic& c) { return between(c[0].y, c[1].y, c[3].y) && between(c[0].y, c[2].y, c[3].y); } bool serpentine(const Cubic& c) { if (!controls_contained_by_ends(c)) { return false; } double wiggle = (c[0].x - c[2].x) * (c[0].y + c[2].y); for (int idx = 0; idx < 2; ++idx){ wiggle += (c[idx + 1].x - c[idx].x) * (c[idx + 1].y + c[idx].y); } double waggle = (c[1].x - c[3].x) * (c[1].y + c[3].y); for (int idx = 1; idx < 3; ++idx){ waggle += (c[idx + 1].x - c[idx].x) * (c[idx + 1].y + c[idx].y); } return wiggle * waggle < 0; } // cubic roots const double PI = 4 * atan(1); // from SkGeometry.cpp (and Numeric Solutions, 5.6) int cubicRootsValidT(double A, double B, double C, double D, double t[3]) { #if 0 if (approximately_zero(A)) { // we're just a quadratic return quadraticRootsValidT(B, C, D, t); } double a, b, c; { double invA = 1 / A; a = B * invA; b = C * invA; c = D * invA; } double a2 = a * a; double Q = (a2 - b * 3) / 9; double R = (2 * a2 * a - 9 * a * b + 27 * c) / 54; double Q3 = Q * Q * Q; double R2MinusQ3 = R * R - Q3; double adiv3 = a / 3; double* roots = t; double r; if (R2MinusQ3 < 0) // we have 3 real roots { double theta = acos(R / sqrt(Q3)); double neg2RootQ = -2 * sqrt(Q); r = neg2RootQ * cos(theta / 3) - adiv3; if (is_unit_interval(r)) *roots++ = r; r = neg2RootQ * cos((theta + 2 * PI) / 3) - adiv3; if (is_unit_interval(r)) *roots++ = r; r = neg2RootQ * cos((theta - 2 * PI) / 3) - adiv3; if (is_unit_interval(r)) *roots++ = r; } else // we have 1 real root { double A = fabs(R) + sqrt(R2MinusQ3); A = cube_root(A); if (R > 0) { A = -A; } if (A != 0) { A += Q / A; } r = A - adiv3; if (is_unit_interval(r)) *roots++ = r; } return (int)(roots - t); #else double s[3]; int realRoots = cubicRootsReal(A, B, C, D, s); int foundRoots = add_valid_ts(s, realRoots, t); return foundRoots; #endif } int cubicRootsReal(double A, double B, double C, double D, double s[3]) { #if SK_DEBUG // create a string mathematica understands // GDB set print repe 15 # if repeated digits is a bother // set print elements 400 # if line doesn't fit char str[1024]; bzero(str, sizeof(str)); sprintf(str, "Solve[%1.19g x^3 + %1.19g x^2 + %1.19g x + %1.19g == 0, x]", A, B, C, D); mathematica_ize(str, sizeof(str)); #if ONE_OFF_DEBUG && ONE_OFF_DEBUG_MATHEMATICA SkDebugf("%s\n", str); #endif #endif if (approximately_zero(A) && approximately_zero_when_compared_to(A, B) && approximately_zero_when_compared_to(A, C) && approximately_zero_when_compared_to(A, D)) { // we're just a quadratic return quadraticRootsReal(B, C, D, s); } if (approximately_zero_when_compared_to(D, A) && approximately_zero_when_compared_to(D, B) && approximately_zero_when_compared_to(D, C)) { // 0 is one root int num = quadraticRootsReal(A, B, C, s); for (int i = 0; i < num; ++i) { if (approximately_zero(s[i])) { return num; } } s[num++] = 0; return num; } if (approximately_zero(A + B + C + D)) { // 1 is one root int num = quadraticRootsReal(A, A + B, -D, s); for (int i = 0; i < num; ++i) { if (AlmostEqualUlps(s[i], 1)) { return num; } } s[num++] = 1; return num; } double a, b, c; { double invA = 1 / A; a = B * invA; b = C * invA; c = D * invA; } double a2 = a * a; double Q = (a2 - b * 3) / 9; double R = (2 * a2 * a - 9 * a * b + 27 * c) / 54; double R2 = R * R; double Q3 = Q * Q * Q; double R2MinusQ3 = R2 - Q3; double adiv3 = a / 3; double r; double* roots = s; #if 0 if (approximately_zero_squared(R2MinusQ3) && AlmostEqualUlps(R2, Q3)) { if (approximately_zero_squared(R)) {/* one triple solution */ *roots++ = -adiv3; } else { /* one single and one double solution */ double u = cube_root(-R); *roots++ = 2 * u - adiv3; *roots++ = -u - adiv3; } } else #endif if (R2MinusQ3 < 0) // we have 3 real roots { double theta = acos(R / sqrt(Q3)); double neg2RootQ = -2 * sqrt(Q); r = neg2RootQ * cos(theta / 3) - adiv3; *roots++ = r; r = neg2RootQ * cos((theta + 2 * PI) / 3) - adiv3; if (!AlmostEqualUlps(s[0], r)) { *roots++ = r; } r = neg2RootQ * cos((theta - 2 * PI) / 3) - adiv3; if (!AlmostEqualUlps(s[0], r) && (roots - s == 1 || !AlmostEqualUlps(s[1], r))) { *roots++ = r; } } else // we have 1 real root { double sqrtR2MinusQ3 = sqrt(R2MinusQ3); double A = fabs(R) + sqrtR2MinusQ3; A = cube_root(A); if (R > 0) { A = -A; } if (A != 0) { A += Q / A; } r = A - adiv3; *roots++ = r; if (AlmostEqualUlps(R2, Q3)) { r = -A / 2 - adiv3; if (!AlmostEqualUlps(s[0], r)) { *roots++ = r; } } } return (int)(roots - s); } // from http://www.cs.sunysb.edu/~qin/courses/geometry/4.pdf // c(t) = a(1-t)^3 + 3bt(1-t)^2 + 3c(1-t)t^2 + dt^3 // c'(t) = -3a(1-t)^2 + 3b((1-t)^2 - 2t(1-t)) + 3c(2t(1-t) - t^2) + 3dt^2 // = 3(b-a)(1-t)^2 + 6(c-b)t(1-t) + 3(d-c)t^2 static double derivativeAtT(const double* cubic, double t) { double one_t = 1 - t; double a = cubic[0]; double b = cubic[2]; double c = cubic[4]; double d = cubic[6]; return 3 * ((b - a) * one_t * one_t + 2 * (c - b) * t * one_t + (d - c) * t * t); } double dx_at_t(const Cubic& cubic, double t) { return derivativeAtT(&cubic[0].x, t); } double dy_at_t(const Cubic& cubic, double t) { return derivativeAtT(&cubic[0].y, t); } // OPTIMIZE? compute t^2, t(1-t), and (1-t)^2 and pass them to another version of derivative at t? _Vector dxdy_at_t(const Cubic& cubic, double t) { _Vector result = { derivativeAtT(&cubic[0].x, t), derivativeAtT(&cubic[0].y, t) }; return result; } // OPTIMIZE? share code with formulate_F1DotF2 int find_cubic_inflections(const Cubic& src, double tValues[]) { double Ax = src[1].x - src[0].x; double Ay = src[1].y - src[0].y; double Bx = src[2].x - 2 * src[1].x + src[0].x; double By = src[2].y - 2 * src[1].y + src[0].y; double Cx = src[3].x + 3 * (src[1].x - src[2].x) - src[0].x; double Cy = src[3].y + 3 * (src[1].y - src[2].y) - src[0].y; return quadraticRootsValidT(Bx * Cy - By * Cx, Ax * Cy - Ay * Cx, Ax * By - Ay * Bx, tValues); } static void formulate_F1DotF2(const double src[], double coeff[4]) { double a = src[2] - src[0]; double b = src[4] - 2 * src[2] + src[0]; double c = src[6] + 3 * (src[2] - src[4]) - src[0]; coeff[0] = c * c; coeff[1] = 3 * b * c; coeff[2] = 2 * b * b + c * a; coeff[3] = a * b; } /* from SkGeometry.cpp Looking for F' dot F'' == 0 A = b - a B = c - 2b + a C = d - 3c + 3b - a F' = 3Ct^2 + 6Bt + 3A F'' = 6Ct + 6B F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB */ int find_cubic_max_curvature(const Cubic& src, double tValues[]) { double coeffX[4], coeffY[4]; int i; formulate_F1DotF2(&src[0].x, coeffX); formulate_F1DotF2(&src[0].y, coeffY); for (i = 0; i < 4; i++) { coeffX[i] = coeffX[i] + coeffY[i]; } return cubicRootsValidT(coeffX[0], coeffX[1], coeffX[2], coeffX[3], tValues); } bool rotate(const Cubic& cubic, int zero, int index, Cubic& rotPath) { double dy = cubic[index].y - cubic[zero].y; double dx = cubic[index].x - cubic[zero].x; if (approximately_zero(dy)) { if (approximately_zero(dx)) { return false; } memcpy(rotPath, cubic, sizeof(Cubic)); return true; } for (int index = 0; index < 4; ++index) { rotPath[index].x = cubic[index].x * dx + cubic[index].y * dy; rotPath[index].y = cubic[index].y * dx - cubic[index].x * dy; } return true; } #if 0 // unused for now double secondDerivativeAtT(const double* cubic, double t) { double a = cubic[0]; double b = cubic[2]; double c = cubic[4]; double d = cubic[6]; return (c - 2 * b + a) * (1 - t) + (d - 2 * c + b) * t; } #endif _Point top(const Cubic& cubic, double startT, double endT) { Cubic sub; sub_divide(cubic, startT, endT, sub); _Point topPt = sub[0]; if (topPt.y > sub[3].y || (topPt.y == sub[3].y && topPt.x > sub[3].x)) { topPt = sub[3]; } double extremeTs[2]; if (!monotonic_in_y(sub)) { int roots = findExtrema(sub[0].y, sub[1].y, sub[2].y, sub[3].y, extremeTs); for (int index = 0; index < roots; ++index) { _Point mid; double t = startT + (endT - startT) * extremeTs[index]; xy_at_t(cubic, t, mid.x, mid.y); if (topPt.y > mid.y || (topPt.y == mid.y && topPt.x > mid.x)) { topPt = mid; } } } return topPt; } // OPTIMIZE: avoid computing the unused half void xy_at_t(const Cubic& cubic, double t, double& x, double& y) { _Point xy = xy_at_t(cubic, t); if (&x) { x = xy.x; } if (&y) { y = xy.y; } } _Point xy_at_t(const Cubic& cubic, double t) { double one_t = 1 - t; double one_t2 = one_t * one_t; double a = one_t2 * one_t; double b = 3 * one_t2 * t; double t2 = t * t; double c = 3 * one_t * t2; double d = t2 * t; _Point result = {a * cubic[0].x + b * cubic[1].x + c * cubic[2].x + d * cubic[3].x, a * cubic[0].y + b * cubic[1].y + c * cubic[2].y + d * cubic[3].y}; return result; }