| #include "CubicUtilities.h" |
| #include "DataTypes.h" |
| #include "QuadraticUtilities.h" |
| |
| 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 |
| } |
| |
| // cubic roots |
| |
| const double PI = 4 * atan(1); |
| |
| static bool is_unit_interval(double x) { |
| return x > 0 && x < 1; |
| } |
| |
| // from SkGeometry.cpp (and Numeric Solutions, 5.6) |
| int cubicRoots(double A, double B, double C, double D, double t[3]) { |
| if (approximately_zero(A)) { // we're just a quadratic |
| return quadraticRoots(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); |
| } |
| |
| // 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 |
| 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 (b - a) * one_t * one_t + 2 * (c - b) * t * one_t + (d - c) * t * t; |
| } |
| |
| // same as derivativeAtT |
| // which is more accurate? which is faster? |
| double derivativeAtT_2(const double* cubic, double t) { |
| double a = cubic[2] - cubic[0]; |
| double b = cubic[4] - 2 * cubic[2] + cubic[0]; |
| double c = cubic[6] + 3 * (cubic[2] - cubic[4]) - cubic[0]; |
| return c * c * t * t + 2 * b * t + a; |
| } |
| |
| void dxdy_at_t(const Cubic& cubic, double t, double& dx, double& dy) { |
| if (&dx) { |
| dx = derivativeAtT(&cubic[0].x, t); |
| } |
| if (&dy) { |
| dy = derivativeAtT(&cubic[0].y, t); |
| } |
| } |
| |
| 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_equal(dy, 0)) { |
| if (approximately_equal(dx, 0)) { |
| 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; |
| } |
| |
| 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; |
| } |
| |
| void xy_at_t(const Cubic& cubic, double t, double& x, double& y) { |
| 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; |
| if (&x) { |
| x = a * cubic[0].x + b * cubic[1].x + c * cubic[2].x + d * cubic[3].x; |
| } |
| if (&y) { |
| y = a * cubic[0].y + b * cubic[1].y + c * cubic[2].y + d * cubic[3].y; |
| } |
| } |