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 /* * Copyright 2012 Google LLC * * Use of this source code is governed by a BSD-style license that can be * found in the LICENSE file. */ #include "src/base/SkBezierCurves.h" #include "include/private/base/SkAssert.h" #include static inline double interpolate(double A, double B, double t) { return A + (B - A) * t; } std::array SkBezierCubic::EvalAt(const double curve[8], double t) { const auto in_X = [&curve](size_t n) { return curve[2*n]; }; const auto in_Y = [&curve](size_t n) { return curve[2*n + 1]; }; // Two semi-common fast paths if (t == 0) { return {in_X(0), in_Y(0)}; } if (t == 1) { return {in_X(3), in_Y(3)}; } // X(t) = X_0*(1-t)^3 + 3*X_1*t(1-t)^2 + 3*X_2*t^2(1-t) + X_3*t^3 // Y(t) = Y_0*(1-t)^3 + 3*Y_1*t(1-t)^2 + 3*Y_2*t^2(1-t) + Y_3*t^3 // Some compilers are smart enough and have sufficient registers/intrinsics to write optimal // code from // double one_minus_t = 1 - t; // double a = one_minus_t * one_minus_t * one_minus_t; // double b = 3 * one_minus_t * one_minus_t * t; // double c = 3 * one_minus_t * t * t; // double d = t * t * t; // However, some (e.g. when compiling for ARM) fail to do so, so we use this form // to help more compilers generate smaller/faster ASM. https://godbolt.org/z/M6jG9x45c double one_minus_t = 1 - t; double one_minus_t_squared = one_minus_t * one_minus_t; double a = (one_minus_t_squared * one_minus_t); double b = 3 * one_minus_t_squared * t; double t_squared = t * t; double c = 3 * one_minus_t * t_squared; double d = t_squared * t; return {a * in_X(0) + b * in_X(1) + c * in_X(2) + d * in_X(3), a * in_Y(0) + b * in_Y(1) + c * in_Y(2) + d * in_Y(3)}; } // Perform subdivision using De Casteljau's algorithm, that is, repeated linear // interpolation between adjacent points. void SkBezierCubic::Subdivide(const double curve[8], double t, double twoCurves[14]) { SkASSERT(0.0 <= t && t <= 1.0); // We split the curve "in" into two curves "alpha" and "beta" const auto in_X = [&curve](size_t n) { return curve[2*n]; }; const auto in_Y = [&curve](size_t n) { return curve[2*n + 1]; }; const auto alpha_X = [&twoCurves](size_t n) -> double& { return twoCurves[2*n]; }; const auto alpha_Y = [&twoCurves](size_t n) -> double& { return twoCurves[2*n + 1]; }; const auto beta_X = [&twoCurves](size_t n) -> double& { return twoCurves[2*n + 6]; }; const auto beta_Y = [&twoCurves](size_t n) -> double& { return twoCurves[2*n + 7]; }; alpha_X(0) = in_X(0); alpha_Y(0) = in_Y(0); beta_X(3) = in_X(3); beta_Y(3) = in_Y(3); double x01 = interpolate(in_X(0), in_X(1), t); double y01 = interpolate(in_Y(0), in_Y(1), t); double x12 = interpolate(in_X(1), in_X(2), t); double y12 = interpolate(in_Y(1), in_Y(2), t); double x23 = interpolate(in_X(2), in_X(3), t); double y23 = interpolate(in_Y(2), in_Y(3), t); alpha_X(1) = x01; alpha_Y(1) = y01; beta_X(2) = x23; beta_Y(2) = y23; alpha_X(2) = interpolate(x01, x12, t); alpha_Y(2) = interpolate(y01, y12, t); beta_X(1) = interpolate(x12, x23, t); beta_Y(1) = interpolate(y12, y23, t); alpha_X(3) /*= beta_X(0) */ = interpolate(alpha_X(2), beta_X(1), t); alpha_Y(3) /*= beta_Y(0) */ = interpolate(alpha_Y(2), beta_Y(1), t); } std::array SkBezierCubic::ConvertToPolynomial(const double curve[8], bool yValues) { const double* offset_curve = yValues ? curve + 1 : curve; const auto P = [&offset_curve](size_t n) { return offset_curve[2*n]; }; // A cubic Bézier curve is interpolated as follows: // c(t) = (1 - t)^3 P_0 + 3t(1 - t)^2 P_1 + 3t^2 (1 - t) P_2 + t^3 P_3 // = (-P_0 + 3P_1 + -3P_2 + P_3) t^3 + (3P_0 - 6P_1 + 3P_2) t^2 + // (-3P_0 + 3P_1) t + P_0 // Where P_N is the Nth point. The second step expands the polynomial and groups // by powers of t. The desired output is a cubic formula, so we just need to // combine the appropriate points to make the coefficients. std::array results; results[0] = -P(0) + 3*P(1) - 3*P(2) + P(3); results[1] = 3*P(0) - 6*P(1) + 3*P(2); results[2] = -3*P(0) + 3*P(1); results[3] = P(0); return results; }