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 /* * Copyright 2018 Google Inc. * * Use of this source code is governed by a BSD-style license that can be * found in the LICENSE file. */ #include "SkCubicMap.h" #include "SkNx.h" //#define CUBICMAP_TRACK_MAX_ERROR #ifdef CUBICMAP_TRACK_MAX_ERROR #include "../../src/pathops/SkPathOpsCubic.h" #endif static float eval_poly3(float a, float b, float c, float d, float t) { return ((a * t + b) * t + c) * t + d; } static float eval_poly2(float a, float b, float c, float t) { return (a * t + b) * t + c; } static float eval_poly1(float a, float b, float t) { return a * t + b; } static float guess_nice_cubic_root(float A, float B, float C, float D) { return -D; } #ifdef SK_DEBUG static bool valid(float r) { return r >= 0 && r <= 1; } #endif static inline bool nearly_zero(SkScalar x) { SkASSERT(x >= 0); return x <= 0.0000000001f; } static inline bool delta_nearly_zero(float delta) { return sk_float_abs(delta) <= 0.0001f; } #ifdef CUBICMAP_TRACK_MAX_ERROR static int max_iters; #endif /* * TODO: will this be faster if we algebraically compute the polynomials for the numer and denom * rather than compute them in parts? */ static float solve_nice_cubic_halley(float A, float B, float C, float D) { const int MAX_ITERS = 8; const float A3 = 3 * A; const float B2 = B + B; float t = guess_nice_cubic_root(A, B, C, D); int iters = 0; for (; iters < MAX_ITERS; ++iters) { float f = eval_poly3(A, B, C, D, t); // f = At^3 + Bt^2 + Ct + D float fp = eval_poly2(A3, B2, C, t); // f' = 3At^2 + 2Bt + C float fpp = eval_poly1(A3 + A3, B2, t); // f'' = 6At + 2B float numer = 2 * fp * f; if (numer == 0) { break; } float denom = 2 * fp * fp - f * fpp; float delta = numer / denom; // SkDebugf("[%d] delta %g t %g\n", iters, delta, t); if (delta_nearly_zero(delta)) { break; } float new_t = t - delta; SkASSERT(valid(new_t)); t = new_t; } SkASSERT(valid(t)); #ifdef CUBICMAP_TRACK_MAX_ERROR if (iters > max_iters) { max_iters = iters; SkDebugf("max_iters %d\n", max_iters); } #endif return t; } // At the moment, this technique does not appear to be better (i.e. faster at same precision) // but the code is left here (at least for a while) to document the attempt. static float solve_nice_cubic_householder(float A, float B, float C, float D) { const int MAX_ITERS = 8; const float A3 = 3 * A; const float B2 = B + B; float t = guess_nice_cubic_root(A, B, C, D); int iters = 0; for (; iters < MAX_ITERS; ++iters) { float f = eval_poly3(A, B, C, D, t); // f = At^3 + Bt^2 + Ct + D float fp = eval_poly2(A3, B2, C, t); // f' = 3At^2 + 2Bt + C float fpp = eval_poly1(A3 + A3, B2, t); // f'' = 6At + 2B float fppp = A3 + A3; // f''' = 6A float f2 = f * f; float fp2 = fp * fp; // float numer = 6 * f * fp * fp - 3 * f * f * fpp; // float denom = 6 * fp * fp * fp - 6 * f * fp * fpp + f * f * fppp; float numer = 6 * f * fp2 - 3 * f2 * fpp; if (numer == 0) { break; } float denom = 6 * (fp2 * fp - f * fp * fpp) + f2 * fppp; float delta = numer / denom; // SkDebugf("[%d] delta %g t %g\n", iters, delta, t); if (delta_nearly_zero(delta)) { break; } float new_t = t - delta; SkASSERT(valid(new_t)); t = new_t; } SkASSERT(valid(t)); #ifdef CUBICMAP_TRACK_MAX_ERROR if (iters > max_iters) { max_iters = iters; SkDebugf("max_iters %d\n", max_iters); } #endif return t; } #ifdef CUBICMAP_TRACK_MAX_ERROR static float compute_slow(float A, float B, float C, float x) { double roots[3]; SkDEBUGCODE(int count =) SkDCubic::RootsValidT(A, B, C, -x, roots); SkASSERT(count == 1); return (float)roots[0]; } static float max_err; #endif static float compute_t_from_x(float A, float B, float C, float x) { #ifdef CUBICMAP_TRACK_MAX_ERROR float answer = compute_slow(A, B, C, x); #endif float answer2 = true ? solve_nice_cubic_halley(A, B, C, -x) : solve_nice_cubic_householder(A, B, C, -x); #ifdef CUBICMAP_TRACK_MAX_ERROR float err = sk_float_abs(answer - answer2); if (err > max_err) { max_err = err; SkDebugf("max error %g\n", max_err); } #endif return answer2; } float SkCubicMap::computeYFromX(float x) const { x = SkScalarPin(x, 0, 1); if (nearly_zero(x) || nearly_zero(1 - x)) { return x; } if (fType == kLine_Type) { return x; } float t; if (fType == kCubeRoot_Type) { t = sk_float_pow(x / fCoeff[0].fX, 1.0f / 3); } else { t = compute_t_from_x(fCoeff[0].fX, fCoeff[1].fX, fCoeff[2].fX, x); } float a = fCoeff[0].fY; float b = fCoeff[1].fY; float c = fCoeff[2].fY; float y = ((a * t + b) * t + c) * t; return y; } static inline bool coeff_nearly_zero(float delta) { return sk_float_abs(delta) <= 0.0000001f; } SkCubicMap::SkCubicMap(SkPoint p1, SkPoint p2) { // Clamp X values only (we allow Ys outside [0..1]). p1.fX = SkTMin(SkTMax(p1.fX, 0.0f), 1.0f); p2.fX = SkTMin(SkTMax(p2.fX, 0.0f), 1.0f); Sk2s s1 = Sk2s::Load(&p1) * 3; Sk2s s2 = Sk2s::Load(&p2) * 3; (Sk2s(1) + s1 - s2).store(&fCoeff[0]); (s2 - s1 - s1).store(&fCoeff[1]); s1.store(&fCoeff[2]); fType = kSolver_Type; if (SkScalarNearlyEqual(p1.fX, p1.fY) && SkScalarNearlyEqual(p2.fX, p2.fY)) { fType = kLine_Type; } else if (coeff_nearly_zero(fCoeff[1].fX) && coeff_nearly_zero(fCoeff[2].fX)) { fType = kCubeRoot_Type; } } SkPoint SkCubicMap::computeFromT(float t) const { Sk2s a = Sk2s::Load(&fCoeff[0]); Sk2s b = Sk2s::Load(&fCoeff[1]); Sk2s c = Sk2s::Load(&fCoeff[2]); SkPoint result; (((a * t + b) * t + c) * t).store(&result); return result; }