/* Solve coeff(t) == 0, returning the number of roots that lie withing 0 < t < 1. coeff[0]t^3 + coeff[1]t^2 + coeff[2]t + coeff[3] Eliminates repeated roots (so that all tValues are distinct, and are always in increasing order. */ static int solve_cubic_polynomial(const SkFP coeff[4], SkScalar tValues[3]) { #ifndef SK_SCALAR_IS_FLOAT return 0; // this is not yet implemented for software float #endif if (SkScalarNearlyZero(coeff[0])) // we're just a quadratic { return SkFindUnitQuadRoots(coeff[1], coeff[2], coeff[3], tValues); } SkFP a, b, c, Q, R; { SkASSERT(coeff[0] != 0); SkFP inva = SkFPInvert(coeff[0]); a = SkFPMul(coeff[1], inva); b = SkFPMul(coeff[2], inva); c = SkFPMul(coeff[3], inva); } Q = SkFPDivInt(SkFPSub(SkFPMul(a,a), SkFPMulInt(b, 3)), 9); // R = (2*a*a*a - 9*a*b + 27*c) / 54; R = SkFPMulInt(SkFPMul(SkFPMul(a, a), a), 2); R = SkFPSub(R, SkFPMulInt(SkFPMul(a, b), 9)); R = SkFPAdd(R, SkFPMulInt(c, 27)); R = SkFPDivInt(R, 54); SkFP Q3 = SkFPMul(SkFPMul(Q, Q), Q); SkFP R2MinusQ3 = SkFPSub(SkFPMul(R,R), Q3); SkFP adiv3 = SkFPDivInt(a, 3); SkScalar* roots = tValues; SkScalar r; if (SkFPLT(R2MinusQ3, 0)) // we have 3 real roots { #ifdef SK_SCALAR_IS_FLOAT float theta = sk_float_acos(R / sk_float_sqrt(Q3)); float neg2RootQ = -2 * sk_float_sqrt(Q); r = neg2RootQ * sk_float_cos(theta/3) - adiv3; if (is_unit_interval(r)) *roots++ = r; r = neg2RootQ * sk_float_cos((theta + 2*SK_ScalarPI)/3) - adiv3; if (is_unit_interval(r)) *roots++ = r; r = neg2RootQ * sk_float_cos((theta - 2*SK_ScalarPI)/3) - adiv3; if (is_unit_interval(r)) *roots++ = r; SkDEBUGCODE(test_collaps_duplicates();)
/* Solve coeff(t) == 0, returning the number of roots that lie withing 0 < t < 1. coeff[0]t^3 + coeff[1]t^2 + coeff[2]t + coeff[3] */ static int solve_cubic_polynomial(const SkFP coeff[4], SkScalar tValues[3]) { #ifndef SK_SCALAR_IS_FLOAT return 0; // this is not yet implemented for software float #endif if (SkScalarNearlyZero(coeff[0])) // we're just a quadratic { return SkFindUnitQuadRoots(coeff[1], coeff[2], coeff[3], tValues); } SkFP a, b, c, Q, R; { SkASSERT(coeff[0] != 0); SkFP inva = SkFPInvert(coeff[0]); a = SkFPMul(coeff[1], inva); b = SkFPMul(coeff[2], inva); c = SkFPMul(coeff[3], inva); } Q = SkFPDivInt(SkFPSub(SkFPMul(a,a), SkFPMulInt(b, 3)), 9); // R = (2*a*a*a - 9*a*b + 27*c) / 54; R = SkFPMulInt(SkFPMul(SkFPMul(a, a), a), 2); R = SkFPSub(R, SkFPMulInt(SkFPMul(a, b), 9)); R = SkFPAdd(R, SkFPMulInt(c, 27)); R = SkFPDivInt(R, 54); SkFP Q3 = SkFPMul(SkFPMul(Q, Q), Q); SkFP R2MinusQ3 = SkFPSub(SkFPMul(R,R), Q3); SkFP adiv3 = SkFPDivInt(a, 3); SkScalar* roots = tValues; SkScalar r; if (SkFPLT(R2MinusQ3, 0)) // we have 3 real roots { #ifdef SK_SCALAR_IS_FLOAT float theta = sk_float_acos(R / sk_float_sqrt(Q3)); float neg2RootQ = -2 * sk_float_sqrt(Q); r = neg2RootQ * sk_float_cos(theta/3) - adiv3; if (is_unit_interval(r)) *roots++ = r; r = neg2RootQ * sk_float_cos((theta + 2*SK_ScalarPI)/3) - adiv3; if (is_unit_interval(r)) *roots++ = r; r = neg2RootQ * sk_float_cos((theta - 2*SK_ScalarPI)/3) - adiv3; if (is_unit_interval(r)) *roots++ = r; // now sort the roots bubble_sort(tValues, (int)(roots - tValues)); #endif } else // we have 1 real root { SkFP A = SkFPAdd(SkFPAbs(R), SkFPSqrt(R2MinusQ3)); A = SkFPCubeRoot(A); if (SkFPGT(R, 0)) A = SkFPNeg(A); if (A != 0) A = SkFPAdd(A, SkFPDiv(Q, A)); r = SkFPToScalar(SkFPSub(A, adiv3)); if (is_unit_interval(r)) *roots++ = r; } return (int)(roots - tValues); }