/* 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);
}
