/** Trim A/B/C down so that they are all <= 32bits
and then call SkFindUnitQuadRoots()
*/
static int Sk64FindFixedQuadRoots(const Sk64& A, const Sk64& B, const Sk64& C, SkFixed roots[2])
{
int na = A.shiftToMake32();
int nb = B.shiftToMake32();
int nc = C.shiftToMake32();
int shift = SkMax32(na, SkMax32(nb, nc));
SkASSERT(shift >= 0);
return SkFindUnitQuadRoots(A.getShiftRight(shift), B.getShiftRight(shift), C.getShiftRight(shift), roots);
}
/** http://www.faculty.idc.ac.il/arik/quality/appendixA.html
Inflection means that curvature is zero.
Curvature is [F' x F''] / [F'^3]
So we solve F'x X F''y - F'y X F''y == 0
After some canceling of the cubic term, we get
A = b - a
B = c - 2b + a
C = d - 3c + 3b - a
(BxCy - ByCx)t^2 + (AxCy - AyCx)t + AxBy - AyBx == 0
*/
int SkFindCubicInflections(const SkPoint src[4], SkScalar tValues[]) {
SkScalar Ax = src[1].fX - src[0].fX;
SkScalar Ay = src[1].fY - src[0].fY;
SkScalar Bx = src[2].fX - 2 * src[1].fX + src[0].fX;
SkScalar By = src[2].fY - 2 * src[1].fY + src[0].fY;
SkScalar Cx = src[3].fX + 3 * (src[1].fX - src[2].fX) - src[0].fX;
SkScalar Cy = src[3].fY + 3 * (src[1].fY - src[2].fY) - src[0].fY;
return SkFindUnitQuadRoots(Bx*Cy - By*Cx,
Ax*Cy - Ay*Cx,
Ax*By - Ay*Bx,
tValues);
}
/* Find t value for quadratic [a, b, c] = d.
Return 0 if there is no solution within [0, 1)
*/
static SkScalar quad_solve(SkScalar a, SkScalar b, SkScalar c, SkScalar d)
{
// At^2 + Bt + C = d
SkScalar A = a - 2 * b + c;
SkScalar B = 2 * (b - a);
SkScalar C = a - d;
SkScalar roots[2];
int count = SkFindUnitQuadRoots(A, B, C, roots);
SkASSERT(count <= 1);
return count == 1 ? roots[0] : 0;
}
/** Cubic'(t) = At^2 + Bt + C, where
A = 3(-a + 3(b - c) + d)
B = 6(a - 2b + c)
C = 3(b - a)
Solve for t, keeping only those that fit betwee 0 < t < 1
*/
int SkFindCubicExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar d, SkScalar tValues[2])
{
#ifdef SK_SCALAR_IS_FIXED
if (!is_not_monotonic(a, b, c, d))
return 0;
#endif
// we divide A,B,C by 3 to simplify
SkScalar A = d - a + 3*(b - c);
SkScalar B = 2*(a - b - b + c);
SkScalar C = b - a;
return SkFindUnitQuadRoots(A, B, C, tValues);
}
static bool chopMonoQuadAtY(SkPoint pts[3], SkScalar y, SkScalar* t) {
/* Solve F(t) = y where F(t) := [0](1-t)^2 + 2[1]t(1-t) + [2]t^2
* We solve for t, using quadratic equation, hence we have to rearrange
* our cooefficents to look like At^2 + Bt + C
*/
SkScalar A = pts[0].fY - pts[1].fY - pts[1].fY + pts[2].fY;
SkScalar B = 2*(pts[1].fY - pts[0].fY);
SkScalar C = pts[0].fY - y;
SkScalar roots[2]; // we only expect one, but make room for 2 for safety
int count = SkFindUnitQuadRoots(A, B, C, roots);
if (count) {
*t = roots[0];
return true;
}
return false;
}
static bool chopMonoQuadAt(SkScalar c0, SkScalar c1, SkScalar c2,
SkScalar target, SkScalar* t) {
/* Solve F(t) = y where F(t) := [0](1-t)^2 + 2[1]t(1-t) + [2]t^2
* We solve for t, using quadratic equation, hence we have to rearrange
* our cooefficents to look like At^2 + Bt + C
*/
SkScalar A = c0 - c1 - c1 + c2;
SkScalar B = 2*(c1 - c0);
SkScalar C = c0 - target;
SkScalar roots[2]; // we only expect one, but make room for 2 for safety
int count = SkFindUnitQuadRoots(A, B, C, roots);
if (count) {
*t = roots[0];
return true;
}
return false;
}
/* 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_poly(const SkScalar coeff[4], SkScalar tValues[3]) {
if (SkScalarNearlyZero(coeff[0])) { // we're just a quadratic
return SkFindUnitQuadRoots(coeff[1], coeff[2], coeff[3], tValues);
}
SkScalar a, b, c, Q, R;
{
SkASSERT(coeff[0] != 0);
SkScalar inva = SkScalarInvert(coeff[0]);
a = coeff[1] * inva;
b = coeff[2] * inva;
c = coeff[3] * inva;
}
Q = (a*a - b*3) / 9;
R = (2*a*a*a - 9*a*b + 27*c) / 54;
SkScalar Q3 = Q * Q * Q;
SkScalar R2MinusQ3 = R * R - Q3;
SkScalar adiv3 = a / 3;
SkScalar* roots = tValues;
SkScalar r;
if (R2MinusQ3 < 0) { // we have 3 real roots
SkScalar theta = SkScalarACos(R / SkScalarSqrt(Q3));
SkScalar neg2RootQ = -2 * SkScalarSqrt(Q);
r = neg2RootQ * SkScalarCos(theta/3) - adiv3;
if (is_unit_interval(r)) {
*roots++ = r;
}
r = neg2RootQ * SkScalarCos((theta + 2*SK_ScalarPI)/3) - adiv3;
if (is_unit_interval(r)) {
*roots++ = r;
}
r = neg2RootQ * SkScalarCos((theta - 2*SK_ScalarPI)/3) - adiv3;
if (is_unit_interval(r)) {
*roots++ = r;
}
SkDEBUGCODE(test_collaps_duplicates();)
static int winding_mono_quad(const SkPoint pts[], SkScalar x, SkScalar y) {
SkScalar y0 = pts[0].fY;
SkScalar y2 = pts[2].fY;
int dir = 1;
if (y0 > y2) {
SkTSwap(y0, y2);
dir = -1;
}
if (y < y0 || y >= y2) {
return 0;
}
// bounds check on X (not required, but maybe faster)
#if 0
if (pts[0].fX > x && pts[1].fX > x && pts[2].fX > x) {
return 0;
}
#endif
SkScalar roots[2];
int n = SkFindUnitQuadRoots(pts[0].fY - 2 * pts[1].fY + pts[2].fY,
2 * (pts[1].fY - pts[0].fY),
pts[0].fY - y,
roots);
SkASSERT(n <= 1);
SkScalar xt;
if (0 == n) {
SkScalar mid = SkScalarAve(y0, y2);
// Need [0] and [2] if dir == 1
// and [2] and [0] if dir == -1
xt = y < mid ? pts[1 - dir].fX : pts[dir - 1].fX;
} else {
SkScalar t = roots[0];
SkScalar C = pts[0].fX;
SkScalar A = pts[2].fX - 2 * pts[1].fX + C;
SkScalar B = 2 * (pts[1].fX - C);
xt = SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C);
}
return xt < x ? dir : 0;
}
/* 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);
}
