/** From Numerical Recipes in C.
Q = -1/2 (B + sign(B) sqrt[B*B - 4*A*C])
x1 = Q / A
x2 = C / Q
*/
int SkFindUnitQuadRoots(SkScalar A, SkScalar B, SkScalar C, SkScalar roots[2]) {
SkASSERT(roots);
if (A == 0) {
return valid_unit_divide(-C, B, roots);
}
SkScalar* r = roots;
SkScalar R = B*B - 4*A*C;
if (R < 0 || SkScalarIsNaN(R)) { // complex roots
return 0;
}
R = SkScalarSqrt(R);
SkScalar Q = (B < 0) ? -(B-R)/2 : -(B+R)/2;
r += valid_unit_divide(Q, A, r);
r += valid_unit_divide(C, Q, r);
if (r - roots == 2) {
if (roots[0] > roots[1])
SkTSwap<SkScalar>(roots[0], roots[1]);
else if (roots[0] == roots[1]) // nearly-equal?
r -= 1; // skip the double root
}
return (int)(r - roots);
}
/** From Numerical Recipes in C.
Q = -1/2 (B + sign(B) sqrt[B*B - 4*A*C])
x1 = Q / A
x2 = C / Q
*/
int SkFindUnitQuadRoots(SkScalar A, SkScalar B, SkScalar C, SkScalar roots[2]) {
SkASSERT(roots);
if (A == 0) {
return valid_unit_divide(-C, B, roots);
}
SkScalar* r = roots;
SkScalar R = B*B - 4*A*C;
if (R < 0 || !SkScalarIsFinite(R)) { // complex roots
// if R is infinite, it's possible that it may still produce
// useful results if the operation was repeated in doubles
// the flipside is determining if the more precise answer
// isn't useful because surrounding machinery (e.g., subtracting
// the axis offset from C) already discards the extra precision
// more investigation and unit tests required...
return 0;
}
R = SkScalarSqrt(R);
SkScalar Q = (B < 0) ? -(B-R)/2 : -(B+R)/2;
r += valid_unit_divide(Q, A, r);
r += valid_unit_divide(C, Q, r);
if (r - roots == 2) {
if (roots[0] > roots[1])
SkTSwap<SkScalar>(roots[0], roots[1]);
else if (roots[0] == roots[1]) // nearly-equal?
r -= 1; // skip the double root
}
return (int)(r - roots);
}
/** Quad'(t) = At + B, where
A = 2(a - 2b + c)
B = 2(b - a)
Solve for t, only if it fits between 0 < t < 1
*/
int SkFindQuadExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar tValue[1])
{
/* At + B == 0
t = -B / A
*/
#ifdef SK_SCALAR_IS_FIXED
return is_not_monotonic(a, b, c) && valid_unit_divide(a - b, a - b - b + c, tValue);
#else
return valid_unit_divide(a - b, a - b - b + c, tValue);
#endif
}
/* Returns 0 for 1 quad, and 1 for two quads, either way the answer is
stored in dst[]. Guarantees that the 1/2 quads will be monotonic.
*/
int SkChopQuadAtXExtrema(const SkPoint src[3], SkPoint dst[5])
{
SkASSERT(src);
SkASSERT(dst);
SkScalar a = src[0].fX;
SkScalar b = src[1].fX;
SkScalar c = src[2].fX;
if (is_not_monotonic(a, b, c)) {
SkScalar tValue;
if (valid_unit_divide(a - b, a - b - b + c, &tValue)) {
SkChopQuadAt(src, dst, tValue);
flatten_double_quad_extrema(&dst[0].fX);
return 1;
}
// if we get here, we need to force dst to be monotonic, even though
// we couldn't compute a unit_divide value (probably underflow).
b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c;
}
dst[0].set(a, src[0].fY);
dst[1].set(b, src[1].fY);
dst[2].set(c, src[2].fY);
return 0;
}
/** Quad'(t) = At + B, where
A = 2(a - 2b + c)
B = 2(b - a)
Solve for t, only if it fits between 0 < t < 1
*/
int SkDQuad::FindExtrema(const double src[], double tValue[1]) {
/* At + B == 0
t = -B / A
*/
double a = src[0];
double b = src[2];
double c = src[4];
return valid_unit_divide(a - b, a - b - b + c, tValue);
}
// F(t) = a (1 - t) ^ 2 + 2 b t (1 - t) + c t ^ 2
// F'(t) = 2 (b - a) + 2 (a - 2b + c) t
// F''(t) = 2 (a - 2b + c)
//
// A = 2 (b - a)
// B = 2 (a - 2b + c)
//
// Maximum curvature for a quadratic means solving
// Fx' Fx'' + Fy' Fy'' = 0
//
// t = - (Ax Bx + Ay By) / (Bx ^ 2 + By ^ 2)
//
SkScalar SkFindQuadMaxCurvature(const SkPoint src[3]) {
SkScalar Ax = src[1].fX - src[0].fX;
SkScalar Ay = src[1].fY - src[0].fY;
SkScalar Bx = src[0].fX - src[1].fX - src[1].fX + src[2].fX;
SkScalar By = src[0].fY - src[1].fY - src[1].fY + src[2].fY;
SkScalar t = 0; // 0 means don't chop
(void)valid_unit_divide(-(Ax * Bx + Ay * By), Bx * Bx + By * By, &t);
return t;
}
/** From Numerical Recipes in C.
Q = -1/2 (B + sign(B) sqrt[B*B - 4*A*C])
x1 = Q / A
x2 = C / Q
*/
int SkFindUnitQuadRoots(SkScalar A, SkScalar B, SkScalar C, SkScalar roots[2])
{
SkASSERT(roots);
if (A == 0)
return valid_unit_divide(-C, B, roots);
SkScalar* r = roots;
#ifdef SK_SCALAR_IS_FLOAT
float R = B*B - 4*A*C;
if (R < 0 || SkScalarIsNaN(R)) { // complex roots
return 0;
}
R = sk_float_sqrt(R);
#else
Sk64 RR, tmp;
RR.setMul(B,B);
tmp.setMul(A,C);
tmp.shiftLeft(2);
RR.sub(tmp);
if (RR.isNeg())
return 0;
SkFixed R = RR.getSqrt();
#endif
SkScalar Q = (B < 0) ? -(B-R)/2 : -(B+R)/2;
r += valid_unit_divide(Q, A, r);
r += valid_unit_divide(C, Q, r);
if (r - roots == 2)
{
if (roots[0] > roots[1])
SkTSwap<SkScalar>(roots[0], roots[1]);
else if (roots[0] == roots[1]) // nearly-equal?
r -= 1; // skip the double root
}
return (int)(r - roots);
}
// F(t) = a (1 - t) ^ 2 + 2 b t (1 - t) + c t ^ 2
// F'(t) = 2 (b - a) + 2 (a - 2b + c) t
// F''(t) = 2 (a - 2b + c)
//
// A = 2 (b - a)
// B = 2 (a - 2b + c)
//
// Maximum curvature for a quadratic means solving
// Fx' Fx'' + Fy' Fy'' = 0
//
// t = - (Ax Bx + Ay By) / (Bx ^ 2 + By ^ 2)
//
int SkChopQuadAtMaxCurvature(const SkPoint src[3], SkPoint dst[5])
{
SkScalar Ax = src[1].fX - src[0].fX;
SkScalar Ay = src[1].fY - src[0].fY;
SkScalar Bx = src[0].fX - src[1].fX - src[1].fX + src[2].fX;
SkScalar By = src[0].fY - src[1].fY - src[1].fY + src[2].fY;
SkScalar t = 0; // 0 means don't chop
#ifdef SK_SCALAR_IS_FLOAT
(void)valid_unit_divide(-(Ax * Bx + Ay * By), Bx * Bx + By * By, &t);
#else
// !!! should I use SkFloat here? seems like it
Sk64 numer, denom, tmp;
numer.setMul(Ax, -Bx);
tmp.setMul(Ay, -By);
numer.add(tmp);
if (numer.isPos()) // do nothing if numer <= 0
{
denom.setMul(Bx, Bx);
tmp.setMul(By, By);
denom.add(tmp);
SkASSERT(!denom.isNeg());
if (numer < denom)
{
t = numer.getFixedDiv(denom);
SkASSERT(t >= 0 && t <= SK_Fixed1); // assert that we're numerically stable (ha!)
if ((unsigned)t >= SK_Fixed1) // runtime check for numerical stability
t = 0; // ignore the chop
}
}
#endif
if (t == 0)
{
memcpy(dst, src, 3 * sizeof(SkPoint));
return 1;
}
else
{
SkChopQuadAt(src, dst, t);
return 2;
}
}
void SkChopCubicAt(const SkPoint src[4], SkPoint dst[], const SkScalar tValues[], int roots)
{
#ifdef SK_DEBUG
{
for (int i = 0; i < roots - 1; i++)
{
SkASSERT(is_unit_interval(tValues[i]));
SkASSERT(is_unit_interval(tValues[i+1]));
SkASSERT(tValues[i] < tValues[i+1]);
}
}
#endif
if (dst)
{
if (roots == 0) // nothing to chop
memcpy(dst, src, 4*sizeof(SkPoint));
else
{
SkScalar t = tValues[0];
SkPoint tmp[4];
for (int i = 0; i < roots; i++)
{
SkChopCubicAt(src, dst, t);
if (i == roots - 1)
break;
dst += 3;
// have src point to the remaining cubic (after the chop)
memcpy(tmp, dst, 4 * sizeof(SkPoint));
src = tmp;
// watch out in case the renormalized t isn't in range
if (!valid_unit_divide(tValues[i+1] - tValues[i],
SK_Scalar1 - tValues[i], &t)) {
// if we can't, just create a degenerate cubic
dst[4] = dst[5] = dst[6] = src[3];
break;
}
}
}
}
}
/* Returns 0 for 1 quad, and 1 for two quads, either way the answer is
stored in dst[]. Guarantees that the 1/2 quads will be monotonic.
*/
int SkChopQuadAtYExtrema(const SkPoint src[3], SkPoint dst[5])
{
SkASSERT(src);
SkASSERT(dst);
#if 0
static bool once = true;
if (once)
{
once = false;
SkPoint s[3] = { 0, 26398, 0, 26331, 0, 20621428 };
SkPoint d[6];
int n = SkChopQuadAtYExtrema(s, d);
SkDebugf("chop=%d, Y=[%x %x %x %x %x %x]\n", n, d[0].fY, d[1].fY, d[2].fY, d[3].fY, d[4].fY, d[5].fY);
}
#endif
SkScalar a = src[0].fY;
SkScalar b = src[1].fY;
SkScalar c = src[2].fY;
if (is_not_monotonic(a, b, c))
{
SkScalar tValue;
if (valid_unit_divide(a - b, a - b - b + c, &tValue))
{
SkChopQuadAt(src, dst, tValue);
flatten_double_quad_extrema(&dst[0].fY);
return 1;
}
// if we get here, we need to force dst to be monotonic, even though
// we couldn't compute a unit_divide value (probably underflow).
b = SkScalarAbs(a - b) < SkScalarAbs(b - c) ? a : c;
}
dst[0].set(src[0].fX, a);
dst[1].set(src[1].fX, b);
dst[2].set(src[2].fX, c);
return 0;
}
/** Quad'(t) = At + B, where
A = 2(a - 2b + c)
B = 2(b - a)
Solve for t, only if it fits between 0 < t < 1
*/
int SkFindQuadExtrema(SkScalar a, SkScalar b, SkScalar c, SkScalar tValue[1]) {
/* At + B == 0
t = -B / A
*/
return valid_unit_divide(a - b, a - b - b + c, tValue);
}
/** Quad'(t) = At + B, where
A = 2(a - 2b + c)
B = 2(b - a)
Solve for t, only if it fits between 0 < t < 1
*/
int SkDQuad::FindExtrema(double a, double b, double c, double tValue[1]) {
/* At + B == 0
t = -B / A
*/
return valid_unit_divide(a - b, a - b - b + c, tValue);
}
