/*
* Copyright 2006 The Android Open Source Project
*
* Use of this source code is governed by a BSD-style license that can be
* found in the LICENSE file.
*/
#include "SkGeometry.h"
#include "Sk64.h"
#include "SkMatrix.h"
bool SkXRayCrossesLine(const SkXRay& pt, const SkPoint pts[2], bool* ambiguous) {
if (ambiguous) {
*ambiguous = false;
}
// Determine quick discards.
// Consider query line going exactly through point 0 to not
// intersect, for symmetry with SkXRayCrossesMonotonicCubic.
if (pt.fY == pts[0].fY) {
if (ambiguous) {
*ambiguous = true;
}
return false;
}
if (pt.fY < pts[0].fY && pt.fY < pts[1].fY)
return false;
if (pt.fY > pts[0].fY && pt.fY > pts[1].fY)
return false;
if (pt.fX > pts[0].fX && pt.fX > pts[1].fX)
return false;
// Determine degenerate cases
if (SkScalarNearlyZero(pts[0].fY - pts[1].fY))
return false;
if (SkScalarNearlyZero(pts[0].fX - pts[1].fX)) {
// We've already determined the query point lies within the
// vertical range of the line segment.
if (pt.fX <= pts[0].fX) {
if (ambiguous) {
*ambiguous = (pt.fY == pts[1].fY);
}
return true;
}
return false;
}
// Ambiguity check
if (pt.fY == pts[1].fY) {
if (pt.fX <= pts[1].fX) {
if (ambiguous) {
*ambiguous = true;
}
return true;
}
return false;
}
// Full line segment evaluation
SkScalar delta_y = pts[1].fY - pts[0].fY;
SkScalar delta_x = pts[1].fX - pts[0].fX;
SkScalar slope = SkScalarDiv(delta_y, delta_x);
SkScalar b = pts[0].fY - SkScalarMul(slope, pts[0].fX);
// Solve for x coordinate at y = pt.fY
SkScalar x = SkScalarDiv(pt.fY - b, slope);
return pt.fX <= x;
}
/** If defined, this makes eval_quad and eval_cubic do more setup (sometimes
involving integer multiplies by 2 or 3, but fewer calls to SkScalarMul.
May also introduce overflow of fixed when we compute our setup.
*/
#ifdef SK_SCALAR_IS_FIXED
#define DIRECT_EVAL_OF_POLYNOMIALS
#endif
////////////////////////////////////////////////////////////////////////
#ifdef SK_SCALAR_IS_FIXED
static int is_not_monotonic(int a, int b, int c, int d)
{
return (((a - b) | (b - c) | (c - d)) & ((b - a) | (c - b) | (d - c))) >> 31;
}
static int is_not_monotonic(int a, int b, int c)
{
return (((a - b) | (b - c)) & ((b - a) | (c - b))) >> 31;
}
#else
static int is_not_monotonic(float a, float b, float c)
{
float ab = a - b;
float bc = b - c;
if (ab < 0)
bc = -bc;
return ab == 0 || bc < 0;
}
#endif
////////////////////////////////////////////////////////////////////////
static bool is_unit_interval(SkScalar x)
{
return x > 0 && x < SK_Scalar1;
}
static int valid_unit_divide(SkScalar numer, SkScalar denom, SkScalar* ratio)
{
SkASSERT(ratio);
if (numer < 0)
{
numer = -numer;
denom = -denom;
}
if (denom == 0 || numer == 0 || numer >= denom)
return 0;
SkScalar r = SkScalarDiv(numer, denom);
if (SkScalarIsNaN(r)) {
return 0;
}
SkASSERT(r >= 0 && r < SK_Scalar1);
if (r == 0) // catch underflow if numer <<<< denom
return 0;
*ratio = r;
return 1;
}
/** 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);
}
#ifdef SK_SCALAR_IS_FIXED
/** 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);
}
#endif
/////////////////////////////////////////////////////////////////////////////////////
/////////////////////////////////////////////////////////////////////////////////////
static SkScalar eval_quad(const SkScalar src[], SkScalar t)
{
SkASSERT(src);
SkASSERT(t >= 0 && t <= SK_Scalar1);
#ifdef DIRECT_EVAL_OF_POLYNOMIALS
SkScalar C = src[0];
SkScalar A = src[4] - 2 * src[2] + C;
SkScalar B = 2 * (src[2] - C);
return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C);
#else
SkScalar ab = SkScalarInterp(src[0], src[2], t);
SkScalar bc = SkScalarInterp(src[2], src[4], t);
return SkScalarInterp(ab, bc, t);
#endif
}
static SkScalar eval_quad_derivative(const SkScalar src[], SkScalar t)
{
SkScalar A = src[4] - 2 * src[2] + src[0];
SkScalar B = src[2] - src[0];
return 2 * SkScalarMulAdd(A, t, B);
}
static SkScalar eval_quad_derivative_at_half(const SkScalar src[])
{
SkScalar A = src[4] - 2 * src[2] + src[0];
SkScalar B = src[2] - src[0];
return A + 2 * B;
}
void SkEvalQuadAt(const SkPoint src[3], SkScalar t, SkPoint* pt, SkVector* tangent)
{
SkASSERT(src);
SkASSERT(t >= 0 && t <= SK_Scalar1);
if (pt)
pt->set(eval_quad(&src[0].fX, t), eval_quad(&src[0].fY, t));
if (tangent)
tangent->set(eval_quad_derivative(&src[0].fX, t),
eval_quad_derivative(&src[0].fY, t));
}
void SkEvalQuadAtHalf(const SkPoint src[3], SkPoint* pt, SkVector* tangent)
{
SkASSERT(src);
if (pt)
{
SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX);
SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY);
SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX);
SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY);
pt->set(SkScalarAve(x01, x12), SkScalarAve(y01, y12));
}
if (tangent)
tangent->set(eval_quad_derivative_at_half(&src[0].fX),
eval_quad_derivative_at_half(&src[0].fY));
}
static void interp_quad_coords(const SkScalar* src, SkScalar* dst, SkScalar t)
{
SkScalar ab = SkScalarInterp(src[0], src[2], t);
SkScalar bc = SkScalarInterp(src[2], src[4], t);
dst[0] = src[0];
dst[2] = ab;
dst[4] = SkScalarInterp(ab, bc, t);
dst[6] = bc;
dst[8] = src[4];
}
void SkChopQuadAt(const SkPoint src[3], SkPoint dst[5], SkScalar t)
{
SkASSERT(t > 0 && t < SK_Scalar1);
interp_quad_coords(&src[0].fX, &dst[0].fX, t);
interp_quad_coords(&src[0].fY, &dst[0].fY, t);
}
void SkChopQuadAtHalf(const SkPoint src[3], SkPoint dst[5])
{
SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX);
SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY);
SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX);
SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY);
dst[0] = src[0];
dst[1].set(x01, y01);
dst[2].set(SkScalarAve(x01, x12), SkScalarAve(y01, y12));
dst[3].set(x12, y12);
dst[4] = src[2];
}
/** 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
}
static inline void flatten_double_quad_extrema(SkScalar coords[14])
{
coords[2] = coords[6] = coords[4];
}
/* 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;
}
/* 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;
}
// 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)
//
float 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
#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
return t;
}
int SkChopQuadAtMaxCurvature(const SkPoint src[3], SkPoint dst[5])
{
SkScalar t = SkFindQuadMaxCurvature(src);
if (t == 0) {
memcpy(dst, src, 3 * sizeof(SkPoint));
return 1;
} else {
SkChopQuadAt(src, dst, t);
return 2;
}
}
#ifdef SK_SCALAR_IS_FLOAT
#define SK_ScalarTwoThirds (0.666666666f)
#else
#define SK_ScalarTwoThirds ((SkFixed)(43691))
#endif
void SkConvertQuadToCubic(const SkPoint src[3], SkPoint dst[4]) {
const SkScalar scale = SK_ScalarTwoThirds;
dst[0] = src[0];
dst[1].set(src[0].fX + SkScalarMul(src[1].fX - src[0].fX, scale),
src[0].fY + SkScalarMul(src[1].fY - src[0].fY, scale));
dst[2].set(src[2].fX + SkScalarMul(src[1].fX - src[2].fX, scale),
src[2].fY + SkScalarMul(src[1].fY - src[2].fY, scale));
dst[3] = src[2];
}
////////////////////////////////////////////////////////////////////////////////////////
///// CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS /////
////////////////////////////////////////////////////////////////////////////////////////
static void get_cubic_coeff(const SkScalar pt[], SkScalar coeff[4])
{
coeff[0] = pt[6] + 3*(pt[2] - pt[4]) - pt[0];
coeff[1] = 3*(pt[4] - pt[2] - pt[2] + pt[0]);
coeff[2] = 3*(pt[2] - pt[0]);
coeff[3] = pt[0];
}
void SkGetCubicCoeff(const SkPoint pts[4], SkScalar cx[4], SkScalar cy[4])
{
SkASSERT(pts);
if (cx)
get_cubic_coeff(&pts[0].fX, cx);
if (cy)
get_cubic_coeff(&pts[0].fY, cy);
}
static SkScalar eval_cubic(const SkScalar src[], SkScalar t)
{
SkASSERT(src);
SkASSERT(t >= 0 && t <= SK_Scalar1);
if (t == 0)
return src[0];
#ifdef DIRECT_EVAL_OF_POLYNOMIALS
SkScalar D = src[0];
SkScalar A = src[6] + 3*(src[2] - src[4]) - D;
SkScalar B = 3*(src[4] - src[2] - src[2] + D);
SkScalar C = 3*(src[2] - D);
return SkScalarMulAdd(SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C), t, D);
#else
SkScalar ab = SkScalarInterp(src[0], src[2], t);
SkScalar bc = SkScalarInterp(src[2], src[4], t);
SkScalar cd = SkScalarInterp(src[4], src[6], t);
SkScalar abc = SkScalarInterp(ab, bc, t);
SkScalar bcd = SkScalarInterp(bc, cd, t);
return SkScalarInterp(abc, bcd, t);
#endif
}
/** return At^2 + Bt + C
*/
static SkScalar eval_quadratic(SkScalar A, SkScalar B, SkScalar C, SkScalar t)
{
SkASSERT(t >= 0 && t <= SK_Scalar1);
return SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C);
}
static SkScalar eval_cubic_derivative(const SkScalar src[], SkScalar t)
{
SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0];
SkScalar B = 2*(src[4] - 2 * src[2] + src[0]);
SkScalar C = src[2] - src[0];
return eval_quadratic(A, B, C, t);
}
static SkScalar eval_cubic_2ndDerivative(const SkScalar src[], SkScalar t)
{
SkScalar A = src[6] + 3*(src[2] - src[4]) - src[0];
SkScalar B = src[4] - 2 * src[2] + src[0];
return SkScalarMulAdd(A, t, B);
}
void SkEvalCubicAt(const SkPoint src[4], SkScalar t, SkPoint* loc, SkVector* tangent, SkVector* curvature)
{
SkASSERT(src);
SkASSERT(t >= 0 && t <= SK_Scalar1);
if (loc)
loc->set(eval_cubic(&src[0].fX, t), eval_cubic(&src[0].fY, t));
if (tangent)
tangent->set(eval_cubic_derivative(&src[0].fX, t),
eval_cubic_derivative(&src[0].fY, t));
if (curvature)
curvature->set(eval_cubic_2ndDerivative(&src[0].fX, t),
eval_cubic_2ndDerivative(&src[0].fY, t));
}
/** 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 void interp_cubic_coords(const SkScalar* src, SkScalar* dst, SkScalar t)
{
SkScalar ab = SkScalarInterp(src[0], src[2], t);
SkScalar bc = SkScalarInterp(src[2], src[4], t);
SkScalar cd = SkScalarInterp(src[4], src[6], t);
SkScalar abc = SkScalarInterp(ab, bc, t);
SkScalar bcd = SkScalarInterp(bc, cd, t);
SkScalar abcd = SkScalarInterp(abc, bcd, t);
dst[0] = src[0];
dst[2] = ab;
dst[4] = abc;
dst[6] = abcd;
dst[8] = bcd;
dst[10] = cd;
dst[12] = src[6];
}
void SkChopCubicAt(const SkPoint src[4], SkPoint dst[7], SkScalar t)
{
SkASSERT(t > 0 && t < SK_Scalar1);
interp_cubic_coords(&src[0].fX, &dst[0].fX, t);
interp_cubic_coords(&src[0].fY, &dst[0].fY, t);
}
/* http://code.google.com/p/skia/issues/detail?id=32
This test code would fail when we didn't check the return result of
valid_unit_divide in SkChopCubicAt(... tValues[], int roots). The reason is
that after the first chop, the parameters to valid_unit_divide are equal
(thanks to finite float precision and rounding in the subtracts). Thus
even though the 2nd tValue looks < 1.0, after we renormalize it, we end
up with 1.0, hence the need to check and just return the last cubic as
a degenerate clump of 4 points in the sampe place.
static void test_cubic() {
SkPoint src[4] = {
{ 556.25000, 523.03003 },
{ 556.23999, 522.96002 },
{ 556.21997, 522.89001 },
{ 556.21997, 522.82001 }
};
SkPoint dst[10];
SkScalar tval[] = { 0.33333334f, 0.99999994f };
SkChopCubicAt(src, dst, tval, 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;
}
}
}
}
}
void SkChopCubicAtHalf(const SkPoint src[4], SkPoint dst[7])
{
SkScalar x01 = SkScalarAve(src[0].fX, src[1].fX);
SkScalar y01 = SkScalarAve(src[0].fY, src[1].fY);
SkScalar x12 = SkScalarAve(src[1].fX, src[2].fX);
SkScalar y12 = SkScalarAve(src[1].fY, src[2].fY);
SkScalar x23 = SkScalarAve(src[2].fX, src[3].fX);
SkScalar y23 = SkScalarAve(src[2].fY, src[3].fY);
SkScalar x012 = SkScalarAve(x01, x12);
SkScalar y012 = SkScalarAve(y01, y12);
SkScalar x123 = SkScalarAve(x12, x23);
SkScalar y123 = SkScalarAve(y12, y23);
dst[0] = src[0];
dst[1].set(x01, y01);
dst[2].set(x012, y012);
dst[3].set(SkScalarAve(x012, x123), SkScalarAve(y012, y123));
dst[4].set(x123, y123);
dst[5].set(x23, y23);
dst[6] = src[3];
}
static void flatten_double_cubic_extrema(SkScalar coords[14])
{
coords[4] = coords[8] = coords[6];
}
/** Given 4 points on a cubic bezier, chop it into 1, 2, 3 beziers such that
the resulting beziers are monotonic in Y. This is called by the scan converter.
Depending on what is returned, dst[] is treated as follows
0 dst[0..3] is the original cubic
1 dst[0..3] and dst[3..6] are the two new cubics
2 dst[0..3], dst[3..6], dst[6..9] are the three new cubics
If dst == null, it is ignored and only the count is returned.
*/
int SkChopCubicAtYExtrema(const SkPoint src[4], SkPoint dst[10]) {
SkScalar tValues[2];
int roots = SkFindCubicExtrema(src[0].fY, src[1].fY, src[2].fY,
src[3].fY, tValues);
SkChopCubicAt(src, dst, tValues, roots);
if (dst && roots > 0) {
// we do some cleanup to ensure our Y extrema are flat
flatten_double_cubic_extrema(&dst[0].fY);
if (roots == 2) {
flatten_double_cubic_extrema(&dst[3].fY);
}
}
return roots;
}
int SkChopCubicAtXExtrema(const SkPoint src[4], SkPoint dst[10]) {
SkScalar tValues[2];
int roots = SkFindCubicExtrema(src[0].fX, src[1].fX, src[2].fX,
src[3].fX, tValues);
SkChopCubicAt(src, dst, tValues, roots);
if (dst && roots > 0) {
// we do some cleanup to ensure our Y extrema are flat
flatten_double_cubic_extrema(&dst[0].fX);
if (roots == 2) {
flatten_double_cubic_extrema(&dst[3].fX);
}
}
return 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;
int count;
#ifdef SK_SCALAR_IS_FLOAT
count = SkFindUnitQuadRoots(Bx*Cy - By*Cx, Ax*Cy - Ay*Cx, Ax*By - Ay*Bx, tValues);
#else
Sk64 A, B, C, tmp;
A.setMul(Bx, Cy);
tmp.setMul(By, Cx);
A.sub(tmp);
B.setMul(Ax, Cy);
tmp.setMul(Ay, Cx);
B.sub(tmp);
C.setMul(Ax, By);
tmp.setMul(Ay, Bx);
C.sub(tmp);
count = Sk64FindFixedQuadRoots(A, B, C, tValues);
#endif
return count;
}
int SkChopCubicAtInflections(const SkPoint src[], SkPoint dst[10])
{
SkScalar tValues[2];
int count = SkFindCubicInflections(src, tValues);
if (dst)
{
if (count == 0)
memcpy(dst, src, 4 * sizeof(SkPoint));
else
SkChopCubicAt(src, dst, tValues, count);
}
return count + 1;
}
template <typename T> void bubble_sort(T array[], int count)
{
for (int i = count - 1; i > 0; --i)
for (int j = i; j > 0; --j)
if (array[j] < array[j-1])
{
T tmp(array[j]);
array[j] = array[j-1];
array[j-1] = tmp;
}
}
#include "SkFP.h"
// newton refinement
#if 0
static SkScalar refine_cubic_root(const SkFP coeff[4], SkScalar root)
{
// x1 = x0 - f(t) / f'(t)
SkFP T = SkScalarToFloat(root);
SkFP N, D;
// f' = 3*coeff[0]*T^2 + 2*coeff[1]*T + coeff[2]
D = SkFPMul(SkFPMul(coeff[0], SkFPMul(T,T)), 3);
D = SkFPAdd(D, SkFPMulInt(SkFPMul(coeff[1], T), 2));
D = SkFPAdd(D, coeff[2]);
if (D == 0)
return root;
// f = coeff[0]*T^3 + coeff[1]*T^2 + coeff[2]*T + coeff[3]
N = SkFPMul(SkFPMul(SkFPMul(T, T), T), coeff[0]);
N = SkFPAdd(N, SkFPMul(SkFPMul(T, T), coeff[1]));
N = SkFPAdd(N, SkFPMul(T, coeff[2]));
N = SkFPAdd(N, coeff[3]);
if (N)
{
SkScalar delta = SkFPToScalar(SkFPDiv(N, D));
if (delta)
root -= delta;
}
return root;
}
#endif
/**
* Given an array and count, remove all pair-wise duplicates from the array,
* keeping the existing sorting, and return the new count
*/
static int collaps_duplicates(float array[], int count) {
for (int n = count; n > 1; --n) {
if (array[0] == array[1]) {
for (int i = 1; i < n; ++i) {
array[i - 1] = array[i];
}
count -= 1;
} else {
array += 1;
}
}
return count;
}
#ifdef SK_DEBUG
#define TEST_COLLAPS_ENTRY(array) array, SK_ARRAY_COUNT(array)
static void test_collaps_duplicates() {
static bool gOnce;
if (gOnce) { return; }
gOnce = true;
const float src0[] = { 0 };
const float src1[] = { 0, 0 };
const float src2[] = { 0, 1 };
const float src3[] = { 0, 0, 0 };
const float src4[] = { 0, 0, 1 };
const float src5[] = { 0, 1, 1 };
const float src6[] = { 0, 1, 2 };
const struct {
const float* fData;
int fCount;
int fCollapsedCount;
} data[] = {
{ TEST_COLLAPS_ENTRY(src0), 1 },
{ TEST_COLLAPS_ENTRY(src1), 1 },
{ TEST_COLLAPS_ENTRY(src2), 2 },
{ TEST_COLLAPS_ENTRY(src3), 1 },
{ TEST_COLLAPS_ENTRY(src4), 2 },
{ TEST_COLLAPS_ENTRY(src5), 2 },
{ TEST_COLLAPS_ENTRY(src6), 3 },
};
for (size_t i = 0; i < SK_ARRAY_COUNT(data); ++i) {
float dst[3];
memcpy(dst, data[i].fData, data[i].fCount * sizeof(dst[0]));
int count = collaps_duplicates(dst, data[i].fCount);
SkASSERT(data[i].fCollapsedCount == count);
for (int j = 1; j < count; ++j) {
SkASSERT(dst[j-1] < dst[j]);
}
}
}
#endif
#if defined _WIN32 && _MSC_VER >= 1300 && defined SK_SCALAR_IS_FIXED // disable warning : unreachable code if building fixed point for windows desktop
#pragma warning ( disable : 4702 )
#endif
/* 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();)
// now sort the roots
int count = (int)(roots - tValues);
SkASSERT((unsigned)count <= 3);
bubble_sort(tValues, count);
count = collaps_duplicates(tValues, count);
roots = tValues + count; // so we compute the proper count below
#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);
}
/* Looking for F' dot F'' == 0
A = b - a
B = c - 2b + a
C = d - 3c + 3b - a
F' = 3Ct^2 + 6Bt + 3A
F'' = 6Ct + 6B
F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB
*/
static void formulate_F1DotF2(const SkScalar src[], SkFP coeff[4])
{
SkScalar a = src[2] - src[0];
SkScalar b = src[4] - 2 * src[2] + src[0];
SkScalar c = src[6] + 3 * (src[2] - src[4]) - src[0];
SkFP A = SkScalarToFP(a);
SkFP B = SkScalarToFP(b);
SkFP C = SkScalarToFP(c);
coeff[0] = SkFPMul(C, C);
coeff[1] = SkFPMulInt(SkFPMul(B, C), 3);
coeff[2] = SkFPMulInt(SkFPMul(B, B), 2);
coeff[2] = SkFPAdd(coeff[2], SkFPMul(C, A));
coeff[3] = SkFPMul(A, B);
}
// EXPERIMENTAL: can set this to zero to accept all t-values 0 < t < 1
//#define kMinTValueForChopping (SK_Scalar1 / 256)
#define kMinTValueForChopping 0
/* Looking for F' dot F'' == 0
A = b - a
B = c - 2b + a
C = d - 3c + 3b - a
F' = 3Ct^2 + 6Bt + 3A
F'' = 6Ct + 6B
F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB
*/
int SkFindCubicMaxCurvature(const SkPoint src[4], SkScalar tValues[3])
{
SkFP coeffX[4], coeffY[4];
int i;
formulate_F1DotF2(&src[0].fX, coeffX);
formulate_F1DotF2(&src[0].fY, coeffY);
for (i = 0; i < 4; i++)
coeffX[i] = SkFPAdd(coeffX[i],coeffY[i]);
SkScalar t[3];
int count = solve_cubic_polynomial(coeffX, t);
int maxCount = 0;
// now remove extrema where the curvature is zero (mins)
// !!!! need a test for this !!!!
for (i = 0; i < count; i++)
{
// if (not_min_curvature())
if (t[i] > kMinTValueForChopping && t[i] < SK_Scalar1 - kMinTValueForChopping)
tValues[maxCount++] = t[i];
}
return maxCount;
}
int SkChopCubicAtMaxCurvature(const SkPoint src[4], SkPoint dst[13], SkScalar tValues[3])
{
SkScalar t_storage[3];
if (tValues == NULL)
tValues = t_storage;
int count = SkFindCubicMaxCurvature(src, tValues);
if (dst) {
if (count == 0)
memcpy(dst, src, 4 * sizeof(SkPoint));
else
SkChopCubicAt(src, dst, tValues, count);
}
return count + 1;
}
bool SkXRayCrossesMonotonicCubic(const SkXRay& pt, const SkPoint cubic[4], bool* ambiguous) {
if (ambiguous) {
*ambiguous = false;
}
// Find the minimum and maximum y of the extrema, which are the
// first and last points since this cubic is monotonic
SkScalar min_y = SkMinScalar(cubic[0].fY, cubic[3].fY);
SkScalar max_y = SkMaxScalar(cubic[0].fY, cubic[3].fY);
if (pt.fY == cubic[0].fY
|| pt.fY < min_y
|| pt.fY > max_y) {
// The query line definitely does not cross the curve
if (ambiguous) {
*ambiguous = (pt.fY == cubic[0].fY);
}
return false;
}
bool pt_at_extremum = (pt.fY == cubic[3].fY);
SkScalar min_x =
SkMinScalar(
SkMinScalar(
SkMinScalar(cubic[0].fX, cubic[1].fX),
cubic[2].fX),
cubic[3].fX);
if (pt.fX < min_x) {
// The query line definitely crosses the curve
if (ambiguous) {
*ambiguous = pt_at_extremum;
}
return true;
}
SkScalar max_x =
SkMaxScalar(
SkMaxScalar(
SkMaxScalar(cubic[0].fX, cubic[1].fX),
cubic[2].fX),
cubic[3].fX);
if (pt.fX > max_x) {
// The query line definitely does not cross the curve
return false;
}
// Do a binary search to find the parameter value which makes y as
// close as possible to the query point. See whether the query
// line's origin is to the left of the associated x coordinate.
// kMaxIter is chosen as the number of mantissa bits for a float,
// since there's no way we are going to get more precision by
// iterating more times than that.
const int kMaxIter = 23;
SkPoint eval;
int iter = 0;
SkScalar upper_t;
SkScalar lower_t;
// Need to invert direction of t parameter if cubic goes up
// instead of down
if (cubic[3].fY > cubic[0].fY) {
upper_t = SK_Scalar1;
lower_t = 0;
} else {
upper_t = 0;
lower_t = SK_Scalar1;
}
do {
SkScalar t = SkScalarAve(upper_t, lower_t);
SkEvalCubicAt(cubic, t, &eval, NULL, NULL);
if (pt.fY > eval.fY) {
lower_t = t;
} else {
upper_t = t;
}
} while (++iter < kMaxIter
&& !SkScalarNearlyZero(eval.fY - pt.fY));
if (pt.fX <= eval.fX) {
if (ambiguous) {
*ambiguous = pt_at_extremum;
}
return true;
}
return false;
}
int SkNumXRayCrossingsForCubic(const SkXRay& pt, const SkPoint cubic[4], bool* ambiguous) {
int num_crossings = 0;
SkPoint monotonic_cubics[10];
int num_monotonic_cubics = SkChopCubicAtYExtrema(cubic, monotonic_cubics);
if (ambiguous) {
*ambiguous = false;
}
bool locally_ambiguous;
if (SkXRayCrossesMonotonicCubic(pt, &monotonic_cubics[0], &locally_ambiguous))
++num_crossings;
if (ambiguous) {
*ambiguous |= locally_ambiguous;
}
if (num_monotonic_cubics > 0)
if (SkXRayCrossesMonotonicCubic(pt, &monotonic_cubics[3], &locally_ambiguous))
++num_crossings;
if (ambiguous) {
*ambiguous |= locally_ambiguous;
}
if (num_monotonic_cubics > 1)
if (SkXRayCrossesMonotonicCubic(pt, &monotonic_cubics[6], &locally_ambiguous))
++num_crossings;
if (ambiguous) {
*ambiguous |= locally_ambiguous;
}
return num_crossings;
}
////////////////////////////////////////////////////////////////////////////////
/* 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;
}
/* given a quad-curve and a point (x,y), chop the quad at that point and place
the new off-curve point and endpoint into 'dest'.
Should only return false if the computed pos is the start of the curve
(i.e. root == 0)
*/
static bool truncate_last_curve(const SkPoint quad[3], SkScalar x, SkScalar y, SkPoint* dest)
{
const SkScalar* base;
SkScalar value;
if (SkScalarAbs(x) < SkScalarAbs(y)) {
base = &quad[0].fX;
value = x;
} else {
base = &quad[0].fY;
value = y;
}
// note: this returns 0 if it thinks value is out of range, meaning the
// root might return something outside of [0, 1)
SkScalar t = quad_solve(base[0], base[2], base[4], value);
if (t > 0)
{
SkPoint tmp[5];
SkChopQuadAt(quad, tmp, t);
dest[0] = tmp[1];
dest[1].set(x, y);
return true;
} else {
/* t == 0 means either the value triggered a root outside of [0, 1)
For our purposes, we can ignore the <= 0 roots, but we want to
catch the >= 1 roots (which given our caller, will basically mean
a root of 1, give-or-take numerical instability). If we are in the
>= 1 case, return the existing offCurve point.
The test below checks to see if we are close to the "end" of the
curve (near base[4]). Rather than specifying a tolerance, I just
check to see if value is on to the right/left of the middle point
(depending on the direction/sign of the end points).
*/
if ((base[0] < base[4] && value > base[2]) ||
(base[0] > base[4] && value < base[2])) // should root have been 1
{
dest[0] = quad[1];
dest[1].set(x, y);
return true;
}
}
return false;
}
static const SkPoint gQuadCirclePts[kSkBuildQuadArcStorage] = {
// The mid point of the quadratic arc approximation is half way between the two
// control points. The float epsilon adjustment moves the on curve point out by
// two bits, distributing the convex test error between the round rect approximation
// and the convex cross product sign equality test.
#define SK_MID_RRECT_OFFSET (SK_Scalar1 + SK_ScalarTanPIOver8 + FLT_EPSILON * 4) / 2
{ SK_Scalar1, 0 },
{ SK_Scalar1, SK_ScalarTanPIOver8 },
{ SK_MID_RRECT_OFFSET, SK_MID_RRECT_OFFSET },
{ SK_ScalarTanPIOver8, SK_Scalar1 },
{ 0, SK_Scalar1 },
{ -SK_ScalarTanPIOver8, SK_Scalar1 },
{ -SK_MID_RRECT_OFFSET, SK_MID_RRECT_OFFSET },
{ -SK_Scalar1, SK_ScalarTanPIOver8 },
{ -SK_Scalar1, 0 },
{ -SK_Scalar1, -SK_ScalarTanPIOver8 },
{ -SK_MID_RRECT_OFFSET, -SK_MID_RRECT_OFFSET },
{ -SK_ScalarTanPIOver8, -SK_Scalar1 },
{ 0, -SK_Scalar1 },
{ SK_ScalarTanPIOver8, -SK_Scalar1 },
{ SK_MID_RRECT_OFFSET, -SK_MID_RRECT_OFFSET },
{ SK_Scalar1, -SK_ScalarTanPIOver8 },
{ SK_Scalar1, 0 }
#undef SK_MID_RRECT_OFFSET
};
int SkBuildQuadArc(const SkVector& uStart, const SkVector& uStop,
SkRotationDirection dir, const SkMatrix* userMatrix,
SkPoint quadPoints[])
{
// rotate by x,y so that uStart is (1.0)
SkScalar x = SkPoint::DotProduct(uStart, uStop);
SkScalar y = SkPoint::CrossProduct(uStart, uStop);
SkScalar absX = SkScalarAbs(x);
SkScalar absY = SkScalarAbs(y);
int pointCount;
// check for (effectively) coincident vectors
// this can happen if our angle is nearly 0 or nearly 180 (y == 0)
// ... we use the dot-prod to distinguish between 0 and 180 (x > 0)
if (absY <= SK_ScalarNearlyZero && x > 0 &&
((y >= 0 && kCW_SkRotationDirection == dir) ||
(y <= 0 && kCCW_SkRotationDirection == dir))) {
// just return the start-point
quadPoints[0].set(SK_Scalar1, 0);
pointCount = 1;
} else {
if (dir == kCCW_SkRotationDirection)
y = -y;
// what octant (quadratic curve) is [xy] in?
int oct = 0;
bool sameSign = true;
if (0 == y)
{
oct = 4; // 180
SkASSERT(SkScalarAbs(x + SK_Scalar1) <= SK_ScalarNearlyZero);
}
else if (0 == x)
{
SkASSERT(absY - SK_Scalar1 <= SK_ScalarNearlyZero);
if (y > 0)
oct = 2; // 90
else
oct = 6; // 270
}
else
{
if (y < 0)
oct += 4;
if ((x < 0) != (y < 0))
{
oct += 2;
sameSign = false;
}
if ((absX < absY) == sameSign)
oct += 1;
}
int wholeCount = oct << 1;
memcpy(quadPoints, gQuadCirclePts, (wholeCount + 1) * sizeof(SkPoint));
const SkPoint* arc = &gQuadCirclePts[wholeCount];
if (truncate_last_curve(arc, x, y, &quadPoints[wholeCount + 1]))
{
wholeCount += 2;
}
pointCount = wholeCount + 1;
}
// now handle counter-clockwise and the initial unitStart rotation
SkMatrix matrix;
matrix.setSinCos(uStart.fY, uStart.fX);
if (dir == kCCW_SkRotationDirection) {
matrix.preScale(SK_Scalar1, -SK_Scalar1);
}
if (userMatrix) {
matrix.postConcat(*userMatrix);
}
matrix.mapPoints(quadPoints, pointCount);
return pointCount;
}
///////////////////////////////////////////////////////////////////////////////
// F = (A (1 - t)^2 + C t^2 + 2 B (1 - t) t w)
// ------------------------------------------
// ((1 - t)^2 + t^2 + 2 (1 - t) t w)
//
// = {t^2 (P0 + P2 - 2 P1 w), t (-2 P0 + 2 P1 w), P0}
// ------------------------------------------------
// {t^2 (2 - 2 w), t (-2 + 2 w), 1}
//
// Take the parametric specification for the conic (either X or Y) and return
// in coeff[] the coefficients for the simple quadratic polynomial
// coeff[0] for t^2
// coeff[1] for t
// coeff[2] for constant term
//
static SkScalar conic_eval_pos(const SkScalar src[], SkScalar w, SkScalar t) {
SkASSERT(src);
SkASSERT(t >= 0 && t <= SK_Scalar1);
SkScalar src2w = SkScalarMul(src[2], w);
SkScalar C = src[0];
SkScalar A = src[4] - 2 * src2w + C;
SkScalar B = 2 * (src2w - C);
SkScalar numer = SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C);
B = 2 * (w - SK_Scalar1);
C = SK_Scalar1;
A = -B;
SkScalar denom = SkScalarMulAdd(SkScalarMulAdd(A, t, B), t, C);
return SkScalarDiv(numer, denom);
}
// F' = 2 (C t (1 + t (-1 + w)) - A (-1 + t) (t (-1 + w) - w) + B (1 - 2 t) w)
//
// t^2 : (2 P0 - 2 P2 - 2 P0 w + 2 P2 w)
// t^1 : (-2 P0 + 2 P2 + 4 P0 w - 4 P1 w)
// t^0 : -2 P0 w + 2 P1 w
//
// We disregard magnitude, so we can freely ignore the denominator of F', and
// divide the numerator by 2
//
// coeff[0] for t^2
// coeff[1] for t^1
// coeff[2] for t^0
//
static void conic_deriv_coeff(const SkScalar src[], SkScalar w, SkScalar coeff[3]) {
const SkScalar P20 = src[4] - src[0];
const SkScalar P10 = src[2] - src[0];
const SkScalar wP10 = w * P10;
coeff[0] = w * P20 - P20;
coeff[1] = P20 - 2 * wP10;
coeff[2] = wP10;
}
static SkScalar conic_eval_tan(const SkScalar coord[], SkScalar w, SkScalar t) {
SkScalar coeff[3];
conic_deriv_coeff(coord, w, coeff);
return t * (t * coeff[0] + coeff[1]) + coeff[2];
}
static bool conic_find_extrema(const SkScalar src[], SkScalar w, SkScalar* t) {
SkScalar coeff[3];
conic_deriv_coeff(src, w, coeff);
SkScalar tValues[2];
int roots = SkFindUnitQuadRoots(coeff[0], coeff[1], coeff[2], tValues);
SkASSERT(0 == roots || 1 == roots);
if (1 == roots) {
*t = tValues[0];
return true;
}
return false;
}
struct SkP3D {
SkScalar fX, fY, fZ;
void set(SkScalar x, SkScalar y, SkScalar z) {
fX = x; fY = y; fZ = z;
}
void projectDown(SkPoint* dst) const {
dst->set(fX / fZ, fY / fZ);
}
};
// we just return the middle 3 points, since the first and last are dups of src
//
static void p3d_interp(const SkScalar src[3], SkScalar dst[3], SkScalar t) {
SkScalar ab = SkScalarInterp(src[0], src[3], t);
SkScalar bc = SkScalarInterp(src[3], src[6], t);
dst[0] = ab;
dst[3] = SkScalarInterp(ab, bc, t);
dst[6] = bc;
}
static void ratquad_mapTo3D(const SkPoint src[3], SkScalar w, SkP3D dst[]) {
dst[0].set(src[0].fX * 1, src[0].fY * 1, 1);
dst[1].set(src[1].fX * w, src[1].fY * w, w);
dst[2].set(src[2].fX * 1, src[2].fY * 1, 1);
}
void SkConic::evalAt(SkScalar t, SkPoint* pt, SkVector* tangent) const {
SkASSERT(t >= 0 && t <= SK_Scalar1);
if (pt) {
pt->set(conic_eval_pos(&fPts[0].fX, fW, t),
conic_eval_pos(&fPts[0].fY, fW, t));
}
if (tangent) {
tangent->set(conic_eval_tan(&fPts[0].fX, fW, t),
conic_eval_tan(&fPts[0].fY, fW, t));
}
}
void SkConic::chopAt(SkScalar t, SkConic dst[2]) const {
SkP3D tmp[3], tmp2[3];
ratquad_mapTo3D(fPts, fW, tmp);
p3d_interp(&tmp[0].fX, &tmp2[0].fX, t);
p3d_interp(&tmp[0].fY, &tmp2[0].fY, t);
p3d_interp(&tmp[0].fZ, &tmp2[0].fZ, t);
dst[0].fPts[0] = fPts[0];
tmp2[0].projectDown(&dst[0].fPts[1]);
tmp2[1].projectDown(&dst[0].fPts[2]); dst[1].fPts[0] = dst[0].fPts[2];
tmp2[2].projectDown(&dst[1].fPts[1]);
dst[1].fPts[2] = fPts[2];
// to put in "standard form", where w0 and w2 are both 1, we compute the
// new w1 as sqrt(w1*w1/w0*w2)
// or
// w1 /= sqrt(w0*w2)
//
// However, in our case, we know that for dst[0], w0 == 1, and for dst[1], w2 == 1
//
SkScalar root = SkScalarSqrt(tmp2[1].fZ);
dst[0].fW = tmp2[0].fZ / root;
dst[1].fW = tmp2[2].fZ / root;
}
static SkScalar subdivide_w_value(SkScalar w) {
return SkScalarSqrt(SK_ScalarHalf + w * SK_ScalarHalf);
}
void SkConic::chop(SkConic dst[2]) const {
SkScalar scale = SkScalarInvert(SK_Scalar1 + fW);
SkScalar p1x = fW * fPts[1].fX;
SkScalar p1y = fW * fPts[1].fY;
SkScalar mx = (fPts[0].fX + 2 * p1x + fPts[2].fX) * scale * SK_ScalarHalf;
SkScalar my = (fPts[0].fY + 2 * p1y + fPts[2].fY) * scale * SK_ScalarHalf;
dst[0].fPts[0] = fPts[0];
dst[0].fPts[1].set((fPts[0].fX + p1x) * scale,
(fPts[0].fY + p1y) * scale);
dst[0].fPts[2].set(mx, my);
dst[1].fPts[0].set(mx, my);
dst[1].fPts[1].set((p1x + fPts[2].fX) * scale,
(p1y + fPts[2].fY) * scale);
dst[1].fPts[2] = fPts[2];
dst[0].fW = dst[1].fW = subdivide_w_value(fW);
}
/*
* "High order approximation of conic sections by quadratic splines"
* by Michael Floater, 1993
*/
#define AS_QUAD_ERROR_SETUP \
SkScalar a = fW - 1; \
SkScalar k = a / (4 * (2 + a)); \
SkScalar x = k * (fPts[0].fX - 2 * fPts[1].fX + fPts[2].fX); \
SkScalar y = k * (fPts[0].fY - 2 * fPts[1].fY + fPts[2].fY);
void SkConic::computeAsQuadError(SkVector* err) const {
AS_QUAD_ERROR_SETUP
err->set(x, y);
}
bool SkConic::asQuadTol(SkScalar tol) const {
AS_QUAD_ERROR_SETUP
return (x * x + y * y) <= tol * tol;
}
int SkConic::computeQuadPOW2(SkScalar tol) const {
AS_QUAD_ERROR_SETUP
SkScalar error = SkScalarSqrt(x * x + y * y) - tol;
if (error <= 0) {
return 0;
}
uint32_t ierr = (uint32_t)error;
return (34 - SkCLZ(ierr)) >> 1;
}
static SkPoint* subdivide(const SkConic& src, SkPoint pts[], int level) {
SkASSERT(level >= 0);
if (0 == level) {
memcpy(pts, &src.fPts[1], 2 * sizeof(SkPoint));
return pts + 2;
} else {
SkConic dst[2];
src.chop(dst);
--level;
pts = subdivide(dst[0], pts, level);
return subdivide(dst[1], pts, level);
}
}
int SkConic::chopIntoQuadsPOW2(SkPoint pts[], int pow2) const {
SkASSERT(pow2 >= 0);
*pts = fPts[0];
SkDEBUGCODE(SkPoint* endPts =) subdivide(*this, pts + 1, pow2);
SkASSERT(endPts - pts == (2 * (1 << pow2) + 1));
return 1 << pow2;
}
bool SkConic::findXExtrema(SkScalar* t) const {
return conic_find_extrema(&fPts[0].fX, fW, t);
}
bool SkConic::findYExtrema(SkScalar* t) const {
return conic_find_extrema(&fPts[0].fY, fW, t);
}
bool SkConic::chopAtXExtrema(SkConic dst[2]) const {
SkScalar t;
if (this->findXExtrema(&t)) {
this->chopAt(t, dst);
// now clean-up the middle, since we know t was meant to be at
// an X-extrema
SkScalar value = dst[0].fPts[2].fX;
dst[0].fPts[1].fX = value;
dst[1].fPts[0].fX = value;
dst[1].fPts[1].fX = value;
return true;
}
return false;
}
bool SkConic::chopAtYExtrema(SkConic dst[2]) const {
SkScalar t;
if (this->findYExtrema(&t)) {
this->chopAt(t, dst);
// now clean-up the middle, since we know t was meant to be at
// an Y-extrema
SkScalar value = dst[0].fPts[2].fY;
dst[0].fPts[1].fY = value;
dst[1].fPts[0].fY = value;
dst[1].fPts[1].fY = value;
return true;
}
return false;
}
void SkConic::computeTightBounds(SkRect* bounds) const {
SkPoint pts[4];
pts[0] = fPts[0];
pts[1] = fPts[2];
int count = 2;
SkScalar t;
if (this->findXExtrema(&t)) {
this->evalAt(t, &pts[count++]);
}
if (this->findYExtrema(&t)) {
this->evalAt(t, &pts[count++]);
}
bounds->set(pts, count);
}
void SkConic::computeFastBounds(SkRect* bounds) const {
bounds->set(fPts, 3);
}