/*
* 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 "SkMatrix.h"
#include "SkNx.h"
static SkVector to_vector(const Sk2s& x) {
SkVector vector;
x.store(&vector);
return vector;
}
/** 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.
*/
// #define DIRECT_EVAL_OF_POLYNOMIALS
////////////////////////////////////////////////////////////////////////
static int is_not_monotonic(SkScalar a, SkScalar b, SkScalar c) {
SkScalar ab = a - b;
SkScalar bc = b - c;
if (ab < 0) {
bc = -bc;
}
return ab == 0 || bc < 0;
}
////////////////////////////////////////////////////////////////////////
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 = numer / denom;
if (SkScalarIsNaN(r)) {
return 0;
}
SkASSERTF(r >= 0 && r < SK_Scalar1, "numer %f, denom %f, r %f", numer, denom, r);
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;
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);
}
///////////////////////////////////////////////////////////////////////////////
///////////////////////////////////////////////////////////////////////////////
void SkEvalQuadAt(const SkPoint src[3], SkScalar t, SkPoint* pt, SkVector* tangent) {
SkASSERT(src);
SkASSERT(t >= 0 && t <= SK_Scalar1);
if (pt) {
*pt = SkEvalQuadAt(src, t);
}
if (tangent) {
*tangent = SkEvalQuadTangentAt(src, t);
}
}
SkPoint SkEvalQuadAt(const SkPoint src[3], SkScalar t) {
return to_point(SkQuadCoeff(src).eval(t));
}
SkVector SkEvalQuadTangentAt(const SkPoint src[3], SkScalar t) {
// The derivative equation is 2(b - a +(a - 2b +c)t). This returns a
// zero tangent vector when t is 0 or 1, and the control point is equal
// to the end point. In this case, use the quad end points to compute the tangent.
if ((t == 0 && src[0] == src[1]) || (t == 1 && src[1] == src[2])) {
return src[2] - src[0];
}
SkASSERT(src);
SkASSERT(t >= 0 && t <= SK_Scalar1);
Sk2s P0 = from_point(src[0]);
Sk2s P1 = from_point(src[1]);
Sk2s P2 = from_point(src[2]);
Sk2s B = P1 - P0;
Sk2s A = P2 - P1 - B;
Sk2s T = A * Sk2s(t) + B;
return to_vector(T + T);
}
static inline Sk2s interp(const Sk2s& v0, const Sk2s& v1, const Sk2s& t) {
return v0 + (v1 - v0) * t;
}
void SkChopQuadAt(const SkPoint src[3], SkPoint dst[5], SkScalar t) {
SkASSERT(t > 0 && t < SK_Scalar1);
Sk2s p0 = from_point(src[0]);
Sk2s p1 = from_point(src[1]);
Sk2s p2 = from_point(src[2]);
Sk2s tt(t);
Sk2s p01 = interp(p0, p1, tt);
Sk2s p12 = interp(p1, p2, tt);
dst[0] = to_point(p0);
dst[1] = to_point(p01);
dst[2] = to_point(interp(p01, p12, tt));
dst[3] = to_point(p12);
dst[4] = to_point(p2);
}
void SkChopQuadAtHalf(const SkPoint src[3], SkPoint dst[5]) {
SkChopQuadAt(src, dst, 0.5f);
}
/** 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);
}
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);
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)
//
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;
}
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;
}
}
void SkConvertQuadToCubic(const SkPoint src[3], SkPoint dst[4]) {
Sk2s scale(SkDoubleToScalar(2.0 / 3.0));
Sk2s s0 = from_point(src[0]);
Sk2s s1 = from_point(src[1]);
Sk2s s2 = from_point(src[2]);
dst[0] = src[0];
dst[1] = to_point(s0 + (s1 - s0) * scale);
dst[2] = to_point(s2 + (s1 - s2) * scale);
dst[3] = src[2];
}
//////////////////////////////////////////////////////////////////////////////
///// CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS /////
//////////////////////////////////////////////////////////////////////////////
#ifdef SK_SUPPORT_LEGACY_EVAL_CUBIC
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
}
#endif
static SkVector eval_cubic_derivative(const SkPoint src[4], SkScalar t) {
SkQuadCoeff coeff;
Sk2s P0 = from_point(src[0]);
Sk2s P1 = from_point(src[1]);
Sk2s P2 = from_point(src[2]);
Sk2s P3 = from_point(src[3]);
coeff.fA = P3 + Sk2s(3) * (P1 - P2) - P0;
coeff.fB = times_2(P2 - times_2(P1) + P0);
coeff.fC = P1 - P0;
return to_vector(coeff.eval(t));
}
static SkVector eval_cubic_2ndDerivative(const SkPoint src[4], SkScalar t) {
Sk2s P0 = from_point(src[0]);
Sk2s P1 = from_point(src[1]);
Sk2s P2 = from_point(src[2]);
Sk2s P3 = from_point(src[3]);
Sk2s A = P3 + Sk2s(3) * (P1 - P2) - P0;
Sk2s B = P2 - times_2(P1) + P0;
return to_vector(A * Sk2s(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) {
#ifdef SK_SUPPORT_LEGACY_EVAL_CUBIC
loc->set(eval_cubic(&src[0].fX, t), eval_cubic(&src[0].fY, t));
#else
*loc = to_point(SkCubicCoeff(src).eval(t));
#endif
}
if (tangent) {
// The derivative equation returns a zero tangent vector when t is 0 or 1, and the
// adjacent control point is equal to the end point. In this case, use the
// next control point or the end points to compute the tangent.
if ((t == 0 && src[0] == src[1]) || (t == 1 && src[2] == src[3])) {
if (t == 0) {
*tangent = src[2] - src[0];
} else {
*tangent = src[3] - src[1];
}
if (!tangent->fX && !tangent->fY) {
*tangent = src[3] - src[0];
}
} else {
*tangent = eval_cubic_derivative(src, t);
}
}
if (curvature) {
*curvature = eval_cubic_2ndDerivative(src, 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]) {
// 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);
}
void SkChopCubicAt(const SkPoint src[4], SkPoint dst[7], SkScalar t) {
SkASSERT(t > 0 && t < SK_Scalar1);
Sk2s p0 = from_point(src[0]);
Sk2s p1 = from_point(src[1]);
Sk2s p2 = from_point(src[2]);
Sk2s p3 = from_point(src[3]);
Sk2s tt(t);
Sk2s ab = interp(p0, p1, tt);
Sk2s bc = interp(p1, p2, tt);
Sk2s cd = interp(p2, p3, tt);
Sk2s abc = interp(ab, bc, tt);
Sk2s bcd = interp(bc, cd, tt);
Sk2s abcd = interp(abc, bcd, tt);
dst[0] = src[0];
dst[1] = to_point(ab);
dst[2] = to_point(abc);
dst[3] = to_point(abcd);
dst[4] = to_point(bcd);
dst[5] = to_point(cd);
dst[6] = src[3];
}
/* 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]) {
SkChopCubicAt(src, dst, 0.5f);
}
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;
return SkFindUnitQuadRoots(Bx*Cy - By*Cx,
Ax*Cy - Ay*Cx,
Ax*By - Ay*Bx,
tValues);
}
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;
}
// See http://http.developer.nvidia.com/GPUGems3/gpugems3_ch25.html (from the book GPU Gems 3)
// discr(I) = d0^2 * (3*d1^2 - 4*d0*d2)
// Classification:
// discr(I) > 0 Serpentine
// discr(I) = 0 Cusp
// discr(I) < 0 Loop
// d0 = d1 = 0 Quadratic
// d0 = d1 = d2 = 0 Line
// p0 = p1 = p2 = p3 Point
static SkCubicType classify_cubic(const SkPoint p[4], const SkScalar d[3]) {
if (p[0] == p[1] && p[0] == p[2] && p[0] == p[3]) {
return kPoint_SkCubicType;
}
const SkScalar discr = d[0] * d[0] * (3.f * d[1] * d[1] - 4.f * d[0] * d[2]);
if (discr > SK_ScalarNearlyZero) {
return kSerpentine_SkCubicType;
} else if (discr < -SK_ScalarNearlyZero) {
return kLoop_SkCubicType;
} else {
if (0.f == d[0] && 0.f == d[1]) {
return (0.f == d[2] ? kLine_SkCubicType : kQuadratic_SkCubicType);
} else {
return kCusp_SkCubicType;
}
}
}
// Assumes the third component of points is 1.
// Calcs p0 . (p1 x p2)
static SkScalar calc_dot_cross_cubic(const SkPoint& p0, const SkPoint& p1, const SkPoint& p2) {
const SkScalar xComp = p0.fX * (p1.fY - p2.fY);
const SkScalar yComp = p0.fY * (p2.fX - p1.fX);
const SkScalar wComp = p1.fX * p2.fY - p1.fY * p2.fX;
return (xComp + yComp + wComp);
}
// Calc coefficients of I(s,t) where roots of I are inflection points of curve
// I(s,t) = t*(3*d0*s^2 - 3*d1*s*t + d2*t^2)
// d0 = a1 - 2*a2+3*a3
// d1 = -a2 + 3*a3
// d2 = 3*a3
// a1 = p0 . (p3 x p2)
// a2 = p1 . (p0 x p3)
// a3 = p2 . (p1 x p0)
// Places the values of d1, d2, d3 in array d passed in
static void calc_cubic_inflection_func(const SkPoint p[4], SkScalar d[3]) {
SkScalar a1 = calc_dot_cross_cubic(p[0], p[3], p[2]);
SkScalar a2 = calc_dot_cross_cubic(p[1], p[0], p[3]);
SkScalar a3 = calc_dot_cross_cubic(p[2], p[1], p[0]);
// need to scale a's or values in later calculations will grow to high
SkScalar max = SkScalarAbs(a1);
max = SkMaxScalar(max, SkScalarAbs(a2));
max = SkMaxScalar(max, SkScalarAbs(a3));
max = 1.f/max;
a1 = a1 * max;
a2 = a2 * max;
a3 = a3 * max;
d[2] = 3.f * a3;
d[1] = d[2] - a2;
d[0] = d[1] - a2 + a1;
}
SkCubicType SkClassifyCubic(const SkPoint src[4], SkScalar d[3]) {
calc_cubic_inflection_func(src, d);
return classify_cubic(src, d);
}
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;
}
}
/**
* 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(SkScalar 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 SkScalar src0[] = { 0 };
const SkScalar src1[] = { 0, 0 };
const SkScalar src2[] = { 0, 1 };
const SkScalar src3[] = { 0, 0, 0 };
const SkScalar src4[] = { 0, 0, 1 };
const SkScalar src5[] = { 0, 1, 1 };
const SkScalar src6[] = { 0, 1, 2 };
const struct {
const SkScalar* 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) {
SkScalar 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
static SkScalar SkScalarCubeRoot(SkScalar x) {
return SkScalarPow(x, 0.3333333f);
}
/* 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();)
// 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
} else { // we have 1 real root
SkScalar A = SkScalarAbs(R) + SkScalarSqrt(R2MinusQ3);
A = SkScalarCubeRoot(A);
if (R > 0) {
A = -A;
}
if (A != 0) {
A += Q / A;
}
r = 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[], SkScalar 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];
coeff[0] = c * c;
coeff[1] = 3 * b * c;
coeff[2] = 2 * b * b + c * a;
coeff[3] = a * b;
}
/* 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]) {
SkScalar 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] += coeffY[i];
}
SkScalar t[3];
int count = solve_cubic_poly(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] > 0 && t[i] < SK_Scalar1) {
tValues[maxCount++] = t[i];
}
}
return maxCount;
}
int SkChopCubicAtMaxCurvature(const SkPoint src[4], SkPoint dst[13],
SkScalar tValues[3]) {
SkScalar t_storage[3];
if (tValues == nullptr) {
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;
}
#include "../pathops/SkPathOpsCubic.h"
typedef int (SkDCubic::*InterceptProc)(double intercept, double roots[3]) const;
static bool cubic_dchop_at_intercept(const SkPoint src[4], SkScalar intercept, SkPoint dst[7],
InterceptProc method) {
SkDCubic cubic;
double roots[3];
int count = (cubic.set(src).*method)(intercept, roots);
if (count > 0) {
SkDCubicPair pair = cubic.chopAt(roots[0]);
for (int i = 0; i < 7; ++i) {
dst[i] = pair.pts[i].asSkPoint();
}
return true;
}
return false;
}
bool SkChopMonoCubicAtY(SkPoint src[4], SkScalar y, SkPoint dst[7]) {
return cubic_dchop_at_intercept(src, y, dst, &SkDCubic::horizontalIntersect);
}
bool SkChopMonoCubicAtX(SkPoint src[4], SkScalar x, SkPoint dst[7]) {
return cubic_dchop_at_intercept(src, x, dst, &SkDCubic::verticalIntersect);
}
///////////////////////////////////////////////////////////////////////////////
//
// NURB representation for conics. Helpful explanations at:
//
// http://citeseerx.ist.psu.edu/viewdoc/
// download?doi=10.1.1.44.5740&rep=rep1&type=ps
// and
// http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/NURBS/RB-conics.html
//
// 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}
//
// 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 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 only interpolate one dimension at a time (the first, at +0, +3, +6).
static void p3d_interp(const SkScalar src[7], SkScalar dst[7], 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::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;
}
void SkConic::chopAt(SkScalar t1, SkScalar t2, SkConic* dst) const {
if (0 == t1 || 1 == t2) {
if (0 == t1 && 1 == t2) {
*dst = *this;
} else {
SkConic pair[2];
this->chopAt(t1 ? t1 : t2, pair);
*dst = pair[SkToBool(t1)];
}
return;
}
SkConicCoeff coeff(*this);
Sk2s tt1(t1);
Sk2s aXY = coeff.fNumer.eval(tt1);
Sk2s aZZ = coeff.fDenom.eval(tt1);
Sk2s midTT((t1 + t2) / 2);
Sk2s dXY = coeff.fNumer.eval(midTT);
Sk2s dZZ = coeff.fDenom.eval(midTT);
Sk2s tt2(t2);
Sk2s cXY = coeff.fNumer.eval(tt2);
Sk2s cZZ = coeff.fDenom.eval(tt2);
Sk2s bXY = times_2(dXY) - (aXY + cXY) * Sk2s(0.5f);
Sk2s bZZ = times_2(dZZ) - (aZZ + cZZ) * Sk2s(0.5f);
dst->fPts[0] = to_point(aXY / aZZ);
dst->fPts[1] = to_point(bXY / bZZ);
dst->fPts[2] = to_point(cXY / cZZ);
Sk2s ww = bZZ / (aZZ * cZZ).sqrt();
dst->fW = ww[0];
}
SkPoint SkConic::evalAt(SkScalar t) const {
return to_point(SkConicCoeff(*this).eval(t));
}
SkVector SkConic::evalTangentAt(SkScalar t) const {
// The derivative equation returns a zero tangent vector when t is 0 or 1,
// and the control point is equal to the end point.
// In this case, use the conic endpoints to compute the tangent.
if ((t == 0 && fPts[0] == fPts[1]) || (t == 1 && fPts[1] == fPts[2])) {
return fPts[2] - fPts[0];
}
Sk2s p0 = from_point(fPts[0]);
Sk2s p1 = from_point(fPts[1]);
Sk2s p2 = from_point(fPts[2]);
Sk2s ww(fW);
Sk2s p20 = p2 - p0;
Sk2s p10 = p1 - p0;
Sk2s C = ww * p10;
Sk2s A = ww * p20 - p20;
Sk2s B = p20 - C - C;
return to_vector(SkQuadCoeff(A, B, C).eval(t));
}
void SkConic::evalAt(SkScalar t, SkPoint* pt, SkVector* tangent) const {
SkASSERT(t >= 0 && t <= SK_Scalar1);
if (pt) {
*pt = this->evalAt(t);
}
if (tangent) {
*tangent = this->evalTangentAt(t);
}
}
static SkScalar subdivide_w_value(SkScalar w) {
return SkScalarSqrt(SK_ScalarHalf + w * SK_ScalarHalf);
}
void SkConic::chop(SkConic * SK_RESTRICT dst) const {
Sk2s scale = Sk2s(SkScalarInvert(SK_Scalar1 + fW));
SkScalar newW = subdivide_w_value(fW);
Sk2s p0 = from_point(fPts[0]);
Sk2s p1 = from_point(fPts[1]);
Sk2s p2 = from_point(fPts[2]);
Sk2s ww(fW);
Sk2s wp1 = ww * p1;
Sk2s m = (p0 + times_2(wp1) + p2) * scale * Sk2s(0.5f);
dst[0].fPts[0] = fPts[0];
dst[0].fPts[1] = to_point((p0 + wp1) * scale);
dst[0].fPts[2] = dst[1].fPts[0] = to_point(m);
dst[1].fPts[1] = to_point((wp1 + p2) * scale);
dst[1].fPts[2] = fPts[2];
dst[0].fW = dst[1].fW = newW;
}
/*
* "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;
}
// Limit the number of suggested quads to approximate a conic
#define kMaxConicToQuadPOW2 5
int SkConic::computeQuadPOW2(SkScalar tol) const {
if (tol < 0 || !SkScalarIsFinite(tol)) {
return 0;
}
AS_QUAD_ERROR_SETUP
SkScalar error = SkScalarSqrt(x * x + y * y);
int pow2;
for (pow2 = 0; pow2 < kMaxConicToQuadPOW2; ++pow2) {
if (error <= tol) {
break;
}
error *= 0.25f;
}
// float version -- using ceil gives the same results as the above.
if (false) {
SkScalar err = SkScalarSqrt(x * x + y * y);
if (err <= tol) {
return 0;
}
SkScalar tol2 = tol * tol;
if (tol2 == 0) {
return kMaxConicToQuadPOW2;
}
SkScalar fpow2 = SkScalarLog2((x * x + y * y) / tol2) * 0.25f;
int altPow2 = SkScalarCeilToInt(fpow2);
if (altPow2 != pow2) {
SkDebugf("pow2 %d altPow2 %d fbits %g err %g tol %g\n", pow2, altPow2, fpow2, err, tol);
}
pow2 = altPow2;
}
return pow2;
}
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);
}
#if 0 // unimplemented
bool SkConic::findMaxCurvature(SkScalar* t) const {
// TODO: Implement me
return false;
}
#endif
SkScalar SkConic::TransformW(const SkPoint pts[], SkScalar w,
const SkMatrix& matrix) {
if (!matrix.hasPerspective()) {
return w;
}
SkP3D src[3], dst[3];
ratquad_mapTo3D(pts, w, src);
matrix.mapHomogeneousPoints(&dst[0].fX, &src[0].fX, 3);
// w' = sqrt(w1*w1/w0*w2)
SkScalar w0 = dst[0].fZ;
SkScalar w1 = dst[1].fZ;
SkScalar w2 = dst[2].fZ;
w = SkScalarSqrt((w1 * w1) / (w0 * w2));
return w;
}
int SkConic::BuildUnitArc(const SkVector& uStart, const SkVector& uStop, SkRotationDirection dir,
const SkMatrix* userMatrix, SkConic dst[kMaxConicsForArc]) {
// rotate by x,y so that uStart is (1.0)
SkScalar x = SkPoint::DotProduct(uStart, uStop);
SkScalar y = SkPoint::CrossProduct(uStart, uStop);
SkScalar absY = SkScalarAbs(y);
// 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))) {
return 0;
}
if (dir == kCCW_SkRotationDirection) {
y = -y;
}
// We decide to use 1-conic per quadrant of a circle. What quadrant does [xy] lie in?
// 0 == [0 .. 90)
// 1 == [90 ..180)
// 2 == [180..270)
// 3 == [270..360)
//
int quadrant = 0;
if (0 == y) {
quadrant = 2; // 180
SkASSERT(SkScalarAbs(x + SK_Scalar1) <= SK_ScalarNearlyZero);
} else if (0 == x) {
SkASSERT(absY - SK_Scalar1 <= SK_ScalarNearlyZero);
quadrant = y > 0 ? 1 : 3; // 90 : 270
} else {
if (y < 0) {
quadrant += 2;
}
if ((x < 0) != (y < 0)) {
quadrant += 1;
}
}
const SkPoint quadrantPts[] = {
{ 1, 0 }, { 1, 1 }, { 0, 1 }, { -1, 1 }, { -1, 0 }, { -1, -1 }, { 0, -1 }, { 1, -1 }
};
const SkScalar quadrantWeight = SK_ScalarRoot2Over2;
int conicCount = quadrant;
for (int i = 0; i < conicCount; ++i) {
dst[i].set(&quadrantPts[i * 2], quadrantWeight);
}
// Now compute any remaing (sub-90-degree) arc for the last conic
const SkPoint finalP = { x, y };
const SkPoint& lastQ = quadrantPts[quadrant * 2]; // will already be a unit-vector
const SkScalar dot = SkVector::DotProduct(lastQ, finalP);
SkASSERT(0 <= dot && dot <= SK_Scalar1 + SK_ScalarNearlyZero);
if (dot < 1) {
SkVector offCurve = { lastQ.x() + x, lastQ.y() + y };
// compute the bisector vector, and then rescale to be the off-curve point.
// we compute its length from cos(theta/2) = length / 1, using half-angle identity we get
// length = sqrt(2 / (1 + cos(theta)). We already have cos() when to computed the dot.
// This is nice, since our computed weight is cos(theta/2) as well!
//
const SkScalar cosThetaOver2 = SkScalarSqrt((1 + dot) / 2);
offCurve.setLength(SkScalarInvert(cosThetaOver2));
dst[conicCount].set(lastQ, offCurve, finalP, cosThetaOver2);
conicCount += 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);
}
for (int i = 0; i < conicCount; ++i) {
matrix.mapPoints(dst[i].fPts, 3);
}
return conicCount;
}