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skia / skia / 8d2fb64247f0f8f5e00f6895464f94ab705c9b83 / . / src / core / SkGeometry.cpp
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/*
* 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 "include/core/SkMatrix.h"
#include "include/core/SkPoint3.h"
#include "include/private/SkNx.h"
#include "include/private/SkTPin.h"
#include "include/private/SkVx.h"
#include "src/core/SkGeometry.h"
#include "src/core/SkPointPriv.h"
#include <algorithm>
#include <tuple>
#include <utility>
static SkVector to_vector(const Sk2s& x) {
SkVector vector;
x.store(&vector);
return vector;
}
////////////////////////////////////////////////////////////////////////
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 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;
}
// Just returns its argument, but makes it easy to set a break-point to know when
// SkFindUnitQuadRoots is going to return 0 (an error).
static int return_check_zero(int value) {
if (value == 0) {
return 0;
}
return value;
}
/** 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 return_check_zero(valid_unit_divide(-C, B, roots));
}
SkScalar* r = roots;
// use doubles so we don't overflow temporarily trying to compute R
double dr = (double)B * B - 4 * (double)A * C;
if (dr < 0) {
return return_check_zero(0);
}
dr = sqrt(dr);
SkScalar R = SkDoubleToScalar(dr);
if (!SkScalarIsFinite(R)) {
return return_check_zero(0);
}
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]) {
using std::swap;
swap(roots[0], roots[1]);
} else if (roots[0] == roots[1]) { // nearly-equal?
r -= 1; // skip the double root
}
}
return return_check_zero((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);
}
float SkMeasureAngleBetweenVectors(SkVector a, SkVector b) {
float cosTheta = sk_ieee_float_divide(a.dot(b), sqrtf(a.dot(a) * b.dot(b)));
// Pin cosTheta such that if it is NaN (e.g., if a or b was 0), then we return acos(1) = 0.
cosTheta = std::max(std::min(1.f, cosTheta), -1.f);
return acosf(cosTheta);
}
SkVector SkFindBisector(SkVector a, SkVector b) {
std::array<SkVector, 2> v;
if (a.dot(b) >= 0) {
// a,b are within +/-90 degrees apart.
v = {a, b};
} else if (a.cross(b) >= 0) {
// a,b are >90 degrees apart. Find the bisector of their interior normals instead. (Above 90
// degrees, the original vectors start cancelling each other out which eventually becomes
// unstable.)
v[0].set(-a.fY, +a.fX);
v[1].set(+b.fY, -b.fX);
} else {
// a,b are <-90 degrees apart. Find the bisector of their interior normals instead. (Below
// -90 degrees, the original vectors start cancelling each other out which eventually
// becomes unstable.)
v[0].set(+a.fY, -a.fX);
v[1].set(-b.fY, +b.fX);
}
// Return "normalize(v[0]) + normalize(v[1])".
Sk2f x0_x1, y0_y1;
Sk2f::Load2(v.data(), &x0_x1, &y0_y1);
Sk2f invLengths = 1.0f / (x0_x1 * x0_x1 + y0_y1 * y0_y1).sqrt();
x0_x1 *= invLengths;
y0_y1 *= invLengths;
return SkPoint{x0_x1[0] + x0_x1[1], y0_y1[0] + y0_y1[1]};
}
float SkFindQuadMidTangent(const SkPoint src[3]) {
// Tangents point in the direction of increasing T, so tan0 and -tan1 both point toward the
// midtangent. The bisector of tan0 and -tan1 is orthogonal to the midtangent:
//
// n dot midtangent = 0
//
SkVector tan0 = src[1] - src[0];
SkVector tan1 = src[2] - src[1];
SkVector bisector = SkFindBisector(tan0, -tan1);
// The midtangent can be found where (F' dot bisector) = 0:
//
// 0 = (F'(T) dot bisector) = |2*T 1| * |p0 - 2*p1 + p2| * |bisector.x|
// |-2*p0 + 2*p1 | |bisector.y|
//
// = |2*T 1| * |tan1 - tan0| * |nx|
// |2*tan0 | |ny|
//
// = 2*T * ((tan1 - tan0) dot bisector) + (2*tan0 dot bisector)
//
// T = (tan0 dot bisector) / ((tan0 - tan1) dot bisector)
float T = sk_ieee_float_divide(tan0.dot(bisector), (tan0 - tan1).dot(bisector));
if (!(T > 0 && T < 1)) { // Use "!(positive_logic)" so T=nan will take this branch.
T = .5; // The quadratic was a line or near-line. Just chop at .5.
}
return T;
}
/** 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 numer = -(Ax * Bx + Ay * By);
SkScalar denom = Bx * Bx + By * By;
if (denom < 0) {
numer = -numer;
denom = -denom;
}
if (numer <= 0) {
return 0;
}
if (numer >= denom) { // Also catches denom=0.
return 1;
}
SkScalar t = numer / denom;
SkASSERT((0 <= t && t < 1) || SkScalarIsNaN(t));
return t;
}
int SkChopQuadAtMaxCurvature(const SkPoint src[3], SkPoint dst[5]) {
SkScalar t = SkFindQuadMaxCurvature(src);
if (t > 0 && t < 1) {
SkChopQuadAt(src, dst, t);
return 2;
} else {
memcpy(dst, src, 3 * sizeof(SkPoint));
return 1;
}
}
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] = to_point(s0);
dst[1] = to_point(s0 + (s1 - s0) * scale);
dst[2] = to_point(s2 + (s1 - s2) * scale);
dst[3] = to_point(s2);
}
//////////////////////////////////////////////////////////////////////////////
///// CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS // CUBICS /////
//////////////////////////////////////////////////////////////////////////////
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) {
*loc = to_point(SkCubicCoeff(src).eval(t));
}
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);
}
// This does not return b when t==1, but it otherwise seems to get better precision than
// "a*(1 - t) + b*t" for things like chopping cubics on exact cusp points.
// The responsibility falls on the caller to check that t != 1 before calling.
template<int N, typename T>
inline static skvx::Vec<N,T> unchecked_mix(const skvx::Vec<N,T>& a, const skvx::Vec<N,T>& b,
const skvx::Vec<N,T>& t) {
return (b - a)*t + a;
}
void SkChopCubicAt(const SkPoint src[4], SkPoint dst[7], SkScalar t) {
using float2 = skvx::Vec<2,float>;
SkASSERT(0 <= t && t <= 1);
if (t == 1) {
memcpy(dst, src, sizeof(SkPoint) * 4);
dst[4] = dst[5] = dst[6] = src[3];
return;
}
float2 p0 = skvx::bit_pun<float2>(src[0]);
float2 p1 = skvx::bit_pun<float2>(src[1]);
float2 p2 = skvx::bit_pun<float2>(src[2]);
float2 p3 = skvx::bit_pun<float2>(src[3]);
float2 T = t;
float2 ab = unchecked_mix(p0, p1, T);
float2 bc = unchecked_mix(p1, p2, T);
float2 cd = unchecked_mix(p2, p3, T);
float2 abc = unchecked_mix(ab, bc, T);
float2 bcd = unchecked_mix(bc, cd, T);
float2 abcd = unchecked_mix(abc, bcd, T);
dst[0] = skvx::bit_pun<SkPoint>(p0);
dst[1] = skvx::bit_pun<SkPoint>(ab);
dst[2] = skvx::bit_pun<SkPoint>(abc);
dst[3] = skvx::bit_pun<SkPoint>(abcd);
dst[4] = skvx::bit_pun<SkPoint>(bcd);
dst[5] = skvx::bit_pun<SkPoint>(cd);
dst[6] = skvx::bit_pun<SkPoint>(p3);
}
void SkChopCubicAt(const SkPoint src[4], SkPoint dst[10], float t0, float t1) {
using float4 = skvx::Vec<4,float>;
using float2 = skvx::Vec<2,float>;
SkASSERT(0 <= t0 && t0 <= t1 && t1 <= 1);
if (t1 == 1) {
SkChopCubicAt(src, dst, t0);
dst[7] = dst[8] = dst[9] = src[3];
return;
}
// Perform both chops in parallel using 4-lane SIMD.
float4 p00, p11, p22, p33, T;
p00.lo = p00.hi = skvx::bit_pun<float2>(src[0]);
p11.lo = p11.hi = skvx::bit_pun<float2>(src[1]);
p22.lo = p22.hi = skvx::bit_pun<float2>(src[2]);
p33.lo = p33.hi = skvx::bit_pun<float2>(src[3]);
T.lo = t0;
T.hi = t1;
float4 ab = unchecked_mix(p00, p11, T);
float4 bc = unchecked_mix(p11, p22, T);
float4 cd = unchecked_mix(p22, p33, T);
float4 abc = unchecked_mix(ab, bc, T);
float4 bcd = unchecked_mix(bc, cd, T);
float4 abcd = unchecked_mix(abc, bcd, T);
float4 middle = unchecked_mix(abc, bcd, skvx::shuffle<2,3,0,1>(T));
dst[0] = skvx::bit_pun<SkPoint>(p00.lo);
dst[1] = skvx::bit_pun<SkPoint>(ab.lo);
dst[2] = skvx::bit_pun<SkPoint>(abc.lo);
dst[3] = skvx::bit_pun<SkPoint>(abcd.lo);
middle.store(dst + 4);
dst[6] = skvx::bit_pun<SkPoint>(abcd.hi);
dst[7] = skvx::bit_pun<SkPoint>(bcd.hi);
dst[8] = skvx::bit_pun<SkPoint>(cd.hi);
dst[9] = skvx::bit_pun<SkPoint>(p33.hi);
}
void SkChopCubicAt(const SkPoint src[4], SkPoint dst[],
const SkScalar tValues[], int tCount) {
using float2 = skvx::Vec<2,float>;
SkASSERT(std::all_of(tValues, tValues + tCount, [](SkScalar t) { return t >= 0 && t <= 1; }));
SkASSERT(std::is_sorted(tValues, tValues + tCount));
if (dst) {
if (tCount == 0) { // nothing to chop
memcpy(dst, src, 4*sizeof(SkPoint));
} else {
int i = 0;
for (; i < tCount - 1; i += 2) {
// Do two chops at once.
float2 tt = float2::Load(tValues + i);
if (i != 0) {
float lastT = tValues[i - 1];
tt = skvx::pin((tt - lastT) / (1 - lastT), float2(0), float2(1));
}
SkChopCubicAt(src, dst, tt[0], tt[1]);
src = dst = dst + 6;
}
if (i < tCount) {
// Chop the final cubic if there was an odd number of chops.
SkASSERT(i + 1 == tCount);
float t = tValues[i];
if (i != 0) {
float lastT = tValues[i - 1];
t = SkTPin(sk_ieee_float_divide(t - lastT, 1 - lastT), 0.f, 1.f);
}
SkChopCubicAt(src, dst, t);
}
}
}
}
void SkChopCubicAtHalf(const SkPoint src[4], SkPoint dst[7]) {
SkChopCubicAt(src, dst, 0.5f);
}
float SkMeasureNonInflectCubicRotation(const SkPoint pts[4]) {
SkVector a = pts[1] - pts[0];
SkVector b = pts[2] - pts[1];
SkVector c = pts[3] - pts[2];
if (a.isZero()) {
return SkMeasureAngleBetweenVectors(b, c);
}
if (b.isZero()) {
return SkMeasureAngleBetweenVectors(a, c);
}
if (c.isZero()) {
return SkMeasureAngleBetweenVectors(a, b);
}
// Postulate: When no points are colocated and there are no inflection points in T=0..1, the
// rotation is: 360 degrees, minus the angle [p0,p1,p2], minus the angle [p1,p2,p3].
return 2*SK_ScalarPI - SkMeasureAngleBetweenVectors(a,-b) - SkMeasureAngleBetweenVectors(b,-c);
}
static Sk4f fma(const Sk4f& f, float m, const Sk4f& a) {
return SkNx_fma(f, Sk4f(m), a);
}
// Finds the root nearest 0.5. Returns 0.5 if the roots are undefined or outside 0..1.
static float solve_quadratic_equation_for_midtangent(float a, float b, float c, float discr) {
// Quadratic formula from Numerical Recipes in C:
float q = -.5f * (b + copysignf(sqrtf(discr), b));
// The roots are q/a and c/q. Pick the midtangent closer to T=.5.
float _5qa = -.5f*q*a;
float T = fabsf(q*q + _5qa) < fabsf(a*c + _5qa) ? sk_ieee_float_divide(q,a)
: sk_ieee_float_divide(c,q);
if (!(T > 0 && T < 1)) { // Use "!(positive_logic)" so T=NaN will take this branch.
// Either the curve is a flat line with no rotation or FP precision failed us. Chop at .5.
T = .5;
}
return T;
}
static float solve_quadratic_equation_for_midtangent(float a, float b, float c) {
return solve_quadratic_equation_for_midtangent(a, b, c, b*b - 4*a*c);
}
float SkFindCubicMidTangent(const SkPoint src[4]) {
// Tangents point in the direction of increasing T, so tan0 and -tan1 both point toward the
// midtangent. The bisector of tan0 and -tan1 is orthogonal to the midtangent:
//
// bisector dot midtangent == 0
//
SkVector tan0 = (src[0] == src[1]) ? src[2] - src[0] : src[1] - src[0];
SkVector tan1 = (src[2] == src[3]) ? src[3] - src[1] : src[3] - src[2];
SkVector bisector = SkFindBisector(tan0, -tan1);
// Find the T value at the midtangent. This is a simple quadratic equation:
//
// midtangent dot bisector == 0, or using a tangent matrix C' in power basis form:
//
// |C'x C'y|
// |T^2 T 1| * |. . | * |bisector.x| == 0
// |. . | |bisector.y|
//
// The coeffs for the quadratic equation we need to solve are therefore: C' * bisector
static const Sk4f kM[4] = {Sk4f(-1, 2, -1, 0),
Sk4f( 3, -4, 1, 0),
Sk4f(-3, 2, 0, 0)};
Sk4f C_x = fma(kM[0], src[0].fX,
fma(kM[1], src[1].fX,
fma(kM[2], src[2].fX, Sk4f(src[3].fX, 0,0,0))));
Sk4f C_y = fma(kM[0], src[0].fY,
fma(kM[1], src[1].fY,
fma(kM[2], src[2].fY, Sk4f(src[3].fY, 0,0,0))));
Sk4f coeffs = C_x * bisector.x() + C_y * bisector.y();
// Now solve the quadratic for T.
float T = 0;
float a=coeffs[0], b=coeffs[1], c=coeffs[2];
float discr = b*b - 4*a*c;
if (discr > 0) { // This will only be false if the curve is a line.
return solve_quadratic_equation_for_midtangent(a, b, c, discr);
} else {
// This is a 0- or 360-degree flat line. It doesn't have single points of midtangent.
// (tangent == midtangent at every point on the curve except the cusp points.)
// Chop in between both cusps instead, if any. There can be up to two cusps on a flat line,
// both where the tangent is perpendicular to the starting tangent:
//
// tangent dot tan0 == 0
//
coeffs = C_x * tan0.x() + C_y * tan0.y();
a = coeffs[0];
b = coeffs[1];
if (a != 0) {
// We want the point in between both cusps. The midpoint of:
//
// (-b +/- sqrt(b^2 - 4*a*c)) / (2*a)
//
// Is equal to:
//
// -b / (2*a)
T = -b / (2*a);
}
if (!(T > 0 && T < 1)) { // Use "!(positive_logic)" so T=NaN will take this branch.
// Either the curve is a flat line with no rotation or FP precision failed us. Chop at
// .5.
T = .5;
}
return T;
}
}
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;
}
// Assumes the third component of points is 1.
// Calcs p0 . (p1 x p2)
static double calc_dot_cross_cubic(const SkPoint& p0, const SkPoint& p1, const SkPoint& p2) {
const double xComp = (double) p0.fX * ((double) p1.fY - (double) p2.fY);
const double yComp = (double) p0.fY * ((double) p2.fX - (double) p1.fX);
const double wComp = (double) p1.fX * (double) p2.fY - (double) p1.fY * (double) p2.fX;
return (xComp + yComp + wComp);
}
// Returns a positive power of 2 that, when multiplied by n, and excepting the two edge cases listed
// below, shifts the exponent of n to yield a magnitude somewhere inside [1..2).
// Returns 2^1023 if abs(n) < 2^-1022 (including 0).
// Returns NaN if n is Inf or NaN.
inline static double previous_inverse_pow2(double n) {
uint64_t bits;
memcpy(&bits, &n, sizeof(double));
bits = ((1023llu*2 << 52) + ((1llu << 52) - 1)) - bits; // exp=-exp
bits &= (0x7ffllu) << 52; // mantissa=1.0, sign=0
memcpy(&n, &bits, sizeof(double));
return n;
}
inline static void write_cubic_inflection_roots(double t0, double s0, double t1, double s1,
double* t, double* s) {
t[0] = t0;
s[0] = s0;
// This copysign/abs business orients the implicit function so positive values are always on the
// "left" side of the curve.
t[1] = -copysign(t1, t1 * s1);
s[1] = -fabs(s1);
// Ensure t[0]/s[0] <= t[1]/s[1] (s[1] is negative from above).
if (copysign(s[1], s[0]) * t[0] > -fabs(s[0]) * t[1]) {
using std::swap;
swap(t[0], t[1]);
swap(s[0], s[1]);
}
}
SkCubicType SkClassifyCubic(const SkPoint P[4], double t[2], double s[2], double d[4]) {
// Find the cubic's inflection function, I = [T^3 -3T^2 3T -1] dot D. (D0 will always be 0
// for integral cubics.)
//
// See "Resolution Independent Curve Rendering using Programmable Graphics Hardware",
// 4.2 Curve Categorization:
//
// https://www.microsoft.com/en-us/research/wp-content/uploads/2005/01/p1000-loop.pdf
double A1 = calc_dot_cross_cubic(P[0], P[3], P[2]);
double A2 = calc_dot_cross_cubic(P[1], P[0], P[3]);
double A3 = calc_dot_cross_cubic(P[2], P[1], P[0]);
double D3 = 3 * A3;
double D2 = D3 - A2;
double D1 = D2 - A2 + A1;
// Shift the exponents in D so the largest magnitude falls somewhere in 1..2. This protects us
// from overflow down the road while solving for roots and KLM functionals.
double Dmax = std::max(std::max(fabs(D1), fabs(D2)), fabs(D3));
double norm = previous_inverse_pow2(Dmax);
D1 *= norm;
D2 *= norm;
D3 *= norm;
if (d) {
d[3] = D3;
d[2] = D2;
d[1] = D1;
d[0] = 0;
}
// Now use the inflection function to classify the cubic.
//
// See "Resolution Independent Curve Rendering using Programmable Graphics Hardware",
// 4.4 Integral Cubics:
//
// https://www.microsoft.com/en-us/research/wp-content/uploads/2005/01/p1000-loop.pdf
if (0 != D1) {
double discr = 3*D2*D2 - 4*D1*D3;
if (discr > 0) { // Serpentine.
if (t && s) {
double q = 3*D2 + copysign(sqrt(3*discr), D2);
write_cubic_inflection_roots(q, 6*D1, 2*D3, q, t, s);
}
return SkCubicType::kSerpentine;
} else if (discr < 0) { // Loop.
if (t && s) {
double q = D2 + copysign(sqrt(-discr), D2);
write_cubic_inflection_roots(q, 2*D1, 2*(D2*D2 - D3*D1), D1*q, t, s);
}
return SkCubicType::kLoop;
} else { // Cusp.
if (t && s) {
write_cubic_inflection_roots(D2, 2*D1, D2, 2*D1, t, s);
}
return SkCubicType::kLocalCusp;
}
} else {
if (0 != D2) { // Cusp at T=infinity.
if (t && s) {
write_cubic_inflection_roots(D3, 3*D2, 1, 0, t, s); // T1=infinity.
}
return SkCubicType::kCuspAtInfinity;
} else { // Degenerate.
if (t && s) {
write_cubic_inflection_roots(1, 0, 1, 0, t, s); // T0=T1=infinity.
}
return 0 != D3 ? SkCubicType::kQuadratic : SkCubicType::kLineOrPoint;
}
}
}
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;
if (R2MinusQ3 < 0) { // we have 3 real roots
// the divide/root can, due to finite precisions, be slightly outside of -1...1
SkScalar theta = SkScalarACos(SkTPin(R / SkScalarSqrt(Q3), -1.0f, 1.0f));
SkScalar neg2RootQ = -2 * SkScalarSqrt(Q);
tValues[0] = SkTPin(neg2RootQ * SkScalarCos(theta/3) - adiv3, 0.0f, 1.0f);
tValues[1] = SkTPin(neg2RootQ * SkScalarCos((theta + 2*SK_ScalarPI)/3) - adiv3, 0.0f, 1.0f);
tValues[2] = SkTPin(neg2RootQ * SkScalarCos((theta - 2*SK_ScalarPI)/3) - adiv3, 0.0f, 1.0f);
SkDEBUGCODE(test_collaps_duplicates();)
// now sort the roots
bubble_sort(tValues, 3);
return collaps_duplicates(tValues, 3);
} 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;
}
tValues[0] = SkTPin(A - adiv3, 0.0f, 1.0f);
return 1;
}
}
/* 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];
}
int numRoots = solve_cubic_poly(coeffX, tValues);
// now remove extrema where the curvature is zero (mins)
// !!!! need a test for this !!!!
return numRoots;
}
int SkChopCubicAtMaxCurvature(const SkPoint src[4], SkPoint dst[13],
SkScalar tValues[3]) {
SkScalar t_storage[3];
if (tValues == nullptr) {
tValues = t_storage;
}
SkScalar roots[3];
int rootCount = SkFindCubicMaxCurvature(src, roots);
// Throw out values not inside 0..1.
int count = 0;
for (int i = 0; i < rootCount; ++i) {
if (0 < roots[i] && roots[i] < 1) {
tValues[count++] = roots[i];
}
}
if (dst) {
if (count == 0) {
memcpy(dst, src, 4 * sizeof(SkPoint));
} else {
SkChopCubicAt(src, dst, tValues, count);
}
}
return count + 1;
}
// Returns a constant proportional to the dimensions of the cubic.
// Constant found through experimentation -- maybe there's a better way....
static SkScalar calc_cubic_precision(const SkPoint src[4]) {
return (SkPointPriv::DistanceToSqd(src[1], src[0]) + SkPointPriv::DistanceToSqd(src[2], src[1])
+ SkPointPriv::DistanceToSqd(src[3], src[2])) * 1e-8f;
}
// Returns true if both points src[testIndex], src[testIndex+1] are in the same half plane defined
// by the line segment src[lineIndex], src[lineIndex+1].
static bool on_same_side(const SkPoint src[4], int testIndex, int lineIndex) {
SkPoint origin = src[lineIndex];
SkVector line = src[lineIndex + 1] - origin;
SkScalar crosses[2];
for (int index = 0; index < 2; ++index) {
SkVector testLine = src[testIndex + index] - origin;
crosses[index] = line.cross(testLine);
}
return crosses[0] * crosses[1] >= 0;
}
// Return location (in t) of cubic cusp, if there is one.
// Note that classify cubic code does not reliably return all cusp'd cubics, so
// it is not called here.
SkScalar SkFindCubicCusp(const SkPoint src[4]) {
// When the adjacent control point matches the end point, it behaves as if
// the cubic has a cusp: there's a point of max curvature where the derivative
// goes to zero. Ideally, this would be where t is zero or one, but math
// error makes not so. It is not uncommon to create cubics this way; skip them.
if (src[0] == src[1]) {
return -1;
}
if (src[2] == src[3]) {
return -1;
}
// Cubics only have a cusp if the line segments formed by the control and end points cross.
// Detect crossing if line ends are on opposite sides of plane formed by the other line.
if (on_same_side(src, 0, 2) || on_same_side(src, 2, 0)) {
return -1;
}
// Cubics may have multiple points of maximum curvature, although at most only
// one is a cusp.
SkScalar maxCurvature[3];
int roots = SkFindCubicMaxCurvature(src, maxCurvature);
for (int index = 0; index < roots; ++index) {
SkScalar testT = maxCurvature[index];
if (0 >= testT || testT >= 1) { // no need to consider max curvature on the end
continue;
}
// A cusp is at the max curvature, and also has a derivative close to zero.
// Choose the 'close to zero' meaning by comparing the derivative length
// with the overall cubic size.
SkVector dPt = eval_cubic_derivative(src, testT);
SkScalar dPtMagnitude = SkPointPriv::LengthSqd(dPt);
SkScalar precision = calc_cubic_precision(src);
if (dPtMagnitude < precision) {
// All three max curvature t values may be close to the cusp;
// return the first one.
return testT;
}
}
return -1;
}
#include "src/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;
}
// 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, SkPoint3 dst[3]) {
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);
}
static SkPoint project_down(const SkPoint3& src) {
return {src.fX / src.fZ, src.fY / src.fZ};
}
// return false if infinity or NaN is generated; caller must check
bool SkConic::chopAt(SkScalar t, SkConic dst[2]) const {
SkPoint3 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];
dst[0].fPts[1] = project_down(tmp2[0]);
dst[0].fPts[2] = project_down(tmp2[1]); dst[1].fPts[0] = dst[0].fPts[2];
dst[1].fPts[1] = project_down(tmp2[2]);
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;
SkASSERT(sizeof(dst[0]) == sizeof(SkScalar) * 7);
SkASSERT(0 == offsetof(SkConic, fPts[0].fX));
return SkScalarsAreFinite(&dst[0].fPts[0].fX, 7 * 2);
}
void SkConic::chopAt(SkScalar t1, SkScalar t2, SkConic* dst) const {
if (0 == t1 || 1 == t2) {
if (0 == t1 && 1 == t2) {
*dst = *this;
return;
} else {
SkConic pair[2];
if (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);
SkPoint mPt = to_point(m);
if (!mPt.isFinite()) {
double w_d = fW;
double w_2 = w_d * 2;
double scale_half = 1 / (1 + w_d) * 0.5;
mPt.fX = SkDoubleToScalar((fPts[0].fX + w_2 * fPts[1].fX + fPts[2].fX) * scale_half);
mPt.fY = SkDoubleToScalar((fPts[0].fY + w_2 * fPts[1].fY + fPts[2].fY) * scale_half);
}
dst[0].fPts[0] = fPts[0];
dst[0].fPts[1] = to_point((p0 + wp1) * scale);
dst[0].fPts[2] = dst[1].fPts[0] = mPt;
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) || !SkPointPriv::AreFinite(fPts, 3)) {
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;
}
// This was originally developed and tested for pathops: see SkOpTypes.h
// returns true if (a <= b <= c) || (a >= b >= c)
static bool between(SkScalar a, SkScalar b, SkScalar c) {
return (a - b) * (c - b) <= 0;
}
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);
const SkScalar startY = src.fPts[0].fY;
SkScalar endY = src.fPts[2].fY;
if (between(startY, src.fPts[1].fY, endY)) {
// If the input is monotonic and the output is not, the scan converter hangs.
// Ensure that the chopped conics maintain their y-order.
SkScalar midY = dst[0].fPts[2].fY;
if (!between(startY, midY, endY)) {
// If the computed midpoint is outside the ends, move it to the closer one.
SkScalar closerY = SkTAbs(midY - startY) < SkTAbs(midY - endY) ? startY : endY;
dst[0].fPts[2].fY = dst[1].fPts[0].fY = closerY;
}
if (!between(startY, dst[0].fPts[1].fY, dst[0].fPts[2].fY)) {
// If the 1st control is not between the start and end, put it at the start.
// This also reduces the quad to a line.
dst[0].fPts[1].fY = startY;
}
if (!between(dst[1].fPts[0].fY, dst[1].fPts[1].fY, endY)) {
// If the 2nd control is not between the start and end, put it at the end.
// This also reduces the quad to a line.
dst[1].fPts[1].fY = endY;
}
// Verify that all five points are in order.
SkASSERT(between(startY, dst[0].fPts[1].fY, dst[0].fPts[2].fY));
SkASSERT(between(dst[0].fPts[1].fY, dst[0].fPts[2].fY, dst[1].fPts[1].fY));
SkASSERT(between(dst[0].fPts[2].fY, dst[1].fPts[1].fY, endY));
}
--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);
if (pow2 == kMaxConicToQuadPOW2) { // If an extreme weight generates many quads ...
SkConic dst[2];
this->chop(dst);
// check to see if the first chop generates a pair of lines
if (SkPointPriv::EqualsWithinTolerance(dst[0].fPts[1], dst[0].fPts[2]) &&
SkPointPriv::EqualsWithinTolerance(dst[1].fPts[0], dst[1].fPts[1])) {
pts[1] = pts[2] = pts[3] = dst[0].fPts[1]; // set ctrl == end to make lines
pts[4] = dst[1].fPts[2];
pow2 = 1;
SkDEBUGCODE(endPts = &pts[5]);
goto commonFinitePtCheck;
}
}
SkDEBUGCODE(endPts = ) subdivide(*this, pts + 1, pow2);
commonFinitePtCheck:
const int quadCount = 1 << pow2;
const int ptCount = 2 * quadCount + 1;
SkASSERT(endPts - pts == ptCount);
if (!SkPointPriv::AreFinite(pts, ptCount)) {
// if we generated a non-finite, pin ourselves to the middle of the hull,
// as our first and last are already on the first/last pts of the hull.
for (int i = 1; i < ptCount - 1; ++i) {
pts[i] = fPts[1];
}
}
return 1 << pow2;
}
float SkConic::findMidTangent() const {
// Tangents point in the direction of increasing T, so tan0 and -tan1 both point toward the
// midtangent. The bisector of tan0 and -tan1 is orthogonal to the midtangent:
//
// bisector dot midtangent = 0
//
SkVector tan0 = fPts[1] - fPts[0];
SkVector tan1 = fPts[2] - fPts[1];
SkVector bisector = SkFindBisector(tan0, -tan1);
// Start by finding the tangent function's power basis coefficients. These define a tangent
// direction (scaled by some uniform value) as:
// |T^2|
// Tangent_Direction(T) = dx,dy = |A B C| * |T |
// |. . .| |1 |
//
// The derivative of a conic has a cumbersome order-4 denominator. However, this isn't necessary
// if we are only interested in a vector in the same *direction* as a given tangent line. Since
// the denominator scales dx and dy uniformly, we can throw it out completely after evaluating
// the derivative with the standard quotient rule. This leaves us with a simpler quadratic
// function that we use to find a tangent.
SkVector A = (fPts[2] - fPts[0]) * (fW - 1);
SkVector B = (fPts[2] - fPts[0]) - (fPts[1] - fPts[0]) * (fW*2);
SkVector C = (fPts[1] - fPts[0]) * fW;
// Now solve for "bisector dot midtangent = 0":
//
// |T^2|
// bisector * |A B C| * |T | = 0
// |. . .| |1 |
//
float a = bisector.dot(A);
float b = bisector.dot(B);
float c = bisector.dot(C);
return solve_quadratic_equation_for_midtangent(a, b, c);
}
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)) {
if (!this->chopAt(t, dst)) {
// if chop can't return finite values, don't chop
return false;
}
// 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)) {
if (!this->chopAt(t, dst)) {
// if chop can't return finite values, don't chop
return false;
}
// 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->setBounds(pts, count);
}
void SkConic::computeFastBounds(SkRect* bounds) const {
bounds->setBounds(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;
}
SkPoint3 src[3], dst[3];
ratquad_mapTo3D(pts, w, src);
matrix.mapHomogeneousPoints(dst, src, 3);
// w' = sqrt(w1*w1/w0*w2)
// use doubles temporarily, to handle small numer/denom
double w0 = dst[0].fZ;
double w1 = dst[1].fZ;
double w2 = dst[2].fZ;
return sk_double_to_float(sqrt(sk_ieee_double_divide(w1 * w1, w0 * w2)));
}
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));
if (!SkPointPriv::EqualsWithinTolerance(lastQ, offCurve)) {
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;
}