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/*
* Copyright 2012 Google Inc.
*
* Use of this source code is governed by a BSD-style license that can be
* found in the LICENSE file.
*/
#include "SkIntersections.h"
#include "SkLineParameters.h"
#include "SkPathOpsCubic.h"
#include "SkPathOpsQuad.h"
#include "SkPathOpsTriangle.h"
// from http://blog.gludion.com/2009/08/distance-to-quadratic-bezier-curve.html
// (currently only used by testing)
double SkDQuad::nearestT(const SkDPoint& pt) const {
SkDVector pos = fPts[0] - pt;
// search points P of bezier curve with PM.(dP / dt) = 0
// a calculus leads to a 3d degree equation :
SkDVector A = fPts[1] - fPts[0];
SkDVector B = fPts[2] - fPts[1];
B -= A;
double a = B.dot(B);
double b = 3 * A.dot(B);
double c = 2 * A.dot(A) + pos.dot(B);
double d = pos.dot(A);
double ts[3];
int roots = SkDCubic::RootsValidT(a, b, c, d, ts);
double d0 = pt.distanceSquared(fPts[0]);
double d2 = pt.distanceSquared(fPts[2]);
double distMin = SkTMin(d0, d2);
int bestIndex = -1;
for (int index = 0; index < roots; ++index) {
SkDPoint onQuad = ptAtT(ts[index]);
double dist = pt.distanceSquared(onQuad);
if (distMin > dist) {
distMin = dist;
bestIndex = index;
}
}
if (bestIndex >= 0) {
return ts[bestIndex];
}
return d0 < d2 ? 0 : 1;
}
bool SkDQuad::pointInHull(const SkDPoint& pt) const {
return ((const SkDTriangle&) fPts).contains(pt);
}
SkDPoint SkDQuad::top(double startT, double endT) const {
SkDQuad sub = subDivide(startT, endT);
SkDPoint topPt = sub[0];
if (topPt.fY > sub[2].fY || (topPt.fY == sub[2].fY && topPt.fX > sub[2].fX)) {
topPt = sub[2];
}
if (!between(sub[0].fY, sub[1].fY, sub[2].fY)) {
double extremeT;
if (FindExtrema(sub[0].fY, sub[1].fY, sub[2].fY, &extremeT)) {
extremeT = startT + (endT - startT) * extremeT;
SkDPoint test = ptAtT(extremeT);
if (topPt.fY > test.fY || (topPt.fY == test.fY && topPt.fX > test.fX)) {
topPt = test;
}
}
}
return topPt;
}
int SkDQuad::AddValidTs(double s[], int realRoots, double* t) {
int foundRoots = 0;
for (int index = 0; index < realRoots; ++index) {
double tValue = s[index];
if (approximately_zero_or_more(tValue) && approximately_one_or_less(tValue)) {
if (approximately_less_than_zero(tValue)) {
tValue = 0;
} else if (approximately_greater_than_one(tValue)) {
tValue = 1;
}
for (int idx2 = 0; idx2 < foundRoots; ++idx2) {
if (approximately_equal(t[idx2], tValue)) {
goto nextRoot;
}
}
t[foundRoots++] = tValue;
}
nextRoot:
{}
}
return foundRoots;
}
// note: caller expects multiple results to be sorted smaller first
// note: http://en.wikipedia.org/wiki/Loss_of_significance has an interesting
// analysis of the quadratic equation, suggesting why the following looks at
// the sign of B -- and further suggesting that the greatest loss of precision
// is in b squared less two a c
int SkDQuad::RootsValidT(double A, double B, double C, double t[2]) {
double s[2];
int realRoots = RootsReal(A, B, C, s);
int foundRoots = AddValidTs(s, realRoots, t);
return foundRoots;
}
/*
Numeric Solutions (5.6) suggests to solve the quadratic by computing
Q = -1/2(B + sgn(B)Sqrt(B^2 - 4 A C))
and using the roots
t1 = Q / A
t2 = C / Q
*/
// this does not discard real roots <= 0 or >= 1
int SkDQuad::RootsReal(const double A, const double B, const double C, double s[2]) {
const double p = B / (2 * A);
const double q = C / A;
if (approximately_zero(A) && (approximately_zero_inverse(p) || approximately_zero_inverse(q))) {
if (approximately_zero(B)) {
s[0] = 0;
return C == 0;
}
s[0] = -C / B;
return 1;
}
/* normal form: x^2 + px + q = 0 */
const double p2 = p * p;
if (!AlmostEqualUlps(p2, q) && p2 < q) {
return 0;
}
double sqrt_D = 0;
if (p2 > q) {
sqrt_D = sqrt(p2 - q);
}
s[0] = sqrt_D - p;
s[1] = -sqrt_D - p;
return 1 + !AlmostEqualUlps(s[0], s[1]);
}
bool SkDQuad::isLinear(int startIndex, int endIndex) const {
SkLineParameters lineParameters;
lineParameters.quadEndPoints(*this, startIndex, endIndex);
// FIXME: maybe it's possible to avoid this and compare non-normalized
lineParameters.normalize();
double distance = lineParameters.controlPtDistance(*this);
return approximately_zero(distance);
}
SkDCubic SkDQuad::toCubic() const {
SkDCubic cubic;
cubic[0] = fPts[0];
cubic[2] = fPts[1];
cubic[3] = fPts[2];
cubic[1].fX = (cubic[0].fX + cubic[2].fX * 2) / 3;
cubic[1].fY = (cubic[0].fY + cubic[2].fY * 2) / 3;
cubic[2].fX = (cubic[3].fX + cubic[2].fX * 2) / 3;
cubic[2].fY = (cubic[3].fY + cubic[2].fY * 2) / 3;
return cubic;
}
SkDVector SkDQuad::dxdyAtT(double t) const {
double a = t - 1;
double b = 1 - 2 * t;
double c = t;
SkDVector result = { a * fPts[0].fX + b * fPts[1].fX + c * fPts[2].fX,
a * fPts[0].fY + b * fPts[1].fY + c * fPts[2].fY };
return result;
}
// OPTIMIZE: assert if caller passes in t == 0 / t == 1 ?
SkDPoint SkDQuad::ptAtT(double t) const {
if (0 == t) {
return fPts[0];
}
if (1 == t) {
return fPts[2];
}
double one_t = 1 - t;
double a = one_t * one_t;
double b = 2 * one_t * t;
double c = t * t;
SkDPoint result = { a * fPts[0].fX + b * fPts[1].fX + c * fPts[2].fX,
a * fPts[0].fY + b * fPts[1].fY + c * fPts[2].fY };
return result;
}
/*
Given a quadratic q, t1, and t2, find a small quadratic segment.
The new quadratic is defined by A, B, and C, where
A = c[0]*(1 - t1)*(1 - t1) + 2*c[1]*t1*(1 - t1) + c[2]*t1*t1
C = c[3]*(1 - t1)*(1 - t1) + 2*c[2]*t1*(1 - t1) + c[1]*t1*t1
To find B, compute the point halfway between t1 and t2:
q(at (t1 + t2)/2) == D
Next, compute where D must be if we know the value of B:
_12 = A/2 + B/2
12_ = B/2 + C/2
123 = A/4 + B/2 + C/4
= D
Group the known values on one side:
B = D*2 - A/2 - C/2
*/
static double interp_quad_coords(const double* src, double t) {
double ab = SkDInterp(src[0], src[2], t);
double bc = SkDInterp(src[2], src[4], t);
double abc = SkDInterp(ab, bc, t);
return abc;
}
bool SkDQuad::monotonicInY() const {
return between(fPts[0].fY, fPts[1].fY, fPts[2].fY);
}
SkDQuad SkDQuad::subDivide(double t1, double t2) const {
SkDQuad dst;
double ax = dst[0].fX = interp_quad_coords(&fPts[0].fX, t1);
double ay = dst[0].fY = interp_quad_coords(&fPts[0].fY, t1);
double dx = interp_quad_coords(&fPts[0].fX, (t1 + t2) / 2);
double dy = interp_quad_coords(&fPts[0].fY, (t1 + t2) / 2);
double cx = dst[2].fX = interp_quad_coords(&fPts[0].fX, t2);
double cy = dst[2].fY = interp_quad_coords(&fPts[0].fY, t2);
/* bx = */ dst[1].fX = 2*dx - (ax + cx)/2;
/* by = */ dst[1].fY = 2*dy - (ay + cy)/2;
return dst;
}
void SkDQuad::align(int endIndex, SkDPoint* dstPt) const {
if (fPts[endIndex].fX == fPts[1].fX) {
dstPt->fX = fPts[endIndex].fX;
}
if (fPts[endIndex].fY == fPts[1].fY) {
dstPt->fY = fPts[endIndex].fY;
}
}
SkDPoint SkDQuad::subDivide(const SkDPoint& a, const SkDPoint& c, double t1, double t2) const {
SkASSERT(t1 != t2);
SkDPoint b;
#if 0
// this approach assumes that the control point computed directly is accurate enough
double dx = interp_quad_coords(&fPts[0].fX, (t1 + t2) / 2);
double dy = interp_quad_coords(&fPts[0].fY, (t1 + t2) / 2);
b.fX = 2 * dx - (a.fX + c.fX) / 2;
b.fY = 2 * dy - (a.fY + c.fY) / 2;
#else
SkDQuad sub = subDivide(t1, t2);
SkDLine b0 = {{a, sub[1] + (a - sub[0])}};
SkDLine b1 = {{c, sub[1] + (c - sub[2])}};
SkIntersections i;
i.intersectRay(b0, b1);
if (i.used() == 1) {
b = i.pt(0);
} else {
SkASSERT(i.used() == 2 || i.used() == 0);
b = SkDPoint::Mid(b0[1], b1[1]);
}
#endif
if (t1 == 0 || t2 == 0) {
align(0, &b);
}
if (t1 == 1 || t2 == 1) {
align(2, &b);
}
if (precisely_subdivide_equal(b.fX, a.fX)) {
b.fX = a.fX;
} else if (precisely_subdivide_equal(b.fX, c.fX)) {
b.fX = c.fX;
}
if (precisely_subdivide_equal(b.fY, a.fY)) {
b.fY = a.fY;
} else if (precisely_subdivide_equal(b.fY, c.fY)) {
b.fY = c.fY;
}
return b;
}
/* classic one t subdivision */
static void interp_quad_coords(const double* src, double* dst, double t) {
double ab = SkDInterp(src[0], src[2], t);
double bc = SkDInterp(src[2], src[4], t);
dst[0] = src[0];
dst[2] = ab;
dst[4] = SkDInterp(ab, bc, t);
dst[6] = bc;
dst[8] = src[4];
}
SkDQuadPair SkDQuad::chopAt(double t) const
{
SkDQuadPair dst;
interp_quad_coords(&fPts[0].fX, &dst.pts[0].fX, t);
interp_quad_coords(&fPts[0].fY, &dst.pts[0].fY, t);
return dst;
}
static int valid_unit_divide(double numer, double denom, double* ratio)
{
if (numer < 0) {
numer = -numer;
denom = -denom;
}
if (denom == 0 || numer == 0 || numer >= denom) {
return 0;
}
double r = numer / denom;
if (r == 0) { // catch underflow if numer <<<< denom
return 0;
}
*ratio = r;
return 1;
}
/** Quad'(t) = At + B, where
A = 2(a - 2b + c)
B = 2(b - a)
Solve for t, only if it fits between 0 < t < 1
*/
int SkDQuad::FindExtrema(double a, double b, double c, double tValue[1]) {
/* At + B == 0
t = -B / A
*/
return valid_unit_divide(a - b, a - b - b + c, tValue);
}
/* Parameterization form, given A*t*t + 2*B*t*(1-t) + C*(1-t)*(1-t)
*
* a = A - 2*B + C
* b = 2*B - 2*C
* c = C
*/
void SkDQuad::SetABC(const double* quad, double* a, double* b, double* c) {
*a = quad[0]; // a = A
*b = 2 * quad[2]; // b = 2*B
*c = quad[4]; // c = C
*b -= *c; // b = 2*B - C
*a -= *b; // a = A - 2*B + C
*b -= *c; // b = 2*B - 2*C
}