/*
* 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 "CubicUtilities.h"
#include "Extrema.h"
#include "QuadraticUtilities.h"
#include "TriangleUtilities.h"
// from http://blog.gludion.com/2009/08/distance-to-quadratic-bezier-curve.html
double nearestT(const Quadratic& quad, const _Point& pt) {
_Vector pos = quad[0] - pt;
// search points P of bezier curve with PM.(dP / dt) = 0
// a calculus leads to a 3d degree equation :
_Vector A = quad[1] - quad[0];
_Vector B = quad[2] - quad[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 = cubicRootsValidT(a, b, c, d, ts);
double d0 = pt.distanceSquared(quad[0]);
double d2 = pt.distanceSquared(quad[2]);
double distMin = SkTMin(d0, d2);
int bestIndex = -1;
for (int index = 0; index < roots; ++index) {
_Point onQuad;
xy_at_t(quad, ts[index], onQuad.x, onQuad.y);
double dist = pt.distanceSquared(onQuad);
if (distMin > dist) {
distMin = dist;
bestIndex = index;
}
}
if (bestIndex >= 0) {
return ts[bestIndex];
}
return d0 < d2 ? 0 : 1;
}
bool point_in_hull(const Quadratic& quad, const _Point& pt) {
return pointInTriangle((const Triangle&) quad, pt);
}
_Point top(const Quadratic& quad, double startT, double endT) {
Quadratic sub;
sub_divide(quad, startT, endT, sub);
_Point topPt = sub[0];
if (topPt.y > sub[2].y || (topPt.y == sub[2].y && topPt.x > sub[2].x)) {
topPt = sub[2];
}
if (!between(sub[0].y, sub[1].y, sub[2].y)) {
double extremeT;
if (findExtrema(sub[0].y, sub[1].y, sub[2].y, &extremeT)) {
extremeT = startT + (endT - startT) * extremeT;
_Point test;
xy_at_t(quad, extremeT, test.x, test.y);
if (topPt.y > test.y || (topPt.y == test.y && topPt.x > test.x)) {
topPt = test;
}
}
}
return topPt;
}
/*
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
*/
int add_valid_ts(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 quadraticRootsValidT(double A, double B, double C, double t[2]) {
#if 0
B *= 2;
double square = B * B - 4 * A * C;
if (approximately_negative(square)) {
if (!approximately_positive(square)) {
return 0;
}
square = 0;
}
double squareRt = sqrt(square);
double Q = (B + (B < 0 ? -squareRt : squareRt)) / -2;
int foundRoots = 0;
double ratio = Q / A;
if (approximately_zero_or_more(ratio) && approximately_one_or_less(ratio)) {
if (approximately_less_than_zero(ratio)) {
ratio = 0;
} else if (approximately_greater_than_one(ratio)) {
ratio = 1;
}
t[0] = ratio;
++foundRoots;
}
ratio = C / Q;
if (approximately_zero_or_more(ratio) && approximately_one_or_less(ratio)) {
if (approximately_less_than_zero(ratio)) {
ratio = 0;
} else if (approximately_greater_than_one(ratio)) {
ratio = 1;
}
if (foundRoots == 0 || !approximately_negative(ratio - t[0])) {
t[foundRoots++] = ratio;
} else if (!approximately_negative(t[0] - ratio)) {
t[foundRoots++] = t[0];
t[0] = ratio;
}
}
#else
double s[2];
int realRoots = quadraticRootsReal(A, B, C, s);
int foundRoots = add_valid_ts(s, realRoots, t);
#endif
return foundRoots;
}
// unlike quadratic roots, this does not discard real roots <= 0 or >= 1
int quadraticRootsReal(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 0
double D = AlmostEqualUlps(p2, q) ? 0 : p2 - q;
if (D <= 0) {
if (D < 0) {
return 0;
}
s[0] = -p;
SkDebugf("[%d] %1.9g\n", 1, s[0]);
return 1;
}
double sqrt_D = sqrt(D);
s[0] = sqrt_D - p;
s[1] = -sqrt_D - p;
SkDebugf("[%d] %1.9g %1.9g\n", 2, s[0], s[1]);
return 2;
#else
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;
#if 0
if (AlmostEqualUlps(s[0], s[1])) {
SkDebugf("[%d] %1.9g\n", 1, s[0]);
} else {
SkDebugf("[%d] %1.9g %1.9g\n", 2, s[0], s[1]);
}
#endif
return 1 + !AlmostEqualUlps(s[0], s[1]);
#endif
}
void toCubic(const Quadratic& quad, Cubic& cubic) {
cubic[0] = quad[0];
cubic[2] = quad[1];
cubic[3] = quad[2];
cubic[1].x = (cubic[0].x + cubic[2].x * 2) / 3;
cubic[1].y = (cubic[0].y + cubic[2].y * 2) / 3;
cubic[2].x = (cubic[3].x + cubic[2].x * 2) / 3;
cubic[2].y = (cubic[3].y + cubic[2].y * 2) / 3;
}
static double derivativeAtT(const double* quad, double t) {
double a = t - 1;
double b = 1 - 2 * t;
double c = t;
return a * quad[0] + b * quad[2] + c * quad[4];
}
double dx_at_t(const Quadratic& quad, double t) {
return derivativeAtT(&quad[0].x, t);
}
double dy_at_t(const Quadratic& quad, double t) {
return derivativeAtT(&quad[0].y, t);
}
_Vector dxdy_at_t(const Quadratic& quad, double t) {
double a = t - 1;
double b = 1 - 2 * t;
double c = t;
_Vector result = { a * quad[0].x + b * quad[1].x + c * quad[2].x,
a * quad[0].y + b * quad[1].y + c * quad[2].y };
return result;
}
void xy_at_t(const Quadratic& quad, double t, double& x, double& y) {
double one_t = 1 - t;
double a = one_t * one_t;
double b = 2 * one_t * t;
double c = t * t;
if (&x) {
x = a * quad[0].x + b * quad[1].x + c * quad[2].x;
}
if (&y) {
y = a * quad[0].y + b * quad[1].y + c * quad[2].y;
}
}
_Point xy_at_t(const Quadratic& quad, double t) {
double one_t = 1 - t;
double a = one_t * one_t;
double b = 2 * one_t * t;
double c = t * t;
_Point result = { a * quad[0].x + b * quad[1].x + c * quad[2].x,
a * quad[0].y + b * quad[1].y + c * quad[2].y };
return result;
}