/*
* 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 "LineUtilities.h"
#include "QuadraticUtilities.h"
const int gPrecisionUnit = 256; // FIXME: arbitrary -- should try different values in test framework
// FIXME: cache keep the bounds and/or precision with the caller?
double calcPrecision(const Cubic& cubic) {
_Rect dRect;
dRect.setBounds(cubic); // OPTIMIZATION: just use setRawBounds ?
double width = dRect.right - dRect.left;
double height = dRect.bottom - dRect.top;
return (width > height ? width : height) / gPrecisionUnit;
}
#if SK_DEBUG
double calcPrecision(const Cubic& cubic, double t, double scale) {
Cubic part;
sub_divide(cubic, SkTMax(0., t - scale), SkTMin(1., t + scale), part);
return calcPrecision(part);
}
#endif
bool clockwise(const Cubic& c) {
double sum = (c[0].x - c[3].x) * (c[0].y + c[3].y);
for (int idx = 0; idx < 3; ++idx){
sum += (c[idx + 1].x - c[idx].x) * (c[idx + 1].y + c[idx].y);
}
return sum <= 0;
}
void coefficients(const double* cubic, double& A, double& B, double& C, double& D) {
A = cubic[6]; // d
B = cubic[4] * 3; // 3*c
C = cubic[2] * 3; // 3*b
D = cubic[0]; // a
A -= D - C + B; // A = -a + 3*b - 3*c + d
B += 3 * D - 2 * C; // B = 3*a - 6*b + 3*c
C -= 3 * D; // C = -3*a + 3*b
}
bool controls_contained_by_ends(const Cubic& c) {
_Vector startTan = c[1] - c[0];
if (startTan.x == 0 && startTan.y == 0) {
startTan = c[2] - c[0];
}
_Vector endTan = c[2] - c[3];
if (endTan.x == 0 && endTan.y == 0) {
endTan = c[1] - c[3];
}
if (startTan.dot(endTan) >= 0) {
return false;
}
_Line startEdge = {c[0], c[0]};
startEdge[1].x -= startTan.y;
startEdge[1].y += startTan.x;
_Line endEdge = {c[3], c[3]};
endEdge[1].x -= endTan.y;
endEdge[1].y += endTan.x;
double leftStart1 = is_left(startEdge, c[1]);
if (leftStart1 * is_left(startEdge, c[2]) < 0) {
return false;
}
double leftEnd1 = is_left(endEdge, c[1]);
if (leftEnd1 * is_left(endEdge, c[2]) < 0) {
return false;
}
return leftStart1 * leftEnd1 >= 0;
}
bool ends_are_extrema_in_x_or_y(const Cubic& c) {
return (between(c[0].x, c[1].x, c[3].x) && between(c[0].x, c[2].x, c[3].x))
|| (between(c[0].y, c[1].y, c[3].y) && between(c[0].y, c[2].y, c[3].y));
}
bool monotonic_in_y(const Cubic& c) {
return between(c[0].y, c[1].y, c[3].y) && between(c[0].y, c[2].y, c[3].y);
}
bool serpentine(const Cubic& c) {
if (!controls_contained_by_ends(c)) {
return false;
}
double wiggle = (c[0].x - c[2].x) * (c[0].y + c[2].y);
for (int idx = 0; idx < 2; ++idx){
wiggle += (c[idx + 1].x - c[idx].x) * (c[idx + 1].y + c[idx].y);
}
double waggle = (c[1].x - c[3].x) * (c[1].y + c[3].y);
for (int idx = 1; idx < 3; ++idx){
waggle += (c[idx + 1].x - c[idx].x) * (c[idx + 1].y + c[idx].y);
}
return wiggle * waggle < 0;
}
// cubic roots
const double PI = 4 * atan(1);
// from SkGeometry.cpp (and Numeric Solutions, 5.6)
int cubicRootsValidT(double A, double B, double C, double D, double t[3]) {
#if 0
if (approximately_zero(A)) { // we're just a quadratic
return quadraticRootsValidT(B, C, D, t);
}
double a, b, c;
{
double invA = 1 / A;
a = B * invA;
b = C * invA;
c = D * invA;
}
double a2 = a * a;
double Q = (a2 - b * 3) / 9;
double R = (2 * a2 * a - 9 * a * b + 27 * c) / 54;
double Q3 = Q * Q * Q;
double R2MinusQ3 = R * R - Q3;
double adiv3 = a / 3;
double* roots = t;
double r;
if (R2MinusQ3 < 0) // we have 3 real roots
{
double theta = acos(R / sqrt(Q3));
double neg2RootQ = -2 * sqrt(Q);
r = neg2RootQ * cos(theta / 3) - adiv3;
if (is_unit_interval(r))
*roots++ = r;
r = neg2RootQ * cos((theta + 2 * PI) / 3) - adiv3;
if (is_unit_interval(r))
*roots++ = r;
r = neg2RootQ * cos((theta - 2 * PI) / 3) - adiv3;
if (is_unit_interval(r))
*roots++ = r;
}
else // we have 1 real root
{
double A = fabs(R) + sqrt(R2MinusQ3);
A = cube_root(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 - t);
#else
double s[3];
int realRoots = cubicRootsReal(A, B, C, D, s);
int foundRoots = add_valid_ts(s, realRoots, t);
return foundRoots;
#endif
}
int cubicRootsReal(double A, double B, double C, double D, double s[3]) {
#if SK_DEBUG
// create a string mathematica understands
// GDB set print repe 15 # if repeated digits is a bother
// set print elements 400 # if line doesn't fit
char str[1024];
bzero(str, sizeof(str));
sprintf(str, "Solve[%1.19g x^3 + %1.19g x^2 + %1.19g x + %1.19g == 0, x]", A, B, C, D);
mathematica_ize(str, sizeof(str));
#if ONE_OFF_DEBUG && ONE_OFF_DEBUG_MATHEMATICA
SkDebugf("%s\n", str);
#endif
#endif
if (approximately_zero(A)
&& approximately_zero_when_compared_to(A, B)
&& approximately_zero_when_compared_to(A, C)
&& approximately_zero_when_compared_to(A, D)) { // we're just a quadratic
return quadraticRootsReal(B, C, D, s);
}
if (approximately_zero_when_compared_to(D, A)
&& approximately_zero_when_compared_to(D, B)
&& approximately_zero_when_compared_to(D, C)) { // 0 is one root
int num = quadraticRootsReal(A, B, C, s);
for (int i = 0; i < num; ++i) {
if (approximately_zero(s[i])) {
return num;
}
}
s[num++] = 0;
return num;
}
if (approximately_zero(A + B + C + D)) { // 1 is one root
int num = quadraticRootsReal(A, A + B, -D, s);
for (int i = 0; i < num; ++i) {
if (AlmostEqualUlps(s[i], 1)) {
return num;
}
}
s[num++] = 1;
return num;
}
double a, b, c;
{
double invA = 1 / A;
a = B * invA;
b = C * invA;
c = D * invA;
}
double a2 = a * a;
double Q = (a2 - b * 3) / 9;
double R = (2 * a2 * a - 9 * a * b + 27 * c) / 54;
double R2 = R * R;
double Q3 = Q * Q * Q;
double R2MinusQ3 = R2 - Q3;
double adiv3 = a / 3;
double r;
double* roots = s;
#if 0
if (approximately_zero_squared(R2MinusQ3) && AlmostEqualUlps(R2, Q3)) {
if (approximately_zero_squared(R)) {/* one triple solution */
*roots++ = -adiv3;
} else { /* one single and one double solution */
double u = cube_root(-R);
*roots++ = 2 * u - adiv3;
*roots++ = -u - adiv3;
}
}
else
#endif
if (R2MinusQ3 < 0) // we have 3 real roots
{
double theta = acos(R / sqrt(Q3));
double neg2RootQ = -2 * sqrt(Q);
r = neg2RootQ * cos(theta / 3) - adiv3;
*roots++ = r;
r = neg2RootQ * cos((theta + 2 * PI) / 3) - adiv3;
if (!AlmostEqualUlps(s[0], r)) {
*roots++ = r;
}
r = neg2RootQ * cos((theta - 2 * PI) / 3) - adiv3;
if (!AlmostEqualUlps(s[0], r) && (roots - s == 1 || !AlmostEqualUlps(s[1], r))) {
*roots++ = r;
}
}
else // we have 1 real root
{
double sqrtR2MinusQ3 = sqrt(R2MinusQ3);
double A = fabs(R) + sqrtR2MinusQ3;
A = cube_root(A);
if (R > 0) {
A = -A;
}
if (A != 0) {
A += Q / A;
}
r = A - adiv3;
*roots++ = r;
if (AlmostEqualUlps(R2, Q3)) {
r = -A / 2 - adiv3;
if (!AlmostEqualUlps(s[0], r)) {
*roots++ = r;
}
}
}
return (int)(roots - s);
}
// from http://www.cs.sunysb.edu/~qin/courses/geometry/4.pdf
// c(t) = a(1-t)^3 + 3bt(1-t)^2 + 3c(1-t)t^2 + dt^3
// c'(t) = -3a(1-t)^2 + 3b((1-t)^2 - 2t(1-t)) + 3c(2t(1-t) - t^2) + 3dt^2
// = 3(b-a)(1-t)^2 + 6(c-b)t(1-t) + 3(d-c)t^2
static double derivativeAtT(const double* cubic, double t) {
double one_t = 1 - t;
double a = cubic[0];
double b = cubic[2];
double c = cubic[4];
double d = cubic[6];
return 3 * ((b - a) * one_t * one_t + 2 * (c - b) * t * one_t + (d - c) * t * t);
}
double dx_at_t(const Cubic& cubic, double t) {
return derivativeAtT(&cubic[0].x, t);
}
double dy_at_t(const Cubic& cubic, double t) {
return derivativeAtT(&cubic[0].y, t);
}
// OPTIMIZE? compute t^2, t(1-t), and (1-t)^2 and pass them to another version of derivative at t?
_Vector dxdy_at_t(const Cubic& cubic, double t) {
_Vector result = { derivativeAtT(&cubic[0].x, t), derivativeAtT(&cubic[0].y, t) };
return result;
}
// OPTIMIZE? share code with formulate_F1DotF2
int find_cubic_inflections(const Cubic& src, double tValues[])
{
double Ax = src[1].x - src[0].x;
double Ay = src[1].y - src[0].y;
double Bx = src[2].x - 2 * src[1].x + src[0].x;
double By = src[2].y - 2 * src[1].y + src[0].y;
double Cx = src[3].x + 3 * (src[1].x - src[2].x) - src[0].x;
double Cy = src[3].y + 3 * (src[1].y - src[2].y) - src[0].y;
return quadraticRootsValidT(Bx * Cy - By * Cx, Ax * Cy - Ay * Cx, Ax * By - Ay * Bx, tValues);
}
static void formulate_F1DotF2(const double src[], double coeff[4])
{
double a = src[2] - src[0];
double b = src[4] - 2 * src[2] + src[0];
double 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;
}
/* from SkGeometry.cpp
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 find_cubic_max_curvature(const Cubic& src, double tValues[])
{
double coeffX[4], coeffY[4];
int i;
formulate_F1DotF2(&src[0].x, coeffX);
formulate_F1DotF2(&src[0].y, coeffY);
for (i = 0; i < 4; i++) {
coeffX[i] = coeffX[i] + coeffY[i];
}
return cubicRootsValidT(coeffX[0], coeffX[1], coeffX[2], coeffX[3], tValues);
}
bool rotate(const Cubic& cubic, int zero, int index, Cubic& rotPath) {
double dy = cubic[index].y - cubic[zero].y;
double dx = cubic[index].x - cubic[zero].x;
if (approximately_zero(dy)) {
if (approximately_zero(dx)) {
return false;
}
memcpy(rotPath, cubic, sizeof(Cubic));
return true;
}
for (int index = 0; index < 4; ++index) {
rotPath[index].x = cubic[index].x * dx + cubic[index].y * dy;
rotPath[index].y = cubic[index].y * dx - cubic[index].x * dy;
}
return true;
}
#if 0 // unused for now
double secondDerivativeAtT(const double* cubic, double t) {
double a = cubic[0];
double b = cubic[2];
double c = cubic[4];
double d = cubic[6];
return (c - 2 * b + a) * (1 - t) + (d - 2 * c + b) * t;
}
#endif
_Point top(const Cubic& cubic, double startT, double endT) {
Cubic sub;
sub_divide(cubic, startT, endT, sub);
_Point topPt = sub[0];
if (topPt.y > sub[3].y || (topPt.y == sub[3].y && topPt.x > sub[3].x)) {
topPt = sub[3];
}
double extremeTs[2];
if (!monotonic_in_y(sub)) {
int roots = findExtrema(sub[0].y, sub[1].y, sub[2].y, sub[3].y, extremeTs);
for (int index = 0; index < roots; ++index) {
_Point mid;
double t = startT + (endT - startT) * extremeTs[index];
xy_at_t(cubic, t, mid.x, mid.y);
if (topPt.y > mid.y || (topPt.y == mid.y && topPt.x > mid.x)) {
topPt = mid;
}
}
}
return topPt;
}
// OPTIMIZE: avoid computing the unused half
void xy_at_t(const Cubic& cubic, double t, double& x, double& y) {
_Point xy = xy_at_t(cubic, t);
if (&x) {
x = xy.x;
}
if (&y) {
y = xy.y;
}
}
_Point xy_at_t(const Cubic& cubic, double t) {
double one_t = 1 - t;
double one_t2 = one_t * one_t;
double a = one_t2 * one_t;
double b = 3 * one_t2 * t;
double t2 = t * t;
double c = 3 * one_t * t2;
double d = t2 * t;
_Point result = {a * cubic[0].x + b * cubic[1].x + c * cubic[2].x + d * cubic[3].x,
a * cubic[0].y + b * cubic[1].y + c * cubic[2].y + d * cubic[3].y};
return result;
}