/*
* 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 "CurveIntersection.h"
#include "Extrema.h"
#include "IntersectionUtilities.h"
#include "LineParameters.h"
static double interp_cubic_coords(const double* src, double t)
{
double ab = interp(src[0], src[2], t);
double bc = interp(src[2], src[4], t);
double cd = interp(src[4], src[6], t);
double abc = interp(ab, bc, t);
double bcd = interp(bc, cd, t);
return interp(abc, bcd, t);
}
static int coincident_line(const Cubic& cubic, Cubic& reduction) {
reduction[0] = reduction[1] = cubic[0];
return 1;
}
static int vertical_line(const Cubic& cubic, ReduceOrder_Styles reduceStyle, Cubic& reduction) {
double tValues[2];
reduction[0] = cubic[0];
reduction[1] = cubic[3];
if (reduceStyle == kReduceOrder_TreatAsFill) {
return 2;
}
int smaller = reduction[1].y > reduction[0].y;
int larger = smaller ^ 1;
int roots = findExtrema(cubic[0].y, cubic[1].y, cubic[2].y, cubic[3].y, tValues);
for (int index = 0; index < roots; ++index) {
double yExtrema = interp_cubic_coords(&cubic[0].y, tValues[index]);
if (reduction[smaller].y > yExtrema) {
reduction[smaller].y = yExtrema;
continue;
}
if (reduction[larger].y < yExtrema) {
reduction[larger].y = yExtrema;
}
}
return 2;
}
static int horizontal_line(const Cubic& cubic, ReduceOrder_Styles reduceStyle, Cubic& reduction) {
double tValues[2];
reduction[0] = cubic[0];
reduction[1] = cubic[3];
if (reduceStyle == kReduceOrder_TreatAsFill) {
return 2;
}
int smaller = reduction[1].x > reduction[0].x;
int larger = smaller ^ 1;
int roots = findExtrema(cubic[0].x, cubic[1].x, cubic[2].x, cubic[3].x, tValues);
for (int index = 0; index < roots; ++index) {
double xExtrema = interp_cubic_coords(&cubic[0].x, tValues[index]);
if (reduction[smaller].x > xExtrema) {
reduction[smaller].x = xExtrema;
continue;
}
if (reduction[larger].x < xExtrema) {
reduction[larger].x = xExtrema;
}
}
return 2;
}
// check to see if it is a quadratic or a line
static int check_quadratic(const Cubic& cubic, Cubic& reduction) {
double dx10 = cubic[1].x - cubic[0].x;
double dx23 = cubic[2].x - cubic[3].x;
double midX = cubic[0].x + dx10 * 3 / 2;
if (!AlmostEqualUlps(midX - cubic[3].x, dx23 * 3 / 2)) {
return 0;
}
double dy10 = cubic[1].y - cubic[0].y;
double dy23 = cubic[2].y - cubic[3].y;
double midY = cubic[0].y + dy10 * 3 / 2;
if (!AlmostEqualUlps(midY - cubic[3].y, dy23 * 3 / 2)) {
return 0;
}
reduction[0] = cubic[0];
reduction[1].x = midX;
reduction[1].y = midY;
reduction[2] = cubic[3];
return 3;
}
static int check_linear(const Cubic& cubic, ReduceOrder_Styles reduceStyle,
int minX, int maxX, int minY, int maxY, Cubic& reduction) {
int startIndex = 0;
int endIndex = 3;
while (cubic[startIndex].approximatelyEqual(cubic[endIndex])) {
--endIndex;
if (endIndex == 0) {
printf("%s shouldn't get here if all four points are about equal\n", __FUNCTION__);
SkASSERT(0);
}
}
if (!isLinear(cubic, startIndex, endIndex)) {
return 0;
}
// four are colinear: return line formed by outside
reduction[0] = cubic[0];
reduction[1] = cubic[3];
if (reduceStyle == kReduceOrder_TreatAsFill) {
return 2;
}
int sameSide1;
int sameSide2;
bool useX = cubic[maxX].x - cubic[minX].x >= cubic[maxY].y - cubic[minY].y;
if (useX) {
sameSide1 = sign(cubic[0].x - cubic[1].x) + sign(cubic[3].x - cubic[1].x);
sameSide2 = sign(cubic[0].x - cubic[2].x) + sign(cubic[3].x - cubic[2].x);
} else {
sameSide1 = sign(cubic[0].y - cubic[1].y) + sign(cubic[3].y - cubic[1].y);
sameSide2 = sign(cubic[0].y - cubic[2].y) + sign(cubic[3].y - cubic[2].y);
}
if (sameSide1 == sameSide2 && (sameSide1 & 3) != 2) {
return 2;
}
double tValues[2];
int roots;
if (useX) {
roots = findExtrema(cubic[0].x, cubic[1].x, cubic[2].x, cubic[3].x, tValues);
} else {
roots = findExtrema(cubic[0].y, cubic[1].y, cubic[2].y, cubic[3].y, tValues);
}
for (int index = 0; index < roots; ++index) {
_Point extrema;
extrema.x = interp_cubic_coords(&cubic[0].x, tValues[index]);
extrema.y = interp_cubic_coords(&cubic[0].y, tValues[index]);
// sameSide > 0 means mid is smaller than either [0] or [3], so replace smaller
int replace;
if (useX) {
if (extrema.x < cubic[0].x ^ extrema.x < cubic[3].x) {
continue;
}
replace = (extrema.x < cubic[0].x | extrema.x < cubic[3].x)
^ (cubic[0].x < cubic[3].x);
} else {
if (extrema.y < cubic[0].y ^ extrema.y < cubic[3].y) {
continue;
}
replace = (extrema.y < cubic[0].y | extrema.y < cubic[3].y)
^ (cubic[0].y < cubic[3].y);
}
reduction[replace] = extrema;
}
return 2;
}
bool isLinear(const Cubic& cubic, int startIndex, int endIndex) {
LineParameters lineParameters;
lineParameters.cubicEndPoints(cubic, startIndex, endIndex);
// FIXME: maybe it's possible to avoid this and compare non-normalized
lineParameters.normalize();
double distance = lineParameters.controlPtDistance(cubic, 1);
if (!approximately_zero(distance)) {
return false;
}
distance = lineParameters.controlPtDistance(cubic, 2);
return approximately_zero(distance);
}
/* food for thought:
http://objectmix.com/graphics/132906-fast-precision-driven-cubic-quadratic-piecewise-degree-reduction-algos-2-a.html
Given points c1, c2, c3 and c4 of a cubic Bezier, the points of the
corresponding quadratic Bezier are (given in convex combinations of
points):
q1 = (11/13)c1 + (3/13)c2 -(3/13)c3 + (2/13)c4
q2 = -c1 + (3/2)c2 + (3/2)c3 - c4
q3 = (2/13)c1 - (3/13)c2 + (3/13)c3 + (11/13)c4
Of course, this curve does not interpolate the end-points, but it would
be interesting to see the behaviour of such a curve in an applet.
--
Kalle Rutanen
http://kaba.hilvi.org
*/
// reduce to a quadratic or smaller
// look for identical points
// look for all four points in a line
// note that three points in a line doesn't simplify a cubic
// look for approximation with single quadratic
// save approximation with multiple quadratics for later
int reduceOrder(const Cubic& cubic, Cubic& reduction, ReduceOrder_Quadratics allowQuadratics,
ReduceOrder_Styles reduceStyle) {
int index, minX, maxX, minY, maxY;
int minXSet, minYSet;
minX = maxX = minY = maxY = 0;
minXSet = minYSet = 0;
for (index = 1; index < 4; ++index) {
if (cubic[minX].x > cubic[index].x) {
minX = index;
}
if (cubic[minY].y > cubic[index].y) {
minY = index;
}
if (cubic[maxX].x < cubic[index].x) {
maxX = index;
}
if (cubic[maxY].y < cubic[index].y) {
maxY = index;
}
}
for (index = 0; index < 4; ++index) {
double cx = cubic[index].x;
double cy = cubic[index].y;
double denom = SkTMax(fabs(cx), SkTMax(fabs(cy),
SkTMax(fabs(cubic[minX].x), fabs(cubic[minY].y))));
if (denom == 0) {
minXSet |= 1 << index;
minYSet |= 1 << index;
continue;
}
double inv = 1 / denom;
if (approximately_equal_half(cx * inv, cubic[minX].x * inv)) {
minXSet |= 1 << index;
}
if (approximately_equal_half(cy * inv, cubic[minY].y * inv)) {
minYSet |= 1 << index;
}
}
if (minXSet == 0xF) { // test for vertical line
if (minYSet == 0xF) { // return 1 if all four are coincident
return coincident_line(cubic, reduction);
}
return vertical_line(cubic, reduceStyle, reduction);
}
if (minYSet == 0xF) { // test for horizontal line
return horizontal_line(cubic, reduceStyle, reduction);
}
int result = check_linear(cubic, reduceStyle, minX, maxX, minY, maxY, reduction);
if (result) {
return result;
}
if (allowQuadratics == kReduceOrder_QuadraticsAllowed
&& (result = check_quadratic(cubic, reduction))) {
return result;
}
memcpy(reduction, cubic, sizeof(Cubic));
return 4;
}