32546db149
git-svn-id: http://skia.googlecode.com/svn/trunk@5376 2bbb7eff-a529-9590-31e7-b0007b416f81
291 lines
10 KiB
C++
291 lines
10 KiB
C++
/*
|
||
* 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 "Intersections.h"
|
||
#include "IntersectionUtilities.h"
|
||
#include "LineIntersection.h"
|
||
#include "LineUtilities.h"
|
||
#include "QuadraticLineSegments.h"
|
||
#include "QuadraticUtilities.h"
|
||
#include <algorithm> // for swap
|
||
|
||
class QuadraticIntersections : public Intersections {
|
||
public:
|
||
|
||
QuadraticIntersections(const Quadratic& q1, const Quadratic& q2, Intersections& i)
|
||
: quad1(q1)
|
||
, quad2(q2)
|
||
, intersections(i)
|
||
, depth(0)
|
||
, splits(0) {
|
||
}
|
||
|
||
bool intersect() {
|
||
double minT1, minT2, maxT1, maxT2;
|
||
if (!bezier_clip(quad2, quad1, minT1, maxT1)) {
|
||
return false;
|
||
}
|
||
if (!bezier_clip(quad1, quad2, minT2, maxT2)) {
|
||
return false;
|
||
}
|
||
quad1Divisions = 1 / subDivisions(quad1);
|
||
quad2Divisions = 1 / subDivisions(quad2);
|
||
int split;
|
||
if (maxT1 - minT1 < maxT2 - minT2) {
|
||
intersections.swap();
|
||
minT2 = 0;
|
||
maxT2 = 1;
|
||
split = maxT1 - minT1 > tClipLimit;
|
||
} else {
|
||
minT1 = 0;
|
||
maxT1 = 1;
|
||
split = (maxT2 - minT2 > tClipLimit) << 1;
|
||
}
|
||
return chop(minT1, maxT1, minT2, maxT2, split);
|
||
}
|
||
|
||
protected:
|
||
|
||
bool intersect(double minT1, double maxT1, double minT2, double maxT2) {
|
||
bool t1IsLine = maxT1 - minT1 <= quad1Divisions;
|
||
bool t2IsLine = maxT2 - minT2 <= quad2Divisions;
|
||
if (t1IsLine | t2IsLine) {
|
||
return intersectAsLine(minT1, maxT1, minT2, maxT2, t1IsLine, t2IsLine);
|
||
}
|
||
Quadratic smaller, larger;
|
||
// FIXME: carry last subdivide and reduceOrder result with quad
|
||
sub_divide(quad1, minT1, maxT1, intersections.swapped() ? larger : smaller);
|
||
sub_divide(quad2, minT2, maxT2, intersections.swapped() ? smaller : larger);
|
||
double minT, maxT;
|
||
if (!bezier_clip(smaller, larger, minT, maxT)) {
|
||
if (approximately_equal(minT, maxT)) {
|
||
double smallT, largeT;
|
||
_Point q2pt, q1pt;
|
||
if (intersections.swapped()) {
|
||
largeT = interp(minT2, maxT2, minT);
|
||
xy_at_t(quad2, largeT, q2pt.x, q2pt.y);
|
||
xy_at_t(quad1, minT1, q1pt.x, q1pt.y);
|
||
if (approximately_equal(q2pt.x, q1pt.x) && approximately_equal(q2pt.y, q1pt.y)) {
|
||
smallT = minT1;
|
||
} else {
|
||
xy_at_t(quad1, maxT1, q1pt.x, q1pt.y); // FIXME: debug code
|
||
assert(approximately_equal(q2pt.x, q1pt.x) && approximately_equal(q2pt.y, q1pt.y));
|
||
smallT = maxT1;
|
||
}
|
||
} else {
|
||
smallT = interp(minT1, maxT1, minT);
|
||
xy_at_t(quad1, smallT, q1pt.x, q1pt.y);
|
||
xy_at_t(quad2, minT2, q2pt.x, q2pt.y);
|
||
if (approximately_equal(q2pt.x, q1pt.x) && approximately_equal(q2pt.y, q1pt.y)) {
|
||
largeT = minT2;
|
||
} else {
|
||
xy_at_t(quad2, maxT2, q2pt.x, q2pt.y); // FIXME: debug code
|
||
assert(approximately_equal(q2pt.x, q1pt.x) && approximately_equal(q2pt.y, q1pt.y));
|
||
largeT = maxT2;
|
||
}
|
||
}
|
||
intersections.add(smallT, largeT);
|
||
return true;
|
||
}
|
||
return false;
|
||
}
|
||
int split;
|
||
if (intersections.swapped()) {
|
||
double newMinT1 = interp(minT1, maxT1, minT);
|
||
double newMaxT1 = interp(minT1, maxT1, maxT);
|
||
split = (newMaxT1 - newMinT1 > (maxT1 - minT1) * tClipLimit) << 1;
|
||
#define VERBOSE 0
|
||
#if VERBOSE
|
||
printf("%s d=%d s=%d new1=(%g,%g) old1=(%g,%g) split=%d\n", __FUNCTION__, depth,
|
||
splits, newMinT1, newMaxT1, minT1, maxT1, split);
|
||
#endif
|
||
minT1 = newMinT1;
|
||
maxT1 = newMaxT1;
|
||
} else {
|
||
double newMinT2 = interp(minT2, maxT2, minT);
|
||
double newMaxT2 = interp(minT2, maxT2, maxT);
|
||
split = newMaxT2 - newMinT2 > (maxT2 - minT2) * tClipLimit;
|
||
#if VERBOSE
|
||
printf("%s d=%d s=%d new2=(%g,%g) old2=(%g,%g) split=%d\n", __FUNCTION__, depth,
|
||
splits, newMinT2, newMaxT2, minT2, maxT2, split);
|
||
#endif
|
||
minT2 = newMinT2;
|
||
maxT2 = newMaxT2;
|
||
}
|
||
return chop(minT1, maxT1, minT2, maxT2, split);
|
||
}
|
||
|
||
bool intersectAsLine(double minT1, double maxT1, double minT2, double maxT2,
|
||
bool treat1AsLine, bool treat2AsLine)
|
||
{
|
||
_Line line1, line2;
|
||
if (intersections.swapped()) {
|
||
std::swap(treat1AsLine, treat2AsLine);
|
||
std::swap(minT1, minT2);
|
||
std::swap(maxT1, maxT2);
|
||
}
|
||
// do line/quadratic or even line/line intersection instead
|
||
if (treat1AsLine) {
|
||
xy_at_t(quad1, minT1, line1[0].x, line1[0].y);
|
||
xy_at_t(quad1, maxT1, line1[1].x, line1[1].y);
|
||
}
|
||
if (treat2AsLine) {
|
||
xy_at_t(quad2, minT2, line2[0].x, line2[0].y);
|
||
xy_at_t(quad2, maxT2, line2[1].x, line2[1].y);
|
||
}
|
||
int pts;
|
||
double smallT, largeT;
|
||
if (treat1AsLine & treat2AsLine) {
|
||
double t1[2], t2[2];
|
||
pts = ::intersect(line1, line2, t1, t2);
|
||
for (int index = 0; index < pts; ++index) {
|
||
smallT = interp(minT1, maxT1, t1[index]);
|
||
largeT = interp(minT2, maxT2, t2[index]);
|
||
if (pts == 2) {
|
||
intersections.addCoincident(smallT, largeT, true);
|
||
} else {
|
||
intersections.add(smallT, largeT);
|
||
}
|
||
}
|
||
} else {
|
||
Intersections lq;
|
||
pts = ::intersect(treat1AsLine ? quad2 : quad1,
|
||
treat1AsLine ? line1 : line2, lq);
|
||
bool coincident = false;
|
||
if (pts == 2) { // if the line and edge are coincident treat differently
|
||
_Point midQuad, midLine;
|
||
double midQuadT = (lq.fT[0][0] + lq.fT[0][1]) / 2;
|
||
xy_at_t(treat1AsLine ? quad2 : quad1, midQuadT, midQuad.x, midQuad.y);
|
||
double lineT = t_at(treat1AsLine ? line1 : line2, midQuad);
|
||
xy_at_t(treat1AsLine ? line1 : line2, lineT, midLine.x, midLine.y);
|
||
coincident = approximately_equal(midQuad.x, midLine.x)
|
||
&& approximately_equal(midQuad.y, midLine.y);
|
||
}
|
||
for (int index = 0; index < pts; ++index) {
|
||
smallT = lq.fT[0][index];
|
||
largeT = lq.fT[1][index];
|
||
if (treat1AsLine) {
|
||
smallT = interp(minT1, maxT1, smallT);
|
||
} else {
|
||
largeT = interp(minT2, maxT2, largeT);
|
||
}
|
||
if (coincident) {
|
||
intersections.addCoincident(smallT, largeT, true);
|
||
} else {
|
||
intersections.add(smallT, largeT);
|
||
}
|
||
}
|
||
}
|
||
return pts > 0;
|
||
}
|
||
|
||
bool chop(double minT1, double maxT1, double minT2, double maxT2, int split) {
|
||
++depth;
|
||
intersections.swap();
|
||
if (split) {
|
||
++splits;
|
||
if (split & 2) {
|
||
double middle1 = (maxT1 + minT1) / 2;
|
||
intersect(minT1, middle1, minT2, maxT2);
|
||
intersect(middle1, maxT1, minT2, maxT2);
|
||
} else {
|
||
double middle2 = (maxT2 + minT2) / 2;
|
||
intersect(minT1, maxT1, minT2, middle2);
|
||
intersect(minT1, maxT1, middle2, maxT2);
|
||
}
|
||
--splits;
|
||
intersections.swap();
|
||
--depth;
|
||
return intersections.intersected();
|
||
}
|
||
bool result = intersect(minT1, maxT1, minT2, maxT2);
|
||
intersections.swap();
|
||
--depth;
|
||
return result;
|
||
}
|
||
|
||
private:
|
||
|
||
static const double tClipLimit = 0.8; // http://cagd.cs.byu.edu/~tom/papers/bezclip.pdf see Multiple intersections
|
||
const Quadratic& quad1;
|
||
const Quadratic& quad2;
|
||
Intersections& intersections;
|
||
int depth;
|
||
int splits;
|
||
double quad1Divisions; // line segments to approximate original within error
|
||
double quad2Divisions;
|
||
};
|
||
|
||
bool intersect(const Quadratic& q1, const Quadratic& q2, Intersections& i) {
|
||
if (implicit_matches(q1, q2)) {
|
||
// FIXME: compute T values
|
||
// compute the intersections of the ends to find the coincident span
|
||
bool useVertical = fabs(q1[0].x - q1[2].x) < fabs(q1[0].y - q1[2].y);
|
||
double t;
|
||
if ((t = axialIntersect(q1, q2[0], useVertical)) >= 0) {
|
||
i.addCoincident(t, 0, false);
|
||
}
|
||
if ((t = axialIntersect(q1, q2[2], useVertical)) >= 0) {
|
||
i.addCoincident(t, 1, false);
|
||
}
|
||
useVertical = fabs(q2[0].x - q2[2].x) < fabs(q2[0].y - q2[2].y);
|
||
if ((t = axialIntersect(q2, q1[0], useVertical)) >= 0) {
|
||
i.addCoincident(0, t, false);
|
||
}
|
||
if ((t = axialIntersect(q2, q1[2], useVertical)) >= 0) {
|
||
i.addCoincident(1, t, false);
|
||
}
|
||
assert(i.fCoincidentUsed <= 2);
|
||
return i.fCoincidentUsed > 0;
|
||
}
|
||
QuadraticIntersections q(q1, q2, i);
|
||
return q.intersect();
|
||
}
|
||
|
||
|
||
// Another approach is to start with the implicit form of one curve and solve
|
||
// by substituting in the parametric form of the other.
|
||
// The downside of this approach is that early rejects are difficult to come by.
|
||
// http://planetmath.org/encyclopedia/GaloisTheoreticDerivationOfTheQuarticFormula.html#step
|
||
/*
|
||
given x^4 + ax^3 + bx^2 + cx + d
|
||
the resolvent cubic is x^3 - 2bx^2 + (b^2 + ac - 4d)x + (c^2 + a^2d - abc)
|
||
use the cubic formula (CubicRoots.cpp) to find the radical expressions t1, t2, and t3.
|
||
|
||
(x - r1 r2) (x - r3 r4) = x^2 - (t2 + t3 - t1) / 2 x + d
|
||
s = r1*r2 = ((t2 + t3 - t1) + sqrt((t2 + t3 - t1)^2 - 16*d)) / 4
|
||
t = r3*r4 = ((t2 + t3 - t1) - sqrt((t2 + t3 - t1)^2 - 16*d)) / 4
|
||
|
||
u = r1+r2 = (-a + sqrt(a^2 - 4*t1)) / 2
|
||
v = r3+r4 = (-a - sqrt(a^2 - 4*t1)) / 2
|
||
|
||
r1 = (u + sqrt(u^2 - 4*s)) / 2
|
||
r2 = (u - sqrt(u^2 - 4*s)) / 2
|
||
r3 = (v + sqrt(v^2 - 4*t)) / 2
|
||
r4 = (v - sqrt(v^2 - 4*t)) / 2
|
||
*/
|
||
|
||
|
||
/* square root of complex number
|
||
http://en.wikipedia.org/wiki/Square_root#Square_roots_of_negative_and_complex_numbers
|
||
Algebraic formula
|
||
When the number is expressed using Cartesian coordinates the following formula
|
||
can be used for the principal square root:[5][6]
|
||
|
||
sqrt(x + iy) = sqrt((r + x) / 2) +/- i*sqrt((r - x) / 2)
|
||
|
||
where the sign of the imaginary part of the root is taken to be same as the sign
|
||
of the imaginary part of the original number, and
|
||
|
||
r = abs(x + iy) = sqrt(x^2 + y^2)
|
||
|
||
is the absolute value or modulus of the original number. The real part of the
|
||
principal value is always non-negative.
|
||
The other square root is simply –1 times the principal square root; in other
|
||
words, the two square roots of a number sum to 0.
|
||
*/
|