198e054b33
git-svn-id: http://skia.googlecode.com/svn/trunk@3566 2bbb7eff-a529-9590-31e7-b0007b416f81
193 lines
6.3 KiB
C++
193 lines
6.3 KiB
C++
#include "CurveIntersection.h"
|
||
#include "Intersections.h"
|
||
#include "IntersectionUtilities.h"
|
||
#include "LineIntersection.h"
|
||
|
||
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;
|
||
}
|
||
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) {
|
||
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);
|
||
Quadratic smallResult;
|
||
if (reduceOrder(smaller, smallResult) <= 2) {
|
||
Quadratic largeResult;
|
||
if (reduceOrder(larger, largeResult) <= 2) {
|
||
double smallT[2], largeT[2];
|
||
const _Line& smallLine = (const _Line&) smallResult;
|
||
const _Line& largeLine = (const _Line&) largeResult;
|
||
// FIXME: this doesn't detect or deal with coincident lines
|
||
if (!::intersect(smallLine, largeLine, smallT, largeT)) {
|
||
return false;
|
||
}
|
||
if (intersections.swapped()) {
|
||
smallT[0] = interp(minT2, maxT2, smallT[0]);
|
||
largeT[0] = interp(minT1, maxT1, largeT[0]);
|
||
} else {
|
||
smallT[0] = interp(minT1, maxT1, smallT[0]);
|
||
largeT[0] = interp(minT2, maxT2, largeT[0]);
|
||
}
|
||
intersections.add(smallT[0], largeT[0]);
|
||
return true;
|
||
}
|
||
}
|
||
double minT, maxT;
|
||
if (!bezier_clip(smaller, larger, minT, maxT)) {
|
||
if (minT == maxT) {
|
||
if (intersections.swapped()) {
|
||
minT1 = (minT1 + maxT1) / 2;
|
||
minT2 = interp(minT2, maxT2, minT);
|
||
} else {
|
||
minT1 = interp(minT1, maxT1, minT);
|
||
minT2 = (minT2 + maxT2) / 2;
|
||
}
|
||
intersections.add(minT1, minT2);
|
||
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;
|
||
#define VERBOSE 0
|
||
#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 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;
|
||
};
|
||
|
||
bool intersect(const Quadratic& q1, const Quadratic& q2, Intersections& i) {
|
||
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.
|
||
*/
|