/*
* 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 "SkIntersections.h"
#include "SkPathOpsLine.h"
#include "SkPathOpsQuad.h"
/*
Find the interection of a line and quadratic by solving for valid t values.
From http://stackoverflow.com/questions/1853637/how-to-find-the-mathematical-function-defining-a-bezier-curve
"A Bezier curve is a parametric function. A quadratic Bezier curve (i.e. three
control points) can be expressed as: F(t) = A(1 - t)^2 + B(1 - t)t + Ct^2 where
A, B and C are points and t goes from zero to one.
This will give you two equations:
x = a(1 - t)^2 + b(1 - t)t + ct^2
y = d(1 - t)^2 + e(1 - t)t + ft^2
If you add for instance the line equation (y = kx + m) to that, you'll end up
with three equations and three unknowns (x, y and t)."
Similar to above, the quadratic is represented as
x = a(1-t)^2 + 2b(1-t)t + ct^2
y = d(1-t)^2 + 2e(1-t)t + ft^2
and the line as
y = g*x + h
Using Mathematica, solve for the values of t where the quadratic intersects the
line:
(in) t1 = Resultant[a*(1 - t)^2 + 2*b*(1 - t)*t + c*t^2 - x,
d*(1 - t)^2 + 2*e*(1 - t)*t + f*t^2 - g*x - h, x]
(out) -d + h + 2 d t - 2 e t - d t^2 + 2 e t^2 - f t^2 +
g (a - 2 a t + 2 b t + a t^2 - 2 b t^2 + c t^2)
(in) Solve[t1 == 0, t]
(out) {
{t -> (-2 d + 2 e + 2 a g - 2 b g -
Sqrt[(2 d - 2 e - 2 a g + 2 b g)^2 -
4 (-d + 2 e - f + a g - 2 b g + c g) (-d + a g + h)]) /
(2 (-d + 2 e - f + a g - 2 b g + c g))
},
{t -> (-2 d + 2 e + 2 a g - 2 b g +
Sqrt[(2 d - 2 e - 2 a g + 2 b g)^2 -
4 (-d + 2 e - f + a g - 2 b g + c g) (-d + a g + h)]) /
(2 (-d + 2 e - f + a g - 2 b g + c g))
}
}
Using the results above (when the line tends towards horizontal)
A = (-(d - 2*e + f) + g*(a - 2*b + c) )
B = 2*( (d - e ) - g*(a - b ) )
C = (-(d ) + g*(a ) + h )
If g goes to infinity, we can rewrite the line in terms of x.
x = g'*y + h'
And solve accordingly in Mathematica:
(in) t2 = Resultant[a*(1 - t)^2 + 2*b*(1 - t)*t + c*t^2 - g'*y - h',
d*(1 - t)^2 + 2*e*(1 - t)*t + f*t^2 - y, y]
(out) a - h' - 2 a t + 2 b t + a t^2 - 2 b t^2 + c t^2 -
g' (d - 2 d t + 2 e t + d t^2 - 2 e t^2 + f t^2)
(in) Solve[t2 == 0, t]
(out) {
{t -> (2 a - 2 b - 2 d g' + 2 e g' -
Sqrt[(-2 a + 2 b + 2 d g' - 2 e g')^2 -
4 (a - 2 b + c - d g' + 2 e g' - f g') (a - d g' - h')]) /
(2 (a - 2 b + c - d g' + 2 e g' - f g'))
},
{t -> (2 a - 2 b - 2 d g' + 2 e g' +
Sqrt[(-2 a + 2 b + 2 d g' - 2 e g')^2 -
4 (a - 2 b + c - d g' + 2 e g' - f g') (a - d g' - h')])/
(2 (a - 2 b + c - d g' + 2 e g' - f g'))
}
}
Thus, if the slope of the line tends towards vertical, we use:
A = ( (a - 2*b + c) - g'*(d - 2*e + f) )
B = 2*(-(a - b ) + g'*(d - e ) )
C = ( (a ) - g'*(d ) - h' )
*/
class LineQuadraticIntersections {
public:
enum PinTPoint {
kPointUninitialized,
kPointInitialized
};
LineQuadraticIntersections(const SkDQuad& q, const SkDLine& l, SkIntersections* i)
: fQuad(q)
, fLine(l)
, fIntersections(i)
, fAllowNear(true) {
i->setMax(2);
}
void allowNear(bool allow) {
fAllowNear = allow;
}
int intersectRay(double roots[2]) {
/*
solve by rotating line+quad so line is horizontal, then finding the roots
set up matrix to rotate quad to x-axis
|cos(a) -sin(a)|
|sin(a) cos(a)|
note that cos(a) = A(djacent) / Hypoteneuse
sin(a) = O(pposite) / Hypoteneuse
since we are computing Ts, we can ignore hypoteneuse, the scale factor:
| A -O |
| O A |
A = line[1].fX - line[0].fX (adjacent side of the right triangle)
O = line[1].fY - line[0].fY (opposite side of the right triangle)
for each of the three points (e.g. n = 0 to 2)
quad[n].fY' = (quad[n].fY - line[0].fY) * A - (quad[n].fX - line[0].fX) * O
*/
double adj = fLine[1].fX - fLine[0].fX;
double opp = fLine[1].fY - fLine[0].fY;
double r[3];
for (int n = 0; n < 3; ++n) {
r[n] = (fQuad[n].fY - fLine[0].fY) * adj - (fQuad[n].fX - fLine[0].fX) * opp;
}
double A = r[2];
double B = r[1];
double C = r[0];
A += C - 2 * B; // A = a - 2*b + c
B -= C; // B = -(b - c)
return SkDQuad::RootsValidT(A, 2 * B, C, roots);
}
int intersect() {
addExactEndPoints();
if (fAllowNear) {
addNearEndPoints();
}
if (fIntersections->used() == 2) {
// FIXME : need sharable code that turns spans into coincident if middle point is on
} else {
double rootVals[2];
int roots = intersectRay(rootVals);
for (int index = 0; index < roots; ++index) {
double quadT = rootVals[index];
double lineT = findLineT(quadT);
SkDPoint pt;
if (pinTs(&quadT, &lineT, &pt, kPointUninitialized)) {
fIntersections->insert(quadT, lineT, pt);
}
}
}
return fIntersections->used();
}
int horizontalIntersect(double axisIntercept, double roots[2]) {
double D = fQuad[2].fY; // f
double E = fQuad[1].fY; // e
double F = fQuad[0].fY; // d
D += F - 2 * E; // D = d - 2*e + f
E -= F; // E = -(d - e)
F -= axisIntercept;
return SkDQuad::RootsValidT(D, 2 * E, F, roots);
}
int horizontalIntersect(double axisIntercept, double left, double right, bool flipped) {
addExactHorizontalEndPoints(left, right, axisIntercept);
if (fAllowNear) {
addNearHorizontalEndPoints(left, right, axisIntercept);
}
double rootVals[2];
int roots = horizontalIntersect(axisIntercept, rootVals);
for (int index = 0; index < roots; ++index) {
double quadT = rootVals[index];
SkDPoint pt = fQuad.ptAtT(quadT);
double lineT = (pt.fX - left) / (right - left);
if (pinTs(&quadT, &lineT, &pt, kPointInitialized)) {
fIntersections->insert(quadT, lineT, pt);
}
}
if (flipped) {
fIntersections->flip();
}
return fIntersections->used();
}
int verticalIntersect(double axisIntercept, double roots[2]) {
double D = fQuad[2].fX; // f
double E = fQuad[1].fX; // e
double F = fQuad[0].fX; // d
D += F - 2 * E; // D = d - 2*e + f
E -= F; // E = -(d - e)
F -= axisIntercept;
return SkDQuad::RootsValidT(D, 2 * E, F, roots);
}
int verticalIntersect(double axisIntercept, double top, double bottom, bool flipped) {
addExactVerticalEndPoints(top, bottom, axisIntercept);
if (fAllowNear) {
addNearVerticalEndPoints(top, bottom, axisIntercept);
}
double rootVals[2];
int roots = verticalIntersect(axisIntercept, rootVals);
for (int index = 0; index < roots; ++index) {
double quadT = rootVals[index];
SkDPoint pt = fQuad.ptAtT(quadT);
double lineT = (pt.fY - top) / (bottom - top);
if (pinTs(&quadT, &lineT, &pt, kPointInitialized)) {
fIntersections->insert(quadT, lineT, pt);
}
}
if (flipped) {
fIntersections->flip();
}
return fIntersections->used();
}
protected:
// add endpoints first to get zero and one t values exactly
void addExactEndPoints() {
for (int qIndex = 0; qIndex < 3; qIndex += 2) {
double lineT = fLine.exactPoint(fQuad[qIndex]);
if (lineT < 0) {
continue;
}
double quadT = (double) (qIndex >> 1);
fIntersections->insert(quadT, lineT, fQuad[qIndex]);
}
}
void addNearEndPoints() {
for (int qIndex = 0; qIndex < 3; qIndex += 2) {
double quadT = (double) (qIndex >> 1);
if (fIntersections->hasT(quadT)) {
continue;
}
double lineT = fLine.nearPoint(fQuad[qIndex]);
if (lineT < 0) {
continue;
}
fIntersections->insert(quadT, lineT, fQuad[qIndex]);
}
// FIXME: see if line end is nearly on quad
}
void addExactHorizontalEndPoints(double left, double right, double y) {
for (int qIndex = 0; qIndex < 3; qIndex += 2) {
double lineT = SkDLine::ExactPointH(fQuad[qIndex], left, right, y);
if (lineT < 0) {
continue;
}
double quadT = (double) (qIndex >> 1);
fIntersections->insert(quadT, lineT, fQuad[qIndex]);
}
}
void addNearHorizontalEndPoints(double left, double right, double y) {
for (int qIndex = 0; qIndex < 3; qIndex += 2) {
double quadT = (double) (qIndex >> 1);
if (fIntersections->hasT(quadT)) {
continue;
}
double lineT = SkDLine::NearPointH(fQuad[qIndex], left, right, y);
if (lineT < 0) {
continue;
}
fIntersections->insert(quadT, lineT, fQuad[qIndex]);
}
// FIXME: see if line end is nearly on quad
}
void addExactVerticalEndPoints(double top, double bottom, double x) {
for (int qIndex = 0; qIndex < 3; qIndex += 2) {
double lineT = SkDLine::ExactPointV(fQuad[qIndex], top, bottom, x);
if (lineT < 0) {
continue;
}
double quadT = (double) (qIndex >> 1);
fIntersections->insert(quadT, lineT, fQuad[qIndex]);
}
}
void addNearVerticalEndPoints(double top, double bottom, double x) {
for (int qIndex = 0; qIndex < 3; qIndex += 2) {
double quadT = (double) (qIndex >> 1);
if (fIntersections->hasT(quadT)) {
continue;
}
double lineT = SkDLine::NearPointV(fQuad[qIndex], top, bottom, x);
if (lineT < 0) {
continue;
}
fIntersections->insert(quadT, lineT, fQuad[qIndex]);
}
// FIXME: see if line end is nearly on quad
}
double findLineT(double t) {
SkDPoint xy = fQuad.ptAtT(t);
double dx = fLine[1].fX - fLine[0].fX;
double dy = fLine[1].fY - fLine[0].fY;
if (fabs(dx) > fabs(dy)) {
return (xy.fX - fLine[0].fX) / dx;
}
return (xy.fY - fLine[0].fY) / dy;
}
bool pinTs(double* quadT, double* lineT, SkDPoint* pt, PinTPoint ptSet) {
if (!approximately_one_or_less(*lineT)) {
return false;
}
if (!approximately_zero_or_more(*lineT)) {
return false;
}
double qT = *quadT = SkPinT(*quadT);
double lT = *lineT = SkPinT(*lineT);
if (lT == 0 || lT == 1 || (ptSet == kPointUninitialized && qT != 0 && qT != 1)) {
*pt = fLine.ptAtT(lT);
} else if (ptSet == kPointUninitialized) {
*pt = fQuad.ptAtT(qT);
}
SkPoint gridPt = pt->asSkPoint();
if (gridPt == fLine[0].asSkPoint()) {
*lineT = 0;
} else if (gridPt == fLine[1].asSkPoint()) {
*lineT = 1;
}
if (gridPt == fQuad[0].asSkPoint()) {
*quadT = 0;
} else if (gridPt == fQuad[2].asSkPoint()) {
*quadT = 1;
}
return true;
}
private:
const SkDQuad& fQuad;
const SkDLine& fLine;
SkIntersections* fIntersections;
bool fAllowNear;
};
// utility for pairs of coincident quads
static double horizontalIntersect(const SkDQuad& quad, const SkDPoint& pt) {
LineQuadraticIntersections q(quad, *(static_cast<SkDLine*>(0)),
static_cast<SkIntersections*>(0));
double rootVals[2];
int roots = q.horizontalIntersect(pt.fY, rootVals);
for (int index = 0; index < roots; ++index) {
double t = rootVals[index];
SkDPoint qPt = quad.ptAtT(t);
if (AlmostEqualUlps(qPt.fX, pt.fX)) {
return t;
}
}
return -1;
}
static double verticalIntersect(const SkDQuad& quad, const SkDPoint& pt) {
LineQuadraticIntersections q(quad, *(static_cast<SkDLine*>(0)),
static_cast<SkIntersections*>(0));
double rootVals[2];
int roots = q.verticalIntersect(pt.fX, rootVals);
for (int index = 0; index < roots; ++index) {
double t = rootVals[index];
SkDPoint qPt = quad.ptAtT(t);
if (AlmostEqualUlps(qPt.fY, pt.fY)) {
return t;
}
}
return -1;
}
double SkIntersections::Axial(const SkDQuad& q1, const SkDPoint& p, bool vertical) {
if (vertical) {
return verticalIntersect(q1, p);
}
return horizontalIntersect(q1, p);
}
int SkIntersections::horizontal(const SkDQuad& quad, double left, double right, double y,
bool flipped) {
SkDLine line = {{{ left, y }, { right, y }}};
LineQuadraticIntersections q(quad, line, this);
return q.horizontalIntersect(y, left, right, flipped);
}
int SkIntersections::vertical(const SkDQuad& quad, double top, double bottom, double x,
bool flipped) {
SkDLine line = {{{ x, top }, { x, bottom }}};
LineQuadraticIntersections q(quad, line, this);
return q.verticalIntersect(x, top, bottom, flipped);
}
int SkIntersections::intersect(const SkDQuad& quad, const SkDLine& line) {
LineQuadraticIntersections q(quad, line, this);
q.allowNear(fAllowNear);
return q.intersect();
}
int SkIntersections::intersectRay(const SkDQuad& quad, const SkDLine& line) {
LineQuadraticIntersections q(quad, line, this);
fUsed = q.intersectRay(fT[0]);
for (int index = 0; index < fUsed; ++index) {
fPt[index] = quad.ptAtT(fT[0][index]);
}
return fUsed;
}