/*
* 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 "CubicUtilities.h"
#include "Intersections.h"
#include "LineUtilities.h"
/*
Find the interection of a line and cubic by solving for valid t values.
Analogous to line-quadratic intersection, solve line-cubic intersection by
representing the cubic as:
x = a(1-t)^3 + 2b(1-t)^2t + c(1-t)t^2 + dt^3
y = e(1-t)^3 + 2f(1-t)^2t + g(1-t)t^2 + ht^3
and the line as:
y = i*x + j (if the line is more horizontal)
or:
x = i*y + j (if the line is more vertical)
Then using Mathematica, solve for the values of t where the cubic intersects the
line:
(in) Resultant[
a*(1 - t)^3 + 3*b*(1 - t)^2*t + 3*c*(1 - t)*t^2 + d*t^3 - x,
e*(1 - t)^3 + 3*f*(1 - t)^2*t + 3*g*(1 - t)*t^2 + h*t^3 - i*x - j, x]
(out) -e + j +
3 e t - 3 f t -
3 e t^2 + 6 f t^2 - 3 g t^2 +
e t^3 - 3 f t^3 + 3 g t^3 - h t^3 +
i ( a -
3 a t + 3 b t +
3 a t^2 - 6 b t^2 + 3 c t^2 -
a t^3 + 3 b t^3 - 3 c t^3 + d t^3 )
if i goes to infinity, we can rewrite the line in terms of x. Mathematica:
(in) Resultant[
a*(1 - t)^3 + 3*b*(1 - t)^2*t + 3*c*(1 - t)*t^2 + d*t^3 - i*y - j,
e*(1 - t)^3 + 3*f*(1 - t)^2*t + 3*g*(1 - t)*t^2 + h*t^3 - y, y]
(out) a - j -
3 a t + 3 b t +
3 a t^2 - 6 b t^2 + 3 c t^2 -
a t^3 + 3 b t^3 - 3 c t^3 + d t^3 -
i ( e -
3 e t + 3 f t +
3 e t^2 - 6 f t^2 + 3 g t^2 -
e t^3 + 3 f t^3 - 3 g t^3 + h t^3 )
Solving this with Mathematica produces an expression with hundreds of terms;
instead, use Numeric Solutions recipe to solve the cubic.
The near-horizontal case, in terms of: Ax^3 + Bx^2 + Cx + D == 0
A = (-(-e + 3*f - 3*g + h) + i*(-a + 3*b - 3*c + d) )
B = 3*(-( e - 2*f + g ) + i*( a - 2*b + c ) )
C = 3*(-(-e + f ) + i*(-a + b ) )
D = (-( e ) + i*( a ) + j )
The near-vertical case, in terms of: Ax^3 + Bx^2 + Cx + D == 0
A = ( (-a + 3*b - 3*c + d) - i*(-e + 3*f - 3*g + h) )
B = 3*( ( a - 2*b + c ) - i*( e - 2*f + g ) )
C = 3*( (-a + b ) - i*(-e + f ) )
D = ( ( a ) - i*( e ) - j )
For horizontal lines:
(in) Resultant[
a*(1 - t)^3 + 3*b*(1 - t)^2*t + 3*c*(1 - t)*t^2 + d*t^3 - j,
e*(1 - t)^3 + 3*f*(1 - t)^2*t + 3*g*(1 - t)*t^2 + h*t^3 - y, y]
(out) e - j -
3 e t + 3 f t +
3 e t^2 - 6 f t^2 + 3 g t^2 -
e t^3 + 3 f t^3 - 3 g t^3 + h t^3
So the cubic coefficients are:
*/
class LineCubicIntersections {
public:
LineCubicIntersections(const Cubic& c, const _Line& l, Intersections& i)
: cubic(c)
, line(l)
, intersections(i) {
}
// see parallel routine in line quadratic intersections
int intersectRay(double roots[3]) {
double adj = line[1].x - line[0].x;
double opp = line[1].y - line[0].y;
Cubic r;
for (int n = 0; n < 4; ++n) {
r[n].x = (cubic[n].y - line[0].y) * adj - (cubic[n].x - line[0].x) * opp;
}
double A, B, C, D;
coefficients(&r[0].x, A, B, C, D);
return cubicRootsValidT(A, B, C, D, roots);
}
int intersect() {
addEndPoints();
double rootVals[3];
int roots = intersectRay(rootVals);
for (int index = 0; index < roots; ++index) {
double cubicT = rootVals[index];
double lineT = findLineT(cubicT);
if (pinTs(cubicT, lineT)) {
_Point pt;
xy_at_t(line, lineT, pt.x, pt.y);
intersections.insert(cubicT, lineT, pt);
}
}
return intersections.fUsed;
}
int horizontalIntersect(double axisIntercept, double roots[3]) {
double A, B, C, D;
coefficients(&cubic[0].y, A, B, C, D);
D -= axisIntercept;
return cubicRootsValidT(A, B, C, D, roots);
}
int horizontalIntersect(double axisIntercept, double left, double right, bool flipped) {
addHorizontalEndPoints(left, right, axisIntercept);
double rootVals[3];
int roots = horizontalIntersect(axisIntercept, rootVals);
for (int index = 0; index < roots; ++index) {
_Point pt;
double cubicT = rootVals[index];
xy_at_t(cubic, cubicT, pt.x, pt.y);
double lineT = (pt.x - left) / (right - left);
if (pinTs(cubicT, lineT)) {
intersections.insert(cubicT, lineT, pt);
}
}
if (flipped) {
flip();
}
return intersections.fUsed;
}
int verticalIntersect(double axisIntercept, double roots[3]) {
double A, B, C, D;
coefficients(&cubic[0].x, A, B, C, D);
D -= axisIntercept;
return cubicRootsValidT(A, B, C, D, roots);
}
int verticalIntersect(double axisIntercept, double top, double bottom, bool flipped) {
addVerticalEndPoints(top, bottom, axisIntercept);
double rootVals[3];
int roots = verticalIntersect(axisIntercept, rootVals);
for (int index = 0; index < roots; ++index) {
_Point pt;
double cubicT = rootVals[index];
xy_at_t(cubic, cubicT, pt.x, pt.y);
double lineT = (pt.y - top) / (bottom - top);
if (pinTs(cubicT, lineT)) {
intersections.insert(cubicT, lineT, pt);
}
}
if (flipped) {
flip();
}
return intersections.fUsed;
}
protected:
void addEndPoints()
{
for (int cIndex = 0; cIndex < 4; cIndex += 3) {
for (int lIndex = 0; lIndex < 2; lIndex++) {
if (cubic[cIndex] == line[lIndex]) {
intersections.insert(cIndex >> 1, lIndex, line[lIndex]);
}
}
}
}
void addHorizontalEndPoints(double left, double right, double y)
{
for (int cIndex = 0; cIndex < 4; cIndex += 3) {
if (cubic[cIndex].y != y) {
continue;
}
if (cubic[cIndex].x == left) {
intersections.insert(cIndex >> 1, 0, cubic[cIndex]);
}
if (cubic[cIndex].x == right) {
intersections.insert(cIndex >> 1, 1, cubic[cIndex]);
}
}
}
void addVerticalEndPoints(double top, double bottom, double x)
{
for (int cIndex = 0; cIndex < 4; cIndex += 3) {
if (cubic[cIndex].x != x) {
continue;
}
if (cubic[cIndex].y == top) {
intersections.insert(cIndex >> 1, 0, cubic[cIndex]);
}
if (cubic[cIndex].y == bottom) {
intersections.insert(cIndex >> 1, 1, cubic[cIndex]);
}
}
}
double findLineT(double t) {
double x, y;
xy_at_t(cubic, t, x, y);
double dx = line[1].x - line[0].x;
double dy = line[1].y - line[0].y;
if (fabs(dx) > fabs(dy)) {
return (x - line[0].x) / dx;
}
return (y - line[0].y) / dy;
}
void flip() {
// OPTIMIZATION: instead of swapping, pass original line, use [1].y - [0].y
int roots = intersections.fUsed;
for (int index = 0; index < roots; ++index) {
intersections.fT[1][index] = 1 - intersections.fT[1][index];
}
}
static bool pinTs(double& cubicT, double& lineT) {
if (!approximately_one_or_less(lineT)) {
return false;
}
if (!approximately_zero_or_more(lineT)) {
return false;
}
if (precisely_less_than_zero(cubicT)) {
cubicT = 0;
} else if (precisely_greater_than_one(cubicT)) {
cubicT = 1;
}
if (precisely_less_than_zero(lineT)) {
lineT = 0;
} else if (precisely_greater_than_one(lineT)) {
lineT = 1;
}
return true;
}
private:
const Cubic& cubic;
const _Line& line;
Intersections& intersections;
};
int horizontalIntersect(const Cubic& cubic, double left, double right, double y,
double tRange[3]) {
LineCubicIntersections c(cubic, *((_Line*) 0), *((Intersections*) 0));
double rootVals[3];
int result = c.horizontalIntersect(y, rootVals);
int tCount = 0;
for (int index = 0; index < result; ++index) {
double x, y;
xy_at_t(cubic, rootVals[index], x, y);
if (x < left || x > right) {
continue;
}
tRange[tCount++] = rootVals[index];
}
return result;
}
int horizontalIntersect(const Cubic& cubic, double left, double right, double y,
bool flipped, Intersections& intersections) {
LineCubicIntersections c(cubic, *((_Line*) 0), intersections);
return c.horizontalIntersect(y, left, right, flipped);
}
int verticalIntersect(const Cubic& cubic, double top, double bottom, double x,
bool flipped, Intersections& intersections) {
LineCubicIntersections c(cubic, *((_Line*) 0), intersections);
return c.verticalIntersect(x, top, bottom, flipped);
}
int intersect(const Cubic& cubic, const _Line& line, Intersections& i) {
LineCubicIntersections c(cubic, line, i);
return c.intersect();
}
int intersectRay(const Cubic& cubic, const _Line& line, Intersections& i) {
LineCubicIntersections c(cubic, line, i);
return c.intersectRay(i.fT[0]);
}