// Another approach is to start with the implicit form of one curve and solve
// (seek implicit coefficients in QuadraticParameter.cpp
// 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
#include "CubicUtilities.h"
#include "CurveIntersection.h"
#include "Intersections.h"
#include "QuadraticParameterization.h"
#include "QuarticRoot.h"
#include "QuadraticUtilities.h"
#include "TSearch.h"
#if SK_DEBUG
#include "LineUtilities.h"
#endif
/* given the implicit form 0 = Ax^2 + Bxy + Cy^2 + Dx + Ey + F
* and given x = at^2 + bt + c (the parameterized form)
* y = dt^2 + et + f
* then
* 0 = A(at^2+bt+c)(at^2+bt+c)+B(at^2+bt+c)(dt^2+et+f)+C(dt^2+et+f)(dt^2+et+f)+D(at^2+bt+c)+E(dt^2+et+f)+F
*/
static int findRoots(const QuadImplicitForm& i, const Quadratic& q2, double roots[4],
bool oneHint, int firstCubicRoot) {
double a, b, c;
set_abc(&q2[0].x, a, b, c);
double d, e, f;
set_abc(&q2[0].y, d, e, f);
const double t4 = i.x2() * a * a
+ i.xy() * a * d
+ i.y2() * d * d;
const double t3 = 2 * i.x2() * a * b
+ i.xy() * (a * e + b * d)
+ 2 * i.y2() * d * e;
const double t2 = i.x2() * (b * b + 2 * a * c)
+ i.xy() * (c * d + b * e + a * f)
+ i.y2() * (e * e + 2 * d * f)
+ i.x() * a
+ i.y() * d;
const double t1 = 2 * i.x2() * b * c
+ i.xy() * (c * e + b * f)
+ 2 * i.y2() * e * f
+ i.x() * b
+ i.y() * e;
const double t0 = i.x2() * c * c
+ i.xy() * c * f
+ i.y2() * f * f
+ i.x() * c
+ i.y() * f
+ i.c();
int rootCount = reducedQuarticRoots(t4, t3, t2, t1, t0, oneHint, roots);
if (rootCount >= 0) {
return rootCount;
}
return quarticRootsReal(firstCubicRoot, t4, t3, t2, t1, t0, roots);
}
static int addValidRoots(const double roots[4], const int count, double valid[4]) {
int result = 0;
int index;
for (index = 0; index < count; ++index) {
if (!approximately_zero_or_more(roots[index]) || !approximately_one_or_less(roots[index])) {
continue;
}
double t = 1 - roots[index];
if (approximately_less_than_zero(t)) {
t = 0;
} else if (approximately_greater_than_one(t)) {
t = 1;
}
valid[result++] = t;
}
return result;
}
static bool onlyEndPtsInCommon(const Quadratic& q1, const Quadratic& q2, Intersections& i) {
// the idea here is to see at minimum do a quick reject by rotating all points
// to either side of the line formed by connecting the endpoints
// if the opposite curves points are on the line or on the other side, the
// curves at most intersect at the endpoints
for (int oddMan = 0; oddMan < 3; ++oddMan) {
const _Point* endPt[2];
for (int opp = 1; opp < 3; ++opp) {
int end = oddMan ^ opp;
if (end == 3) {
end = opp;
}
endPt[opp - 1] = &q1[end];
}
double origX = endPt[0]->x;
double origY = endPt[0]->y;
double adj = endPt[1]->x - origX;
double opp = endPt[1]->y - origY;
double sign = (q1[oddMan].y - origY) * adj - (q1[oddMan].x - origX) * opp;
if (approximately_zero(sign)) {
goto tryNextHalfPlane;
}
for (int n = 0; n < 3; ++n) {
double test = (q2[n].y - origY) * adj - (q2[n].x - origX) * opp;
if (test * sign > 0) {
goto tryNextHalfPlane;
}
}
for (int i1 = 0; i1 < 3; i1 += 2) {
for (int i2 = 0; i2 < 3; i2 += 2) {
if (q1[i1] == q2[i2]) {
i.insert(i1 >> 1, i2 >> 1, q1[i1]);
}
}
}
SkASSERT(i.fUsed < 3);
return true;
tryNextHalfPlane:
;
}
return false;
}
// returns false if there's more than one intercept or the intercept doesn't match the point
// returns true if the intercept was successfully added or if the
// original quads need to be subdivided
static bool addIntercept(const Quadratic& q1, const Quadratic& q2, double tMin, double tMax,
Intersections& i, bool* subDivide) {
double tMid = (tMin + tMax) / 2;
_Point mid;
xy_at_t(q2, tMid, mid.x, mid.y);
_Line line;
line[0] = line[1] = mid;
_Vector dxdy = dxdy_at_t(q2, tMid);
line[0] -= dxdy;
line[1] += dxdy;
Intersections rootTs;
int roots = intersect(q1, line, rootTs);
if (roots == 0) {
if (subDivide) {
*subDivide = true;
}
return true;
}
if (roots == 2) {
return false;
}
_Point pt2;
xy_at_t(q1, rootTs.fT[0][0], pt2.x, pt2.y);
if (!pt2.approximatelyEqualHalf(mid)) {
return false;
}
i.insertSwap(rootTs.fT[0][0], tMid, pt2);
return true;
}
static bool isLinearInner(const Quadratic& q1, double t1s, double t1e, const Quadratic& q2,
double t2s, double t2e, Intersections& i, bool* subDivide) {
Quadratic hull;
sub_divide(q1, t1s, t1e, hull);
_Line line = {hull[2], hull[0]};
const _Line* testLines[] = { &line, (const _Line*) &hull[0], (const _Line*) &hull[1] };
size_t testCount = sizeof(testLines) / sizeof(testLines[0]);
SkTDArray<double> tsFound;
for (size_t index = 0; index < testCount; ++index) {
Intersections rootTs;
int roots = intersect(q2, *testLines[index], rootTs);
for (int idx2 = 0; idx2 < roots; ++idx2) {
double t = rootTs.fT[0][idx2];
#if SK_DEBUG
_Point qPt, lPt;
xy_at_t(q2, t, qPt.x, qPt.y);
xy_at_t(*testLines[index], rootTs.fT[1][idx2], lPt.x, lPt.y);
SkASSERT(qPt.approximatelyEqual(lPt));
#endif
if (approximately_negative(t - t2s) || approximately_positive(t - t2e)) {
continue;
}
*tsFound.append() = rootTs.fT[0][idx2];
}
}
int tCount = tsFound.count();
if (!tCount) {
return true;
}
double tMin, tMax;
if (tCount == 1) {
tMin = tMax = tsFound[0];
} else if (tCount > 1) {
QSort<double>(tsFound.begin(), tsFound.end() - 1);
tMin = tsFound[0];
tMax = tsFound[tsFound.count() - 1];
}
_Point end;
xy_at_t(q2, t2s, end.x, end.y);
bool startInTriangle = point_in_hull(hull, end);
if (startInTriangle) {
tMin = t2s;
}
xy_at_t(q2, t2e, end.x, end.y);
bool endInTriangle = point_in_hull(hull, end);
if (endInTriangle) {
tMax = t2e;
}
int split = 0;
_Vector dxy1, dxy2;
if (tMin != tMax || tCount > 2) {
dxy2 = dxdy_at_t(q2, tMin);
for (int index = 1; index < tCount; ++index) {
dxy1 = dxy2;
dxy2 = dxdy_at_t(q2, tsFound[index]);
double dot = dxy1.dot(dxy2);
if (dot < 0) {
split = index - 1;
break;
}
}
}
if (split == 0) { // there's one point
if (addIntercept(q1, q2, tMin, tMax, i, subDivide)) {
return true;
}
i.swap();
return isLinearInner(q2, tMin, tMax, q1, t1s, t1e, i, subDivide);
}
// At this point, we have two ranges of t values -- treat each separately at the split
bool result;
if (addIntercept(q1, q2, tMin, tsFound[split - 1], i, subDivide)) {
result = true;
} else {
i.swap();
result = isLinearInner(q2, tMin, tsFound[split - 1], q1, t1s, t1e, i, subDivide);
}
if (addIntercept(q1, q2, tsFound[split], tMax, i, subDivide)) {
result = true;
} else {
i.swap();
result |= isLinearInner(q2, tsFound[split], tMax, q1, t1s, t1e, i, subDivide);
}
return result;
}
static double flatMeasure(const Quadratic& q) {
_Vector mid = q[1] - q[0];
_Vector dxy = q[2] - q[0];
double length = dxy.length(); // OPTIMIZE: get rid of sqrt
return fabs(mid.cross(dxy) / length);
}
// FIXME ? should this measure both and then use the quad that is the flattest as the line?
static bool isLinear(const Quadratic& q1, const Quadratic& q2, Intersections& i) {
double measure = flatMeasure(q1);
// OPTIMIZE: (get rid of sqrt) use approximately_zero
if (!approximately_zero_sqrt(measure)) {
return false;
}
return isLinearInner(q1, 0, 1, q2, 0, 1, i, NULL);
}
// FIXME: if flat measure is sufficiently large, then probably the quartic solution failed
static void relaxedIsLinear(const Quadratic& q1, const Quadratic& q2, Intersections& i) {
double m1 = flatMeasure(q1);
double m2 = flatMeasure(q2);
#if SK_DEBUG
double min = SkTMin(m1, m2);
if (min > 5) {
SkDebugf("%s maybe not flat enough.. %1.9g\n", __FUNCTION__, min);
}
#endif
i.reset();
const Quadratic& rounder = m2 < m1 ? q1 : q2;
const Quadratic& flatter = m2 < m1 ? q2 : q1;
bool subDivide = false;
isLinearInner(flatter, 0, 1, rounder, 0, 1, i, &subDivide);
if (subDivide) {
QuadraticPair pair;
chop_at(flatter, pair, 0.5);
Intersections firstI, secondI;
relaxedIsLinear(pair.first(), rounder, firstI);
for (int index = 0; index < firstI.used(); ++index) {
i.insert(firstI.fT[0][index] * 0.5, firstI.fT[1][index], firstI.fPt[index]);
}
relaxedIsLinear(pair.second(), rounder, secondI);
for (int index = 0; index < secondI.used(); ++index) {
i.insert(0.5 + secondI.fT[0][index] * 0.5, secondI.fT[1][index], secondI.fPt[index]);
}
}
if (m2 < m1) {
i.swapPts();
}
}
#if 0
static void unsortableExpanse(const Quadratic& q1, const Quadratic& q2, Intersections& i) {
const Quadratic* qs[2] = { &q1, &q2 };
// need t values for start and end of unsortable expanse on both curves
// try projecting lines parallel to the end points
i.fT[0][0] = 0;
i.fT[0][1] = 1;
int flip = -1; // undecided
for (int qIdx = 0; qIdx < 2; qIdx++) {
for (int t = 0; t < 2; t++) {
_Point dxdy;
dxdy_at_t(*qs[qIdx], t, dxdy);
_Line perp;
perp[0] = perp[1] = (*qs[qIdx])[t == 0 ? 0 : 2];
perp[0].x += dxdy.y;
perp[0].y -= dxdy.x;
perp[1].x -= dxdy.y;
perp[1].y += dxdy.x;
Intersections hitData;
int hits = intersectRay(*qs[qIdx ^ 1], perp, hitData);
SkASSERT(hits <= 1);
if (hits) {
if (flip < 0) {
_Point dxdy2;
dxdy_at_t(*qs[qIdx ^ 1], hitData.fT[0][0], dxdy2);
double dot = dxdy.dot(dxdy2);
flip = dot < 0;
i.fT[1][0] = flip;
i.fT[1][1] = !flip;
}
i.fT[qIdx ^ 1][t ^ flip] = hitData.fT[0][0];
}
}
}
i.fUnsortable = true; // failed, probably coincident or near-coincident
i.fUsed = 2;
}
#endif
// each time through the loop, this computes values it had from the last loop
// if i == j == 1, the center values are still good
// otherwise, for i != 1 or j != 1, four of the values are still good
// and if i == 1 ^ j == 1, an additional value is good
static bool binarySearch(const Quadratic& quad1, const Quadratic& quad2, double& t1Seed,
double& t2Seed, _Point& pt) {
double tStep = ROUGH_EPSILON;
_Point t1[3], t2[3];
int calcMask = ~0;
do {
if (calcMask & (1 << 1)) t1[1] = xy_at_t(quad1, t1Seed);
if (calcMask & (1 << 4)) t2[1] = xy_at_t(quad2, t2Seed);
if (t1[1].approximatelyEqual(t2[1])) {
pt = t1[1];
#if ONE_OFF_DEBUG
SkDebugf("%s t1=%1.9g t2=%1.9g (%1.9g,%1.9g) == (%1.9g,%1.9g)\n", __FUNCTION__,
t1Seed, t2Seed, t1[1].x, t1[1].y, t1[2].x, t1[2].y);
#endif
return true;
}
if (calcMask & (1 << 0)) t1[0] = xy_at_t(quad1, t1Seed - tStep);
if (calcMask & (1 << 2)) t1[2] = xy_at_t(quad1, t1Seed + tStep);
if (calcMask & (1 << 3)) t2[0] = xy_at_t(quad2, t2Seed - tStep);
if (calcMask & (1 << 5)) t2[2] = xy_at_t(quad2, t2Seed + tStep);
double dist[3][3];
// OPTIMIZE: using calcMask value permits skipping some distance calcuations
// if prior loop's results are moved to correct slot for reuse
dist[1][1] = t1[1].distanceSquared(t2[1]);
int best_i = 1, best_j = 1;
for (int i = 0; i < 3; ++i) {
for (int j = 0; j < 3; ++j) {
if (i == 1 && j == 1) {
continue;
}
dist[i][j] = t1[i].distanceSquared(t2[j]);
if (dist[best_i][best_j] > dist[i][j]) {
best_i = i;
best_j = j;
}
}
}
if (best_i == 1 && best_j == 1) {
tStep /= 2;
if (tStep < FLT_EPSILON_HALF) {
break;
}
calcMask = (1 << 0) | (1 << 2) | (1 << 3) | (1 << 5);
continue;
}
if (best_i == 0) {
t1Seed -= tStep;
t1[2] = t1[1];
t1[1] = t1[0];
calcMask = 1 << 0;
} else if (best_i == 2) {
t1Seed += tStep;
t1[0] = t1[1];
t1[1] = t1[2];
calcMask = 1 << 2;
} else {
calcMask = 0;
}
if (best_j == 0) {
t2Seed -= tStep;
t2[2] = t2[1];
t2[1] = t2[0];
calcMask |= 1 << 3;
} else if (best_j == 2) {
t2Seed += tStep;
t2[0] = t2[1];
t2[1] = t2[2];
calcMask |= 1 << 5;
}
} while (true);
#if ONE_OFF_DEBUG
SkDebugf("%s t1=%1.9g t2=%1.9g (%1.9g,%1.9g) != (%1.9g,%1.9g) %s\n", __FUNCTION__,
t1Seed, t2Seed, t1[1].x, t1[1].y, t1[2].x, t1[2].y);
#endif
return false;
}
bool intersect2(const Quadratic& q1, const Quadratic& q2, Intersections& i) {
// if the quads share an end point, check to see if they overlap
if (onlyEndPtsInCommon(q1, q2, i)) {
return i.intersected();
}
if (onlyEndPtsInCommon(q2, q1, i)) {
i.swapPts();
return i.intersected();
}
// see if either quad is really a line
if (isLinear(q1, q2, i)) {
return i.intersected();
}
if (isLinear(q2, q1, i)) {
i.swapPts();
return i.intersected();
}
QuadImplicitForm i1(q1);
QuadImplicitForm i2(q2);
if (i1.implicit_match(i2)) {
// 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.insertCoincident(t, 0, q2[0]);
}
if ((t = axialIntersect(q1, q2[2], useVertical)) >= 0) {
i.insertCoincident(t, 1, q2[2]);
}
useVertical = fabs(q2[0].x - q2[2].x) < fabs(q2[0].y - q2[2].y);
if ((t = axialIntersect(q2, q1[0], useVertical)) >= 0) {
i.insertCoincident(0, t, q1[0]);
}
if ((t = axialIntersect(q2, q1[2], useVertical)) >= 0) {
i.insertCoincident(1, t, q1[2]);
}
SkASSERT(i.coincidentUsed() <= 2);
return i.coincidentUsed() > 0;
}
int index;
bool useCubic = q1[0] == q2[0] || q1[0] == q2[2] || q1[2] == q2[0];
double roots1[4];
int rootCount = findRoots(i2, q1, roots1, useCubic, 0);
// OPTIMIZATION: could short circuit here if all roots are < 0 or > 1
double roots1Copy[4];
int r1Count = addValidRoots(roots1, rootCount, roots1Copy);
_Point pts1[4];
for (index = 0; index < r1Count; ++index) {
xy_at_t(q1, roots1Copy[index], pts1[index].x, pts1[index].y);
}
double roots2[4];
int rootCount2 = findRoots(i1, q2, roots2, useCubic, 0);
double roots2Copy[4];
int r2Count = addValidRoots(roots2, rootCount2, roots2Copy);
_Point pts2[4];
for (index = 0; index < r2Count; ++index) {
xy_at_t(q2, roots2Copy[index], pts2[index].x, pts2[index].y);
}
if (r1Count == r2Count && r1Count <= 1) {
if (r1Count == 1) {
if (pts1[0].approximatelyEqualHalf(pts2[0])) {
i.insert(roots1Copy[0], roots2Copy[0], pts1[0]);
} else if (pts1[0].moreRoughlyEqual(pts2[0])) {
// experiment: see if a different cubic solution provides the correct quartic answer
#if 0
for (int cu1 = 0; cu1 < 3; ++cu1) {
rootCount = findRoots(i2, q1, roots1, useCubic, cu1);
r1Count = addValidRoots(roots1, rootCount, roots1Copy);
if (r1Count == 0) {
continue;
}
for (int cu2 = 0; cu2 < 3; ++cu2) {
if (cu1 == 0 && cu2 == 0) {
continue;
}
rootCount2 = findRoots(i1, q2, roots2, useCubic, cu2);
r2Count = addValidRoots(roots2, rootCount2, roots2Copy);
if (r2Count == 0) {
continue;
}
SkASSERT(r1Count == 1 && r2Count == 1);
SkDebugf("*** [%d,%d] (%1.9g,%1.9g) %s (%1.9g,%1.9g)\n", cu1, cu2,
pts1[0].x, pts1[0].y, pts1[0].approximatelyEqualHalf(pts2[0])
? "==" : "!=", pts2[0].x, pts2[0].y);
}
}
#endif
// experiment: try to find intersection by chasing t
rootCount = findRoots(i2, q1, roots1, useCubic, 0);
r1Count = addValidRoots(roots1, rootCount, roots1Copy);
rootCount2 = findRoots(i1, q2, roots2, useCubic, 0);
r2Count = addValidRoots(roots2, rootCount2, roots2Copy);
if (binarySearch(q1, q2, roots1Copy[0], roots2Copy[0], pts1[0])) {
i.insert(roots1Copy[0], roots2Copy[0], pts1[0]);
}
}
}
return i.intersected();
}
int closest[4];
double dist[4];
bool foundSomething = false;
for (index = 0; index < r1Count; ++index) {
dist[index] = DBL_MAX;
closest[index] = -1;
for (int ndex2 = 0; ndex2 < r2Count; ++ndex2) {
if (!pts2[ndex2].approximatelyEqualHalf(pts1[index])) {
continue;
}
double dx = pts2[ndex2].x - pts1[index].x;
double dy = pts2[ndex2].y - pts1[index].y;
double distance = dx * dx + dy * dy;
if (dist[index] <= distance) {
continue;
}
for (int outer = 0; outer < index; ++outer) {
if (closest[outer] != ndex2) {
continue;
}
if (dist[outer] < distance) {
goto next;
}
closest[outer] = -1;
}
dist[index] = distance;
closest[index] = ndex2;
foundSomething = true;
next:
;
}
}
if (r1Count && r2Count && !foundSomething) {
relaxedIsLinear(q1, q2, i);
return i.intersected();
}
int used = 0;
do {
double lowest = DBL_MAX;
int lowestIndex = -1;
for (index = 0; index < r1Count; ++index) {
if (closest[index] < 0) {
continue;
}
if (roots1Copy[index] < lowest) {
lowestIndex = index;
lowest = roots1Copy[index];
}
}
if (lowestIndex < 0) {
break;
}
i.insert(roots1Copy[lowestIndex], roots2Copy[closest[lowestIndex]],
pts1[lowestIndex]);
closest[lowestIndex] = -1;
} while (++used < r1Count);
i.fFlip = false;
return i.intersected();
}