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// 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 "SkDQuadImplicit.h"
#include "SkIntersections.h"
#include "SkPathOpsLine.h"
#include "SkQuarticRoot.h"
#include "SkTArray.h"
#include "SkTSort.h"
/* 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 SkDQuadImplicit& i, const SkDQuad& quad, double roots[4],
bool oneHint, bool flip, int firstCubicRoot) {
SkDQuad flipped;
const SkDQuad& q = flip ? (flipped = quad.flip()) : quad;
double a, b, c;
SkDQuad::SetABC(&q[0].fX, &a, &b, &c);
double d, e, f;
SkDQuad::SetABC(&q[0].fY, &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 = SkReducedQuarticRoots(t4, t3, t2, t1, t0, oneHint, roots);
if (rootCount < 0) {
rootCount = SkQuarticRootsReal(firstCubicRoot, t4, t3, t2, t1, t0, roots);
}
if (flip) {
for (int index = 0; index < rootCount; ++index) {
roots[index] = 1 - roots[index];
}
}
return rootCount;
}
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 only_end_pts_in_common(const SkDQuad& q1, const SkDQuad& q2) {
// 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 SkDPoint* endPt[2];
for (int opp = 1; opp < 3; ++opp) {
int end = oddMan ^ opp; // choose a value not equal to oddMan
if (3 == end) { // and correct so that largest value is 1 or 2
end = opp;
}
endPt[opp - 1] = &q1[end];
}
double origX = endPt[0]->fX;
double origY = endPt[0]->fY;
double adj = endPt[1]->fX - origX;
double opp = endPt[1]->fY - origY;
double sign = (q1[oddMan].fY - origY) * adj - (q1[oddMan].fX - origX) * opp;
if (approximately_zero(sign)) {
goto tryNextHalfPlane;
}
for (int n = 0; n < 3; ++n) {
double test = (q2[n].fY - origY) * adj - (q2[n].fX - origX) * opp;
if (test * sign > 0 && !precisely_zero(test)) {
goto tryNextHalfPlane;
}
}
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 add_intercept(const SkDQuad& q1, const SkDQuad& q2, double tMin, double tMax,
SkIntersections* i, bool* subDivide) {
double tMid = (tMin + tMax) / 2;
SkDPoint mid = q2.ptAtT(tMid);
SkDLine line;
line[0] = line[1] = mid;
SkDVector dxdy = q2.dxdyAtT(tMid);
line[0] -= dxdy;
line[1] += dxdy;
SkIntersections rootTs;
rootTs.allowNear(false);
int roots = rootTs.intersect(q1, line);
if (roots == 0) {
if (subDivide) {
*subDivide = true;
}
return true;
}
if (roots == 2) {
return false;
}
SkDPoint pt2 = q1.ptAtT(rootTs[0][0]);
if (!pt2.approximatelyEqualHalf(mid)) {
return false;
}
i->insertSwap(rootTs[0][0], tMid, pt2);
return true;
}
static bool is_linear_inner(const SkDQuad& q1, double t1s, double t1e, const SkDQuad& q2,
double t2s, double t2e, SkIntersections* i, bool* subDivide) {
SkDQuad hull = q1.subDivide(t1s, t1e);
SkDLine line = {{hull[2], hull[0]}};
const SkDLine* testLines[] = { &line, (const SkDLine*) &hull[0], (const SkDLine*) &hull[1] };
const size_t kTestCount = SK_ARRAY_COUNT(testLines);
SkSTArray<kTestCount * 2, double, true> tsFound;
for (size_t index = 0; index < kTestCount; ++index) {
SkIntersections rootTs;
rootTs.allowNear(false);
int roots = rootTs.intersect(q2, *testLines[index]);
for (int idx2 = 0; idx2 < roots; ++idx2) {
double t = rootTs[0][idx2];
#ifdef SK_DEBUG
SkDPoint qPt = q2.ptAtT(t);
SkDPoint lPt = testLines[index]->ptAtT(rootTs[1][idx2]);
SkASSERT(qPt.approximatelyEqual(lPt));
#endif
if (approximately_negative(t - t2s) || approximately_positive(t - t2e)) {
continue;
}
tsFound.push_back(rootTs[0][idx2]);
}
}
int tCount = tsFound.count();
if (tCount <= 0) {
return true;
}
double tMin, tMax;
if (tCount == 1) {
tMin = tMax = tsFound[0];
} else {
SkASSERT(tCount > 1);
SkTQSort<double>(tsFound.begin(), tsFound.end() - 1);
tMin = tsFound[0];
tMax = tsFound[tsFound.count() - 1];
}
SkDPoint end = q2.ptAtT(t2s);
bool startInTriangle = hull.pointInHull(end);
if (startInTriangle) {
tMin = t2s;
}
end = q2.ptAtT(t2e);
bool endInTriangle = hull.pointInHull(end);
if (endInTriangle) {
tMax = t2e;
}
int split = 0;
SkDVector dxy1, dxy2;
if (tMin != tMax || tCount > 2) {
dxy2 = q2.dxdyAtT(tMin);
for (int index = 1; index < tCount; ++index) {
dxy1 = dxy2;
dxy2 = q2.dxdyAtT(tsFound[index]);
double dot = dxy1.dot(dxy2);
if (dot < 0) {
split = index - 1;
break;
}
}
}
if (split == 0) { // there's one point
if (add_intercept(q1, q2, tMin, tMax, i, subDivide)) {
return true;
}
i->swap();
return is_linear_inner(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 (add_intercept(q1, q2, tMin, tsFound[split - 1], i, subDivide)) {
result = true;
} else {
i->swap();
result = is_linear_inner(q2, tMin, tsFound[split - 1], q1, t1s, t1e, i, subDivide);
}
if (add_intercept(q1, q2, tsFound[split], tMax, i, subDivide)) {
result = true;
} else {
i->swap();
result |= is_linear_inner(q2, tsFound[split], tMax, q1, t1s, t1e, i, subDivide);
}
return result;
}
static double flat_measure(const SkDQuad& q) {
SkDVector mid = q[1] - q[0];
SkDVector 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 is_linear(const SkDQuad& q1, const SkDQuad& q2, SkIntersections* i) {
double measure = flat_measure(q1);
// OPTIMIZE: (get rid of sqrt) use approximately_zero
if (!approximately_zero_sqrt(measure)) {
return false;
}
return is_linear_inner(q1, 0, 1, q2, 0, 1, i, NULL);
}
// FIXME: if flat measure is sufficiently large, then probably the quartic solution failed
static void relaxed_is_linear(const SkDQuad& q1, const SkDQuad& q2, SkIntersections* i) {
double m1 = flat_measure(q1);
double m2 = flat_measure(q2);
#if DEBUG_FLAT_QUADS
double min = SkTMin(m1, m2);
if (min > 5) {
SkDebugf("%s maybe not flat enough.. %1.9g\n", __FUNCTION__, min);
}
#endif
i->reset();
const SkDQuad& rounder = m2 < m1 ? q1 : q2;
const SkDQuad& flatter = m2 < m1 ? q2 : q1;
bool subDivide = false;
is_linear_inner(flatter, 0, 1, rounder, 0, 1, i, &subDivide);
if (subDivide) {
SkDQuadPair pair = flatter.chopAt(0.5);
SkIntersections firstI, secondI;
relaxed_is_linear(pair.first(), rounder, &firstI);
for (int index = 0; index < firstI.used(); ++index) {
i->insert(firstI[0][index] * 0.5, firstI[1][index], firstI.pt(index));
}
relaxed_is_linear(pair.second(), rounder, &secondI);
for (int index = 0; index < secondI.used(); ++index) {
i->insert(0.5 + secondI[0][index] * 0.5, secondI[1][index], secondI.pt(index));
}
}
if (m2 < m1) {
i->swapPts();
}
}
// 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 binary_search(const SkDQuad& quad1, const SkDQuad& quad2, double* t1Seed,
double* t2Seed, SkDPoint* pt) {
double tStep = ROUGH_EPSILON;
SkDPoint t1[3], t2[3];
int calcMask = ~0;
do {
if (calcMask & (1 << 1)) t1[1] = quad1.ptAtT(*t1Seed);
if (calcMask & (1 << 4)) t2[1] = quad2.ptAtT(*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].fX, t1[1].fY, t1[2].fX, t1[2].fY);
#endif
return true;
}
if (calcMask & (1 << 0)) t1[0] = quad1.ptAtT(*t1Seed - tStep);
if (calcMask & (1 << 2)) t1[2] = quad1.ptAtT(*t1Seed + tStep);
if (calcMask & (1 << 3)) t2[0] = quad2.ptAtT(*t2Seed - tStep);
if (calcMask & (1 << 5)) t2[2] = quad2.ptAtT(*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].fX, t1[1].fY, t1[2].fX, t1[2].fY);
#endif
return false;
}
static void lookNearEnd(const SkDQuad& q1, const SkDQuad& q2, int testT,
const SkIntersections& orig, bool swap, SkIntersections* i) {
if (orig.used() == 1 && orig[!swap][0] == testT) {
return;
}
if (orig.used() == 2 && orig[!swap][1] == testT) {
return;
}
SkDLine tmpLine;
int testTIndex = testT << 1;
tmpLine[0] = tmpLine[1] = q2[testTIndex];
tmpLine[1].fX += q2[1].fY - q2[testTIndex].fY;
tmpLine[1].fY -= q2[1].fX - q2[testTIndex].fX;
SkIntersections impTs;
impTs.intersectRay(q1, tmpLine);
for (int index = 0; index < impTs.used(); ++index) {
SkDPoint realPt = impTs.pt(index);
if (!tmpLine[0].approximatelyEqualHalf(realPt)) {
continue;
}
if (swap) {
i->insert(testT, impTs[0][index], tmpLine[0]);
} else {
i->insert(impTs[0][index], testT, tmpLine[0]);
}
}
}
int SkIntersections::intersect(const SkDQuad& q1, const SkDQuad& q2) {
// if the quads share an end point, check to see if they overlap
for (int i1 = 0; i1 < 3; i1 += 2) {
for (int i2 = 0; i2 < 3; i2 += 2) {
if (q1[i1].approximatelyEqualHalf(q2[i2])) {
insert(i1 >> 1, i2 >> 1, q1[i1]);
}
}
}
SkASSERT(fUsed < 3);
if (only_end_pts_in_common(q1, q2)) {
return fUsed;
}
if (only_end_pts_in_common(q2, q1)) {
return fUsed;
}
// see if either quad is really a line
// FIXME: figure out why reduce step didn't find this earlier
if (is_linear(q1, q2, this)) {
return fUsed;
}
SkIntersections swapped;
if (is_linear(q2, q1, &swapped)) {
swapped.swapPts();
set(swapped);
return fUsed;
}
SkIntersections copyI(*this);
lookNearEnd(q1, q2, 0, *this, false, &copyI);
lookNearEnd(q1, q2, 1, *this, false, &copyI);
lookNearEnd(q2, q1, 0, *this, true, &copyI);
lookNearEnd(q2, q1, 1, *this, true, &copyI);
int innerEqual = 0;
if (copyI.fUsed >= 2) {
SkASSERT(copyI.fUsed <= 4);
double width = copyI[0][1] - copyI[0][0];
int midEnd = 1;
for (int index = 2; index < copyI.fUsed; ++index) {
double testWidth = copyI[0][index] - copyI[0][index - 1];
if (testWidth <= width) {
continue;
}
midEnd = index;
}
for (int index = 0; index < 2; ++index) {
double testT = (copyI[0][midEnd] * (index + 1)
+ copyI[0][midEnd - 1] * (2 - index)) / 3;
SkDPoint testPt1 = q1.ptAtT(testT);
testT = (copyI[1][midEnd] * (index + 1) + copyI[1][midEnd - 1] * (2 - index)) / 3;
SkDPoint testPt2 = q2.ptAtT(testT);
innerEqual += testPt1.approximatelyEqual(testPt2);
}
}
bool expectCoincident = copyI.fUsed >= 2 && innerEqual == 2;
if (expectCoincident) {
reset();
insertCoincident(copyI[0][0], copyI[1][0], copyI.fPt[0]);
int last = copyI.fUsed - 1;
insertCoincident(copyI[0][last], copyI[1][last], copyI.fPt[last]);
return fUsed;
}
SkDQuadImplicit i1(q1);
SkDQuadImplicit i2(q2);
int index;
bool flip1 = q1[2] == q2[0];
bool flip2 = q1[0] == q2[2];
bool useCubic = q1[0] == q2[0];
double roots1[4];
int rootCount = findRoots(i2, q1, roots1, useCubic, flip1, 0);
// OPTIMIZATION: could short circuit here if all roots are < 0 or > 1
double roots1Copy[4];
int r1Count = addValidRoots(roots1, rootCount, roots1Copy);
SkDPoint pts1[4];
for (index = 0; index < r1Count; ++index) {
pts1[index] = q1.ptAtT(roots1Copy[index]);
}
double roots2[4];
int rootCount2 = findRoots(i1, q2, roots2, useCubic, flip2, 0);
double roots2Copy[4];
int r2Count = addValidRoots(roots2, rootCount2, roots2Copy);
SkDPoint pts2[4];
for (index = 0; index < r2Count; ++index) {
pts2[index] = q2.ptAtT(roots2Copy[index]);
}
if (r1Count == r2Count && r1Count <= 1) {
if (r1Count == 1) {
if (pts1[0].approximatelyEqualHalf(pts2[0])) {
insert(roots1Copy[0], roots2Copy[0], pts1[0]);
} else if (pts1[0].moreRoughlyEqual(pts2[0])) {
// experiment: try to find intersection by chasing t
rootCount = findRoots(i2, q1, roots1, useCubic, flip1, 0);
(void) addValidRoots(roots1, rootCount, roots1Copy);
rootCount2 = findRoots(i1, q2, roots2, useCubic, flip2, 0);
(void) addValidRoots(roots2, rootCount2, roots2Copy);
if (binary_search(q1, q2, roots1Copy, roots2Copy, pts1)) {
insert(roots1Copy[0], roots2Copy[0], pts1[0]);
}
}
}
return fUsed;
}
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].fX - pts1[index].fX;
double dy = pts2[ndex2].fY - pts1[index].fY;
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) {
relaxed_is_linear(q1, q2, this);
return fUsed;
}
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;
}
insert(roots1Copy[lowestIndex], roots2Copy[closest[lowestIndex]],
pts1[lowestIndex]);
closest[lowestIndex] = -1;
} while (++used < r1Count);
return fUsed;
}