michael@0: /*
michael@0:  * Copyright 2012 Google Inc.
michael@0:  *
michael@0:  * Use of this source code is governed by a BSD-style license that can be
michael@0:  * found in the LICENSE file.
michael@0:  */
michael@0: #include "SkIntersections.h"
michael@0: #include "SkLineParameters.h"
michael@0: #include "SkPathOpsCubic.h"
michael@0: #include "SkPathOpsQuad.h"
michael@0: #include "SkPathOpsTriangle.h"
michael@0: 
michael@0: // from http://blog.gludion.com/2009/08/distance-to-quadratic-bezier-curve.html
michael@0: // (currently only used by testing)
michael@0: double SkDQuad::nearestT(const SkDPoint& pt) const {
michael@0:     SkDVector pos = fPts[0] - pt;
michael@0:     // search points P of bezier curve with PM.(dP / dt) = 0
michael@0:     // a calculus leads to a 3d degree equation :
michael@0:     SkDVector A = fPts[1] - fPts[0];
michael@0:     SkDVector B = fPts[2] - fPts[1];
michael@0:     B -= A;
michael@0:     double a = B.dot(B);
michael@0:     double b = 3 * A.dot(B);
michael@0:     double c = 2 * A.dot(A) + pos.dot(B);
michael@0:     double d = pos.dot(A);
michael@0:     double ts[3];
michael@0:     int roots = SkDCubic::RootsValidT(a, b, c, d, ts);
michael@0:     double d0 = pt.distanceSquared(fPts[0]);
michael@0:     double d2 = pt.distanceSquared(fPts[2]);
michael@0:     double distMin = SkTMin(d0, d2);
michael@0:     int bestIndex = -1;
michael@0:     for (int index = 0; index < roots; ++index) {
michael@0:         SkDPoint onQuad = ptAtT(ts[index]);
michael@0:         double dist = pt.distanceSquared(onQuad);
michael@0:         if (distMin > dist) {
michael@0:             distMin = dist;
michael@0:             bestIndex = index;
michael@0:         }
michael@0:     }
michael@0:     if (bestIndex >= 0) {
michael@0:         return ts[bestIndex];
michael@0:     }
michael@0:     return d0 < d2 ? 0 : 1;
michael@0: }
michael@0: 
michael@0: bool SkDQuad::pointInHull(const SkDPoint& pt) const {
michael@0:     return ((const SkDTriangle&) fPts).contains(pt);
michael@0: }
michael@0: 
michael@0: SkDPoint SkDQuad::top(double startT, double endT) const {
michael@0:     SkDQuad sub = subDivide(startT, endT);
michael@0:     SkDPoint topPt = sub[0];
michael@0:     if (topPt.fY > sub[2].fY || (topPt.fY == sub[2].fY && topPt.fX > sub[2].fX)) {
michael@0:         topPt = sub[2];
michael@0:     }
michael@0:     if (!between(sub[0].fY, sub[1].fY, sub[2].fY)) {
michael@0:         double extremeT;
michael@0:         if (FindExtrema(sub[0].fY, sub[1].fY, sub[2].fY, &extremeT)) {
michael@0:             extremeT = startT + (endT - startT) * extremeT;
michael@0:             SkDPoint test = ptAtT(extremeT);
michael@0:             if (topPt.fY > test.fY || (topPt.fY == test.fY && topPt.fX > test.fX)) {
michael@0:                 topPt = test;
michael@0:             }
michael@0:         }
michael@0:     }
michael@0:     return topPt;
michael@0: }
michael@0: 
michael@0: int SkDQuad::AddValidTs(double s[], int realRoots, double* t) {
michael@0:     int foundRoots = 0;
michael@0:     for (int index = 0; index < realRoots; ++index) {
michael@0:         double tValue = s[index];
michael@0:         if (approximately_zero_or_more(tValue) && approximately_one_or_less(tValue)) {
michael@0:             if (approximately_less_than_zero(tValue)) {
michael@0:                 tValue = 0;
michael@0:             } else if (approximately_greater_than_one(tValue)) {
michael@0:                 tValue = 1;
michael@0:             }
michael@0:             for (int idx2 = 0; idx2 < foundRoots; ++idx2) {
michael@0:                 if (approximately_equal(t[idx2], tValue)) {
michael@0:                     goto nextRoot;
michael@0:                 }
michael@0:             }
michael@0:             t[foundRoots++] = tValue;
michael@0:         }
michael@0: nextRoot:
michael@0:         {}
michael@0:     }
michael@0:     return foundRoots;
michael@0: }
michael@0: 
michael@0: // note: caller expects multiple results to be sorted smaller first
michael@0: // note: http://en.wikipedia.org/wiki/Loss_of_significance has an interesting
michael@0: //  analysis of the quadratic equation, suggesting why the following looks at
michael@0: //  the sign of B -- and further suggesting that the greatest loss of precision
michael@0: //  is in b squared less two a c
michael@0: int SkDQuad::RootsValidT(double A, double B, double C, double t[2]) {
michael@0:     double s[2];
michael@0:     int realRoots = RootsReal(A, B, C, s);
michael@0:     int foundRoots = AddValidTs(s, realRoots, t);
michael@0:     return foundRoots;
michael@0: }
michael@0: 
michael@0: /*
michael@0: Numeric Solutions (5.6) suggests to solve the quadratic by computing
michael@0:        Q = -1/2(B + sgn(B)Sqrt(B^2 - 4 A C))
michael@0: and using the roots
michael@0:       t1 = Q / A
michael@0:       t2 = C / Q
michael@0: */
michael@0: // this does not discard real roots <= 0 or >= 1
michael@0: int SkDQuad::RootsReal(const double A, const double B, const double C, double s[2]) {
michael@0:     const double p = B / (2 * A);
michael@0:     const double q = C / A;
michael@0:     if (approximately_zero(A) && (approximately_zero_inverse(p) || approximately_zero_inverse(q))) {
michael@0:         if (approximately_zero(B)) {
michael@0:             s[0] = 0;
michael@0:             return C == 0;
michael@0:         }
michael@0:         s[0] = -C / B;
michael@0:         return 1;
michael@0:     }
michael@0:     /* normal form: x^2 + px + q = 0 */
michael@0:     const double p2 = p * p;
michael@0:     if (!AlmostDequalUlps(p2, q) && p2 < q) {
michael@0:         return 0;
michael@0:     }
michael@0:     double sqrt_D = 0;
michael@0:     if (p2 > q) {
michael@0:         sqrt_D = sqrt(p2 - q);
michael@0:     }
michael@0:     s[0] = sqrt_D - p;
michael@0:     s[1] = -sqrt_D - p;
michael@0:     return 1 + !AlmostDequalUlps(s[0], s[1]);
michael@0: }
michael@0: 
michael@0: bool SkDQuad::isLinear(int startIndex, int endIndex) const {
michael@0:     SkLineParameters lineParameters;
michael@0:     lineParameters.quadEndPoints(*this, startIndex, endIndex);
michael@0:     // FIXME: maybe it's possible to avoid this and compare non-normalized
michael@0:     lineParameters.normalize();
michael@0:     double distance = lineParameters.controlPtDistance(*this);
michael@0:     return approximately_zero(distance);
michael@0: }
michael@0: 
michael@0: SkDCubic SkDQuad::toCubic() const {
michael@0:     SkDCubic cubic;
michael@0:     cubic[0] = fPts[0];
michael@0:     cubic[2] = fPts[1];
michael@0:     cubic[3] = fPts[2];
michael@0:     cubic[1].fX = (cubic[0].fX + cubic[2].fX * 2) / 3;
michael@0:     cubic[1].fY = (cubic[0].fY + cubic[2].fY * 2) / 3;
michael@0:     cubic[2].fX = (cubic[3].fX + cubic[2].fX * 2) / 3;
michael@0:     cubic[2].fY = (cubic[3].fY + cubic[2].fY * 2) / 3;
michael@0:     return cubic;
michael@0: }
michael@0: 
michael@0: SkDVector SkDQuad::dxdyAtT(double t) const {
michael@0:     double a = t - 1;
michael@0:     double b = 1 - 2 * t;
michael@0:     double c = t;
michael@0:     SkDVector result = { a * fPts[0].fX + b * fPts[1].fX + c * fPts[2].fX,
michael@0:             a * fPts[0].fY + b * fPts[1].fY + c * fPts[2].fY };
michael@0:     return result;
michael@0: }
michael@0: 
michael@0: // OPTIMIZE: assert if caller passes in t == 0 / t == 1 ?
michael@0: SkDPoint SkDQuad::ptAtT(double t) const {
michael@0:     if (0 == t) {
michael@0:         return fPts[0];
michael@0:     }
michael@0:     if (1 == t) {
michael@0:         return fPts[2];
michael@0:     }
michael@0:     double one_t = 1 - t;
michael@0:     double a = one_t * one_t;
michael@0:     double b = 2 * one_t * t;
michael@0:     double c = t * t;
michael@0:     SkDPoint result = { a * fPts[0].fX + b * fPts[1].fX + c * fPts[2].fX,
michael@0:             a * fPts[0].fY + b * fPts[1].fY + c * fPts[2].fY };
michael@0:     return result;
michael@0: }
michael@0: 
michael@0: /*
michael@0: Given a quadratic q, t1, and t2, find a small quadratic segment.
michael@0: 
michael@0: The new quadratic is defined by A, B, and C, where
michael@0:  A = c[0]*(1 - t1)*(1 - t1) + 2*c[1]*t1*(1 - t1) + c[2]*t1*t1
michael@0:  C = c[3]*(1 - t1)*(1 - t1) + 2*c[2]*t1*(1 - t1) + c[1]*t1*t1
michael@0: 
michael@0: To find B, compute the point halfway between t1 and t2:
michael@0: 
michael@0: q(at (t1 + t2)/2) == D
michael@0: 
michael@0: Next, compute where D must be if we know the value of B:
michael@0: 
michael@0: _12 = A/2 + B/2
michael@0: 12_ = B/2 + C/2
michael@0: 123 = A/4 + B/2 + C/4
michael@0:     = D
michael@0: 
michael@0: Group the known values on one side:
michael@0: 
michael@0: B   = D*2 - A/2 - C/2
michael@0: */
michael@0: 
michael@0: static double interp_quad_coords(const double* src, double t) {
michael@0:     double ab = SkDInterp(src[0], src[2], t);
michael@0:     double bc = SkDInterp(src[2], src[4], t);
michael@0:     double abc = SkDInterp(ab, bc, t);
michael@0:     return abc;
michael@0: }
michael@0: 
michael@0: bool SkDQuad::monotonicInY() const {
michael@0:     return between(fPts[0].fY, fPts[1].fY, fPts[2].fY);
michael@0: }
michael@0: 
michael@0: SkDQuad SkDQuad::subDivide(double t1, double t2) const {
michael@0:     SkDQuad dst;
michael@0:     double ax = dst[0].fX = interp_quad_coords(&fPts[0].fX, t1);
michael@0:     double ay = dst[0].fY = interp_quad_coords(&fPts[0].fY, t1);
michael@0:     double dx = interp_quad_coords(&fPts[0].fX, (t1 + t2) / 2);
michael@0:     double dy = interp_quad_coords(&fPts[0].fY, (t1 + t2) / 2);
michael@0:     double cx = dst[2].fX = interp_quad_coords(&fPts[0].fX, t2);
michael@0:     double cy = dst[2].fY = interp_quad_coords(&fPts[0].fY, t2);
michael@0:     /* bx = */ dst[1].fX = 2*dx - (ax + cx)/2;
michael@0:     /* by = */ dst[1].fY = 2*dy - (ay + cy)/2;
michael@0:     return dst;
michael@0: }
michael@0: 
michael@0: void SkDQuad::align(int endIndex, SkDPoint* dstPt) const {
michael@0:     if (fPts[endIndex].fX == fPts[1].fX) {
michael@0:         dstPt->fX = fPts[endIndex].fX;
michael@0:     }
michael@0:     if (fPts[endIndex].fY == fPts[1].fY) {
michael@0:         dstPt->fY = fPts[endIndex].fY;
michael@0:     }
michael@0: }
michael@0: 
michael@0: SkDPoint SkDQuad::subDivide(const SkDPoint& a, const SkDPoint& c, double t1, double t2) const {
michael@0:     SkASSERT(t1 != t2);
michael@0:     SkDPoint b;
michael@0: #if 0
michael@0:     // this approach assumes that the control point computed directly is accurate enough
michael@0:     double dx = interp_quad_coords(&fPts[0].fX, (t1 + t2) / 2);
michael@0:     double dy = interp_quad_coords(&fPts[0].fY, (t1 + t2) / 2);
michael@0:     b.fX = 2 * dx - (a.fX + c.fX) / 2;
michael@0:     b.fY = 2 * dy - (a.fY + c.fY) / 2;
michael@0: #else
michael@0:     SkDQuad sub = subDivide(t1, t2);
michael@0:     SkDLine b0 = {{a, sub[1] + (a - sub[0])}};
michael@0:     SkDLine b1 = {{c, sub[1] + (c - sub[2])}};
michael@0:     SkIntersections i;
michael@0:     i.intersectRay(b0, b1);
michael@0:     if (i.used() == 1) {
michael@0:         b = i.pt(0);
michael@0:     } else {
michael@0:         SkASSERT(i.used() == 2 || i.used() == 0);
michael@0:         b = SkDPoint::Mid(b0[1], b1[1]);
michael@0:     }
michael@0: #endif
michael@0:     if (t1 == 0 || t2 == 0) {
michael@0:         align(0, &b);
michael@0:     }
michael@0:     if (t1 == 1 || t2 == 1) {
michael@0:         align(2, &b);
michael@0:     }
michael@0:     if (precisely_subdivide_equal(b.fX, a.fX)) {
michael@0:         b.fX = a.fX;
michael@0:     } else if (precisely_subdivide_equal(b.fX, c.fX)) {
michael@0:         b.fX = c.fX;
michael@0:     }
michael@0:     if (precisely_subdivide_equal(b.fY, a.fY)) {
michael@0:         b.fY = a.fY;
michael@0:     } else if (precisely_subdivide_equal(b.fY, c.fY)) {
michael@0:         b.fY = c.fY;
michael@0:     }
michael@0:     return b;
michael@0: }
michael@0: 
michael@0: /* classic one t subdivision */
michael@0: static void interp_quad_coords(const double* src, double* dst, double t) {
michael@0:     double ab = SkDInterp(src[0], src[2], t);
michael@0:     double bc = SkDInterp(src[2], src[4], t);
michael@0:     dst[0] = src[0];
michael@0:     dst[2] = ab;
michael@0:     dst[4] = SkDInterp(ab, bc, t);
michael@0:     dst[6] = bc;
michael@0:     dst[8] = src[4];
michael@0: }
michael@0: 
michael@0: SkDQuadPair SkDQuad::chopAt(double t) const
michael@0: {
michael@0:     SkDQuadPair dst;
michael@0:     interp_quad_coords(&fPts[0].fX, &dst.pts[0].fX, t);
michael@0:     interp_quad_coords(&fPts[0].fY, &dst.pts[0].fY, t);
michael@0:     return dst;
michael@0: }
michael@0: 
michael@0: static int valid_unit_divide(double numer, double denom, double* ratio)
michael@0: {
michael@0:     if (numer < 0) {
michael@0:         numer = -numer;
michael@0:         denom = -denom;
michael@0:     }
michael@0:     if (denom == 0 || numer == 0 || numer >= denom) {
michael@0:         return 0;
michael@0:     }
michael@0:     double r = numer / denom;
michael@0:     if (r == 0) {  // catch underflow if numer <<<< denom
michael@0:         return 0;
michael@0:     }
michael@0:     *ratio = r;
michael@0:     return 1;
michael@0: }
michael@0: 
michael@0: /** Quad'(t) = At + B, where
michael@0:     A = 2(a - 2b + c)
michael@0:     B = 2(b - a)
michael@0:     Solve for t, only if it fits between 0 < t < 1
michael@0: */
michael@0: int SkDQuad::FindExtrema(double a, double b, double c, double tValue[1]) {
michael@0:     /*  At + B == 0
michael@0:         t = -B / A
michael@0:     */
michael@0:     return valid_unit_divide(a - b, a - b - b + c, tValue);
michael@0: }
michael@0: 
michael@0: /* Parameterization form, given A*t*t + 2*B*t*(1-t) + C*(1-t)*(1-t)
michael@0:  *
michael@0:  * a = A - 2*B +   C
michael@0:  * b =     2*B - 2*C
michael@0:  * c =             C
michael@0:  */
michael@0: void SkDQuad::SetABC(const double* quad, double* a, double* b, double* c) {
michael@0:     *a = quad[0];      // a = A
michael@0:     *b = 2 * quad[2];  // b =     2*B
michael@0:     *c = quad[4];      // c =             C
michael@0:     *b -= *c;          // b =     2*B -   C
michael@0:     *a -= *b;          // a = A - 2*B +   C
michael@0:     *b -= *c;          // b =     2*B - 2*C
michael@0: }
michael@0: 
michael@0: #ifdef SK_DEBUG
michael@0: void SkDQuad::dump() {
michael@0:     SkDebugf("{{");
michael@0:     int index = 0;
michael@0:     do {
michael@0:         fPts[index].dump();
michael@0:         SkDebugf(", ");
michael@0:     } while (++index < 2);
michael@0:     fPts[index].dump();
michael@0:     SkDebugf("}}\n");
michael@0: }
michael@0: #endif