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These rights are described in the Digia Qt LGPL Exception ** version 1.1, included in the file LGPL_EXCEPTION.txt in this package. ** ** GNU General Public License Usage ** Alternatively, this file may be used under the terms of the GNU ** General Public License version 3.0 as published by the Free Software ** Foundation and appearing in the file LICENSE.GPL included in the ** packaging of this file. Please review the following information to ** ensure the GNU General Public License version 3.0 requirements will be ** met: http://www.gnu.org/copyleft/gpl.html. ** ** ** $QT_END_LICENSE$ ** ****************************************************************************/ #include "qbezier_p.h" #include #include #include #include #include #include #include #include QT_BEGIN_NAMESPACE //#define QDEBUG_BEZIER #ifdef FLOAT_ACCURACY #define INV_EPS (1L<<23) #else /* The value of 1.0 / (1L<<14) is enough for most applications */ #define INV_EPS (1L<<14) #endif #ifndef M_SQRT2 #define M_SQRT2 1.41421356237309504880 #endif #define log2(x) (qLn(x)/qLn(2.)) static inline qreal log4(qreal x) { return qreal(0.5) * log2(x); } /*! \internal */ QBezier QBezier::fromPoints(const QPointF &p1, const QPointF &p2, const QPointF &p3, const QPointF &p4) { QBezier b; b.x1 = p1.x(); b.y1 = p1.y(); b.x2 = p2.x(); b.y2 = p2.y(); b.x3 = p3.x(); b.y3 = p3.y(); b.x4 = p4.x(); b.y4 = p4.y(); return b; } /*! \internal */ QPolygonF QBezier::toPolygon() const { // flattening is done by splitting the bezier until we can replace the segment by a straight // line. We split further until the control points are close enough to the line connecting the // boundary points. // // the Distance of a point p from a line given by the points (a,b) is given by: // // d = abs( (bx - ax)(ay - py) - (by - ay)(ax - px) ) / line_length // // We can stop splitting if both control points are close enough to the line. // To make the algorithm faster we use the manhattan length of the line. QPolygonF polygon; polygon.append(QPointF(x1, y1)); addToPolygon(&polygon); return polygon; } //0.5 is really low static const qreal flatness = 0.5; //based on "Fast, precise flattening of cubic Bezier path and offset curves" // by T. F. Hain, A. L. Ahmad, S. V. R. Racherla and D. D. Langan static inline void flattenBezierWithoutInflections(QBezier &bez, QPolygonF *&p) { QBezier left; while (1) { qreal dx = bez.x2 - bez.x1; qreal dy = bez.y2 - bez.y1; qreal normalized = qSqrt(dx * dx + dy * dy); if (qFuzzyCompare(normalized + 1, 1)) break; qreal d = qAbs(dx * (bez.y3 - bez.y2) - dy * (bez.x3 - bez.x2)); qreal t = qSqrt(4. / 3. * normalized * flatness / d); if (t > 1 || qFuzzyCompare(t, (qreal)1.)) break; bez.parameterSplitLeft(t, &left); p->append(bez.pt1()); } } static inline int quadraticRoots(qreal a, qreal b, qreal c, qreal *x1, qreal *x2) { if (qFuzzyCompare(a + 1, 1)) { if (qFuzzyCompare(b + 1, 1)) return 0; *x1 = *x2 = (-c / b); return 1; } else { const qreal det = b * b - 4 * a * c; if (qFuzzyCompare(det + 1, 1)) { *x1 = *x2 = -b / (2 * a); return 1; } if (det > 0) { if (qFuzzyCompare(b + 1, 1)) { *x2 = qSqrt(-c / a); *x1 = -(*x2); return 2; } const qreal stableA = b / (2 * a); const qreal stableB = c / (a * stableA * stableA); const qreal stableC = -1 - qSqrt(1 - stableB); *x2 = stableA * stableC; *x1 = (stableA * stableB) / stableC; return 2; } else return 0; } } static inline bool findInflections(qreal a, qreal b, qreal c, qreal *t1 , qreal *t2, qreal *tCups) { qreal r1 = 0, r2 = 0; short rootsCount = quadraticRoots(a, b, c, &r1, &r2); if (rootsCount >= 1) { if (r1 < r2) { *t1 = r1; *t2 = r2; } else { *t1 = r2; *t2 = r1; } if (!qFuzzyCompare(a + 1, 1)) *tCups = 0.5 * (-b / a); else *tCups = 2; return true; } return false; } void QBezier::addToPolygon(QPolygonF *polygon) const { QBezier beziers[32]; beziers[0] = *this; QBezier *b = beziers; while (b >= beziers) { // check if we can pop the top bezier curve from the stack qreal y4y1 = b->y4 - b->y1; qreal x4x1 = b->x4 - b->x1; qreal l = qAbs(x4x1) + qAbs(y4y1); qreal d; if (l > 1.) { d = qAbs( (x4x1)*(b->y1 - b->y2) - (y4y1)*(b->x1 - b->x2) ) + qAbs( (x4x1)*(b->y1 - b->y3) - (y4y1)*(b->x1 - b->x3) ); } else { d = qAbs(b->x1 - b->x2) + qAbs(b->y1 - b->y2) + qAbs(b->x1 - b->x3) + qAbs(b->y1 - b->y3); l = 1.; } if (d < flatness*l || b == beziers + 31) { // good enough, we pop it off and add the endpoint polygon->append(QPointF(b->x4, b->y4)); --b; } else { // split, second half of the polygon goes lower into the stack b->split(b+1, b); ++b; } } } void QBezier::addToPolygonMixed(QPolygonF *polygon) const { qreal ax = -x1 + 3*x2 - 3*x3 + x4; qreal ay = -y1 + 3*y2 - 3*y3 + y4; qreal bx = 3*x1 - 6*x2 + 3*x3; qreal by = 3*y1 - 6*y2 + 3*y3; qreal cx = -3*x1 + 3*x2; qreal cy = -3*y1 + 2*y2; qreal a = 6 * (ay * bx - ax * by); qreal b = 6 * (ay * cx - ax * cy); qreal c = 2 * (by * cx - bx * cy); if ((qFuzzyCompare(a + 1, 1) && qFuzzyCompare(b + 1, 1)) || (b * b - 4 * a *c) < 0) { QBezier bez(*this); flattenBezierWithoutInflections(bez, polygon); polygon->append(QPointF(x4, y4)); } else { QBezier beziers[32]; beziers[0] = *this; QBezier *b = beziers; while (b >= beziers) { // check if we can pop the top bezier curve from the stack qreal y4y1 = b->y4 - b->y1; qreal x4x1 = b->x4 - b->x1; qreal l = qAbs(x4x1) + qAbs(y4y1); qreal d; if (l > 1.) { d = qAbs( (x4x1)*(b->y1 - b->y2) - (y4y1)*(b->x1 - b->x2) ) + qAbs( (x4x1)*(b->y1 - b->y3) - (y4y1)*(b->x1 - b->x3) ); } else { d = qAbs(b->x1 - b->x2) + qAbs(b->y1 - b->y2) + qAbs(b->x1 - b->x3) + qAbs(b->y1 - b->y3); l = 1.; } if (d < .5*l || b == beziers + 31) { // good enough, we pop it off and add the endpoint polygon->append(QPointF(b->x4, b->y4)); --b; } else { // split, second half of the polygon goes lower into the stack b->split(b+1, b); ++b; } } } } QRectF QBezier::bounds() const { qreal xmin = x1; qreal xmax = x1; if (x2 < xmin) xmin = x2; else if (x2 > xmax) xmax = x2; if (x3 < xmin) xmin = x3; else if (x3 > xmax) xmax = x3; if (x4 < xmin) xmin = x4; else if (x4 > xmax) xmax = x4; qreal ymin = y1; qreal ymax = y1; if (y2 < ymin) ymin = y2; else if (y2 > ymax) ymax = y2; if (y3 < ymin) ymin = y3; else if (y3 > ymax) ymax = y3; if (y4 < ymin) ymin = y4; else if (y4 > ymax) ymax = y4; return QRectF(xmin, ymin, xmax-xmin, ymax-ymin); } enum ShiftResult { Ok, Discard, Split, Circle }; static ShiftResult good_offset(const QBezier *b1, const QBezier *b2, qreal offset, qreal threshold) { const qreal o2 = offset*offset; const qreal max_dist_line = threshold*offset*offset; const qreal max_dist_normal = threshold*offset; const qreal spacing = 0.25; for (qreal i = spacing; i < 0.99; i += spacing) { QPointF p1 = b1->pointAt(i); QPointF p2 = b2->pointAt(i); qreal d = (p1.x() - p2.x())*(p1.x() - p2.x()) + (p1.y() - p2.y())*(p1.y() - p2.y()); if (qAbs(d - o2) > max_dist_line) return Split; QPointF normalPoint = b1->normalVector(i); qreal l = qAbs(normalPoint.x()) + qAbs(normalPoint.y()); if (l != 0.) { d = qAbs( normalPoint.x()*(p1.y() - p2.y()) - normalPoint.y()*(p1.x() - p2.x()) ) / l; if (d > max_dist_normal) return Split; } } return Ok; } static inline QLineF qline_shifted(const QPointF &p1, const QPointF &p2, qreal offset) { QLineF l(p1, p2); QLineF ln = l.normalVector().unitVector(); l.translate(ln.dx() * offset, ln.dy() * offset); return l; } static bool qbezier_is_line(QPointF *points, int pointCount) { Q_ASSERT(pointCount > 2); qreal dx13 = points[2].x() - points[0].x(); qreal dy13 = points[2].y() - points[0].y(); qreal dx12 = points[1].x() - points[0].x(); qreal dy12 = points[1].y() - points[0].y(); if (pointCount == 3) { return qFuzzyCompare(dx12 * dy13, dx13 * dy12); } else if (pointCount == 4) { qreal dx14 = points[3].x() - points[0].x(); qreal dy14 = points[3].y() - points[0].y(); return (qFuzzyCompare(dx12 * dy13, dx13 * dy12) && qFuzzyCompare(dx12 * dy14, dx14 * dy12)); } return false; } static ShiftResult shift(const QBezier *orig, QBezier *shifted, qreal offset, qreal threshold) { int map[4]; bool p1_p2_equal = (orig->x1 == orig->x2 && orig->y1 == orig->y2); bool p2_p3_equal = (orig->x2 == orig->x3 && orig->y2 == orig->y3); bool p3_p4_equal = (orig->x3 == orig->x4 && orig->y3 == orig->y4); QPointF points[4]; int np = 0; points[np] = QPointF(orig->x1, orig->y1); map[0] = 0; ++np; if (!p1_p2_equal) { points[np] = QPointF(orig->x2, orig->y2); ++np; } map[1] = np - 1; if (!p2_p3_equal) { points[np] = QPointF(orig->x3, orig->y3); ++np; } map[2] = np - 1; if (!p3_p4_equal) { points[np] = QPointF(orig->x4, orig->y4); ++np; } map[3] = np - 1; if (np == 1) return Discard; // We need to specialcase lines of 3 or 4 points due to numerical // instability in intersections below if (np > 2 && qbezier_is_line(points, np)) { if (points[0] == points[np-1]) return Discard; QLineF l = qline_shifted(points[0], points[np-1], offset); *shifted = QBezier::fromPoints(l.p1(), l.pointAt(qreal(0.33)), l.pointAt(qreal(0.66)), l.p2()); return Ok; } QRectF b = orig->bounds(); if (np == 4 && b.width() < .1*offset && b.height() < .1*offset) { qreal l = (orig->x1 - orig->x2)*(orig->x1 - orig->x2) + (orig->y1 - orig->y2)*(orig->y1 - orig->y1) * (orig->x3 - orig->x4)*(orig->x3 - orig->x4) + (orig->y3 - orig->y4)*(orig->y3 - orig->y4); qreal dot = (orig->x1 - orig->x2)*(orig->x3 - orig->x4) + (orig->y1 - orig->y2)*(orig->y3 - orig->y4); if (dot < 0 && dot*dot < 0.8*l) // the points are close and reverse dirction. Approximate the whole // thing by a semi circle return Circle; } QPointF points_shifted[4]; QLineF prev = QLineF(QPointF(), points[1] - points[0]); QPointF prev_normal = prev.normalVector().unitVector().p2(); points_shifted[0] = points[0] + offset * prev_normal; for (int i = 1; i < np - 1; ++i) { QLineF next = QLineF(QPointF(), points[i + 1] - points[i]); QPointF next_normal = next.normalVector().unitVector().p2(); QPointF normal_sum = prev_normal + next_normal; qreal r = 1.0 + prev_normal.x() * next_normal.x() + prev_normal.y() * next_normal.y(); if (qFuzzyCompare(r + 1, 1)) { points_shifted[i] = points[i] + offset * prev_normal; } else { qreal k = offset / r; points_shifted[i] = points[i] + k * normal_sum; } prev_normal = next_normal; } points_shifted[np - 1] = points[np - 1] + offset * prev_normal; *shifted = QBezier::fromPoints(points_shifted[map[0]], points_shifted[map[1]], points_shifted[map[2]], points_shifted[map[3]]); return good_offset(orig, shifted, offset, threshold); } // This value is used to determine the length of control point vectors // when approximating arc segments as curves. The factor is multiplied // with the radius of the circle. #define KAPPA 0.5522847498 static bool addCircle(const QBezier *b, qreal offset, QBezier *o) { QPointF normals[3]; normals[0] = QPointF(b->y2 - b->y1, b->x1 - b->x2); qreal dist = qSqrt(normals[0].x()*normals[0].x() + normals[0].y()*normals[0].y()); if (qFuzzyCompare(dist + 1, 1)) return false; normals[0] /= dist; normals[2] = QPointF(b->y4 - b->y3, b->x3 - b->x4); dist = qSqrt(normals[2].x()*normals[2].x() + normals[2].y()*normals[2].y()); if (qFuzzyCompare(dist + 1, 1)) return false; normals[2] /= dist; normals[1] = QPointF(b->x1 - b->x2 - b->x3 + b->x4, b->y1 - b->y2 - b->y3 + b->y4); normals[1] /= -1*qSqrt(normals[1].x()*normals[1].x() + normals[1].y()*normals[1].y()); qreal angles[2]; qreal sign = 1.; for (int i = 0; i < 2; ++i) { qreal cos_a = normals[i].x()*normals[i+1].x() + normals[i].y()*normals[i+1].y(); if (cos_a > 1.) cos_a = 1.; if (cos_a < -1.) cos_a = -1; angles[i] = acos(cos_a)/Q_PI; } if (angles[0] + angles[1] > 1.) { // more than 180 degrees normals[1] = -normals[1]; angles[0] = 1. - angles[0]; angles[1] = 1. - angles[1]; sign = -1.; } QPointF circle[3]; circle[0] = QPointF(b->x1, b->y1) + normals[0]*offset; circle[1] = QPointF(0.5*(b->x1 + b->x4), 0.5*(b->y1 + b->y4)) + normals[1]*offset; circle[2] = QPointF(b->x4, b->y4) + normals[2]*offset; for (int i = 0; i < 2; ++i) { qreal kappa = 2.*KAPPA * sign * offset * angles[i]; o->x1 = circle[i].x(); o->y1 = circle[i].y(); o->x2 = circle[i].x() - normals[i].y()*kappa; o->y2 = circle[i].y() + normals[i].x()*kappa; o->x3 = circle[i+1].x() + normals[i+1].y()*kappa; o->y3 = circle[i+1].y() - normals[i+1].x()*kappa; o->x4 = circle[i+1].x(); o->y4 = circle[i+1].y(); ++o; } return true; } int QBezier::shifted(QBezier *curveSegments, int maxSegments, qreal offset, float threshold) const { Q_ASSERT(curveSegments); Q_ASSERT(maxSegments > 0); if (x1 == x2 && x1 == x3 && x1 == x4 && y1 == y2 && y1 == y3 && y1 == y4) return 0; --maxSegments; QBezier beziers[10]; redo: beziers[0] = *this; QBezier *b = beziers; QBezier *o = curveSegments; while (b >= beziers) { int stack_segments = b - beziers + 1; if ((stack_segments == 10) || (o - curveSegments == maxSegments - stack_segments)) { threshold *= 1.5; if (threshold > 2.) goto give_up; goto redo; } ShiftResult res = shift(b, o, offset, threshold); if (res == Discard) { --b; } else if (res == Ok) { ++o; --b; continue; } else if (res == Circle && maxSegments - (o - curveSegments) >= 2) { // add semi circle if (addCircle(b, offset, o)) o += 2; --b; } else { b->split(b+1, b); ++b; } } give_up: while (b >= beziers) { ShiftResult res = shift(b, o, offset, threshold); // if res isn't Ok or Split then *o is undefined if (res == Ok || res == Split) ++o; --b; } Q_ASSERT(o - curveSegments <= maxSegments); return o - curveSegments; } #if 0 static inline bool IntersectBB(const QBezier &a, const QBezier &b) { return a.bounds().intersects(b.bounds()); } #else static int IntersectBB(const QBezier &a, const QBezier &b) { // Compute bounding box for a qreal minax, maxax, minay, maxay; if (a.x1 > a.x4) // These are the most likely to be extremal minax = a.x4, maxax = a.x1; else minax = a.x1, maxax = a.x4; if (a.x3 < minax) minax = a.x3; else if (a.x3 > maxax) maxax = a.x3; if (a.x2 < minax) minax = a.x2; else if (a.x2 > maxax) maxax = a.x2; if (a.y1 > a.y4) minay = a.y4, maxay = a.y1; else minay = a.y1, maxay = a.y4; if (a.y3 < minay) minay = a.y3; else if (a.y3 > maxay) maxay = a.y3; if (a.y2 < minay) minay = a.y2; else if (a.y2 > maxay) maxay = a.y2; // Compute bounding box for b qreal minbx, maxbx, minby, maxby; if (b.x1 > b.x4) minbx = b.x4, maxbx = b.x1; else minbx = b.x1, maxbx = b.x4; if (b.x3 < minbx) minbx = b.x3; else if (b.x3 > maxbx) maxbx = b.x3; if (b.x2 < minbx) minbx = b.x2; else if (b.x2 > maxbx) maxbx = b.x2; if (b.y1 > b.y4) minby = b.y4, maxby = b.y1; else minby = b.y1, maxby = b.y4; if (b.y3 < minby) minby = b.y3; else if (b.y3 > maxby) maxby = b.y3; if (b.y2 < minby) minby = b.y2; else if (b.y2 > maxby) maxby = b.y2; // Test bounding box of b against bounding box of a if ((minax > maxbx) || (minay > maxby) // Not >= : need boundary case || (minbx > maxax) || (minby > maxay)) return 0; // they don't intersect else return 1; // they intersect } #endif #ifdef QDEBUG_BEZIER static QDebug operator<<(QDebug dbg, const QBezier &bz) { dbg <<"["< > *t) { #ifdef QDEBUG_BEZIER static int I = 0; int currentD = I; fprintf(stderr, "%d) t0 = %lf, t1 = %lf, deptha = %d\n" "u0 = %lf, u1 = %lf, depthb = %d\n", I++, t0, t1, deptha, u0, u1, depthb); #endif if (deptha > 0) { QBezier A[2]; a.split(&A[0], &A[1]); qreal tmid = (t0+t1)*0.5; //qDebug()<<"\t1)"< 0) { QBezier B[2]; b.split(&B[0], &B[1]); //qDebug()<<"\t3)"<isEmpty() : false; } else { if (IntersectBB(A[0], b)) { //fprintf(stderr, "\t 5 from %d\n", currentD); if (RecursivelyIntersect(A[0], t0, tmid, deptha, b, u0, u1, depthb, t) && !t) return true; } if (IntersectBB(A[1], b)) { //fprintf(stderr, "\t 6 from %d\n", currentD); if (RecursivelyIntersect(A[1], tmid, t1, deptha, b, u0, u1, depthb, t) && !t) return true; } return t ? !t->isEmpty() : false; } } else { if (depthb > 0) { QBezier B[2]; b.split(&B[0], &B[1]); qreal umid = (u0 + u1)*0.5; depthb--; if (IntersectBB(a, B[0])) { //fprintf(stderr, "\t 7 from %d\n", currentD); if (RecursivelyIntersect(a, t0, t1, deptha, B[0], u0, umid, depthb, t) && !t) return true; } if (IntersectBB(a, B[1])) { //fprintf(stderr, "\t 8 from %d\n", currentD); if (RecursivelyIntersect(a, t0, t1, deptha, B[1], umid, u1, depthb, t) && !t) return true; } return t ? !t->isEmpty() : false; } else { // Both segments are fully subdivided; now do line segments qreal xlk = a.x4 - a.x1; qreal ylk = a.y4 - a.y1; qreal xnm = b.x4 - b.x1; qreal ynm = b.y4 - b.y1; qreal xmk = b.x1 - a.x1; qreal ymk = b.y1 - a.y1; qreal det = xnm * ylk - ynm * xlk; if (1.0 + det == 1.0) { return false; } else { qreal detinv = 1.0 / det; qreal rs = (xnm * ymk - ynm *xmk) * detinv; qreal rt = (xlk * ymk - ylk * xmk) * detinv; if ((rs < 0.0) || (rs > 1.0) || (rt < 0.0) || (rt > 1.0)) return false; if (t) { const qreal alpha_a = t0 + rs * (t1 - t0); const qreal alpha_b = u0 + rt * (u1 - u0); *t << qMakePair(alpha_a, alpha_b); } return true; } } } } QVector< QPair > QBezier::findIntersections(const QBezier &a, const QBezier &b) { QVector< QPair > v(2); findIntersections(a, b, &v); return v; } bool QBezier::findIntersections(const QBezier &a, const QBezier &b, QVector > *t) { if (IntersectBB(a, b)) { QPointF la1(fabs((a.x3 - a.x2) - (a.x2 - a.x1)), fabs((a.y3 - a.y2) - (a.y2 - a.y1))); QPointF la2(fabs((a.x4 - a.x3) - (a.x3 - a.x2)), fabs((a.y4 - a.y3) - (a.y3 - a.y2))); QPointF la; if (la1.x() > la2.x()) la.setX(la1.x()); else la.setX(la2.x()); if (la1.y() > la2.y()) la.setY(la1.y()); else la.setY(la2.y()); QPointF lb1(fabs((b.x3 - b.x2) - (b.x2 - b.x1)), fabs((b.y3 - b.y2) - (b.y2 - b.y1))); QPointF lb2(fabs((b.x4 - b.x3) - (b.x3 - b.x2)), fabs((b.y4 - b.y3) - (b.y3 - b.y2))); QPointF lb; if (lb1.x() > lb2.x()) lb.setX(lb1.x()); else lb.setX(lb2.x()); if (lb1.y() > lb2.y()) lb.setY(lb1.y()); else lb.setY(lb2.y()); qreal l0; if (la.x() > la.y()) l0 = la.x(); else l0 = la.y(); int ra; if (l0 * 0.75 * M_SQRT2 + 1.0 == 1.0) ra = 0; else ra = qCeil(log4(M_SQRT2 * 6.0 / 8.0 * INV_EPS * l0)); if (lb.x() > lb.y()) l0 = lb.x(); else l0 = lb.y(); int rb; if (l0 * 0.75 * M_SQRT2 + 1.0 == 1.0) rb = 0; else rb = qCeil(log4(M_SQRT2 * 6.0 / 8.0 * INV_EPS * l0)); // if qreal is float then halve the number of subdivisions if (sizeof(qreal) == 4) { ra /= 2; rb /= 2; } return RecursivelyIntersect(a, 0., 1., ra, b, 0., 1., rb, t); } //Don't sort here because it breaks the orders of corresponding // intersections points. this way t's at the same locations correspond // to the same intersection point. //qSort(parameters[0].begin(), parameters[0].end(), qLess()); //qSort(parameters[1].begin(), parameters[1].end(), qLess()); return false; } static inline void splitBezierAt(const QBezier &bez, qreal t, QBezier *left, QBezier *right) { left->x1 = bez.x1; left->y1 = bez.y1; left->x2 = bez.x1 + t * ( bez.x2 - bez.x1 ); left->y2 = bez.y1 + t * ( bez.y2 - bez.y1 ); left->x3 = bez.x2 + t * ( bez.x3 - bez.x2 ); // temporary holding spot left->y3 = bez.y2 + t * ( bez.y3 - bez.y2 ); // temporary holding spot right->x3 = bez.x3 + t * ( bez.x4 - bez.x3 ); right->y3 = bez.y3 + t * ( bez.y4 - bez.y3 ); right->x2 = left->x3 + t * ( right->x3 - left->x3); right->y2 = left->y3 + t * ( right->y3 - left->y3); left->x3 = left->x2 + t * ( left->x3 - left->x2 ); left->y3 = left->y2 + t * ( left->y3 - left->y2 ); left->x4 = right->x1 = left->x3 + t * (right->x2 - left->x3); left->y4 = right->y1 = left->y3 + t * (right->y2 - left->y3); right->x4 = bez.x4; right->y4 = bez.y4; } QVector< QList > QBezier::splitAtIntersections(QBezier &b) { QVector< QList > curves(2); QVector< QPair > allInters = findIntersections(*this, b); QList inters1; QList inters2; for (int i = 0; i < allInters.size(); ++i) { inters1 << allInters[i].first; inters2 << allInters[i].second; } qSort(inters1.begin(), inters1.end(), qLess()); qSort(inters2.begin(), inters2.end(), qLess()); Q_ASSERT(inters1.count() == inters2.count()); int i; for (i = 0; i < inters1.count(); ++i) { qreal t1 = inters1.at(i); qreal t2 = inters2.at(i); QBezier curve1, curve2; parameterSplitLeft(t1, &curve1); b.parameterSplitLeft(t2, &curve2); curves[0].append(curve1); curves[0].append(curve2); } curves[0].append(*this); curves[1].append(b); return curves; } qreal QBezier::length(qreal error) const { qreal length = 0.0; addIfClose(&length, error); return length; } void QBezier::addIfClose(qreal *length, qreal error) const { QBezier left, right; /* bez poly splits */ qreal len = 0.0; /* arc length */ qreal chord; /* chord length */ len = len + QLineF(QPointF(x1, y1),QPointF(x2, y2)).length(); len = len + QLineF(QPointF(x2, y2),QPointF(x3, y3)).length(); len = len + QLineF(QPointF(x3, y3),QPointF(x4, y4)).length(); chord = QLineF(QPointF(x1, y1),QPointF(x4, y4)).length(); if((len-chord) > error) { split(&left, &right); /* split in two */ left.addIfClose(length, error); /* try left side */ right.addIfClose(length, error); /* try right side */ return; } *length = *length + len; return; } qreal QBezier::tForY(qreal t0, qreal t1, qreal y) const { qreal py0 = pointAt(t0).y(); qreal py1 = pointAt(t1).y(); if (py0 > py1) { qSwap(py0, py1); qSwap(t0, t1); } Q_ASSERT(py0 <= py1); if (py0 >= y) return t0; else if (py1 <= y) return t1; Q_ASSERT(py0 < y && y < py1); qreal lt = t0; qreal dt; do { qreal t = 0.5 * (t0 + t1); qreal a, b, c, d; QBezier::coefficients(t, a, b, c, d); qreal yt = a * y1 + b * y2 + c * y3 + d * y4; if (yt < y) { t0 = t; py0 = yt; } else { t1 = t; py1 = yt; } dt = lt - t; lt = t; } while (qAbs(dt) > 1e-7); return t0; } int QBezier::stationaryYPoints(qreal &t0, qreal &t1) const { // y(t) = (1 - t)^3 * y1 + 3 * (1 - t)^2 * t * y2 + 3 * (1 - t) * t^2 * y3 + t^3 * y4 // y'(t) = 3 * (-(1-2t+t^2) * y1 + (1 - 4 * t + 3 * t^2) * y2 + (2 * t - 3 * t^2) * y3 + t^2 * y4) // y'(t) = 3 * ((-y1 + 3 * y2 - 3 * y3 + y4)t^2 + (2 * y1 - 4 * y2 + 2 * y3)t + (-y1 + y2)) const qreal a = -y1 + 3 * y2 - 3 * y3 + y4; const qreal b = 2 * y1 - 4 * y2 + 2 * y3; const qreal c = -y1 + y2; qreal reciprocal = b * b - 4 * a * c; QList result; if (qFuzzyCompare(reciprocal + 1, 1)) { t0 = -b / (2 * a); return 1; } else if (reciprocal > 0) { qreal temp = qSqrt(reciprocal); t0 = (-b - temp)/(2*a); t1 = (-b + temp)/(2*a); if (t1 < t0) qSwap(t0, t1); int count = 0; qreal t[2] = { 0, 1 }; if (t0 > 0 && t0 < 1) t[count++] = t0; if (t1 > 0 && t1 < 1) t[count++] = t1; t0 = t[0]; t1 = t[1]; return count; } return 0; } qreal QBezier::tAtLength(qreal l) const { qreal len = length(); qreal t = 1.0; const qreal error = (qreal)0.01; if (l > len || qFuzzyCompare(l, len)) return t; t *= 0.5; //int iters = 0; //qDebug()<<"LEN is "<= 1) { *tMinus = *tPlus = 2; return; } QBezier left, right; splitBezierAt(bez, t, &left, &right); qreal ax = -right.x1 + 3*right.x2 - 3*right.x3 + right.x4; qreal ay = -right.y1 + 3*right.y2 - 3*right.y3 + right.y4; qreal ex = 3 * (right.x2 - right.x3); qreal ey = 3 * (right.y2 - right.y3); qreal s4 = qAbs(6 * (ey * ax - ex * ay) / qSqrt(ex * ex + ey * ey)) + 0.00001f; qreal tf = pow(qreal(9 * flatness / s4), qreal(1./3.)); *tMinus = t - (1 - t) * tf; *tPlus = t + (1 - t) * tf; } void QBezier::addToPolygonIterative(QPolygonF *p) const { qreal t1, t2, tcusp; qreal t1min, t1plus, t2min, t2plus; qreal ax = -x1 + 3*x2 - 3*x3 + x4; qreal ay = -y1 + 3*y2 - 3*y3 + y4; qreal bx = 3*x1 - 6*x2 + 3*x3; qreal by = 3*y1 - 6*y2 + 3*y3; qreal cx = -3*x1 + 3*x2; qreal cy = -3*y1 + 2*y2; if (findInflections(6 * (ay * bx - ax * by), 6 * (ay * cx - ax * cy), 2 * (by * cx - bx * cy), &t1, &t2, &tcusp)) { bindInflectionPoint(*this, t1, &t1min, &t1plus); bindInflectionPoint(*this, t2, &t2min, &t2plus); QBezier tmpBez = *this; QBezier left, right, bez1, bez2, bez3; if (t1min > 0) { if (t1min >= 1) { flattenBezierWithoutInflections(tmpBez, p); } else { splitBezierAt(tmpBez, t1min, &left, &right); flattenBezierWithoutInflections(left, p); p->append(tmpBez.pointAt(t1min)); if (t2min < t1plus) { if (tcusp < 1) { p->append(tmpBez.pointAt(tcusp)); } if (t2plus < 1) { splitBezierAt(tmpBez, t2plus, &left, &right); flattenBezierWithoutInflections(right, p); } } else if (t1plus < 1) { if (t2min < 1) { splitBezierAt(tmpBez, t2min, &bez3, &right); splitBezierAt(bez3, t1plus, &left, &bez2); flattenBezierWithoutInflections(bez2, p); p->append(tmpBez.pointAt(t2min)); if (t2plus < 1) { splitBezierAt(tmpBez, t2plus, &left, &bez2); flattenBezierWithoutInflections(bez2, p); } } else { splitBezierAt(tmpBez, t1plus, &left, &bez2); flattenBezierWithoutInflections(bez2, p); } } } } else if (t1plus > 0) { p->append(QPointF(x1, y1)); if (t2min < t1plus) { if (tcusp < 1) { p->append(tmpBez.pointAt(tcusp)); } if (t2plus < 1) { splitBezierAt(tmpBez, t2plus, &left, &bez2); flattenBezierWithoutInflections(bez2, p); } } else if (t1plus < 1) { if (t2min < 1) { splitBezierAt(tmpBez, t2min, &bez3, &right); splitBezierAt(bez3, t1plus, &left, &bez2); flattenBezierWithoutInflections(bez2, p); p->append(tmpBez.pointAt(t2min)); if (t2plus < 1) { splitBezierAt(tmpBez, t2plus, &left, &bez2); flattenBezierWithoutInflections(bez2, p); } } else { splitBezierAt(tmpBez, t1plus, &left, &bez2); flattenBezierWithoutInflections(bez2, p); } } } else if (t2min > 0) { if (t2min < 1) { splitBezierAt(tmpBez, t2min, &bez1, &right); flattenBezierWithoutInflections(bez1, p); p->append(tmpBez.pointAt(t2min)); if (t2plus < 1) { splitBezierAt(tmpBez, t2plus, &left, &bez2); flattenBezierWithoutInflections(bez2, p); } } else { //### in here we should check whether the area of the // triangle formed between pt1/pt2/pt3 is smaller // or equal to 0 and then do iterative flattening // if not we should fallback and do the recursive // flattening. flattenBezierWithoutInflections(tmpBez, p); } } else if (t2plus > 0) { p->append(QPointF(x1, y1)); if (t2plus < 1) { splitBezierAt(tmpBez, t2plus, &left, &bez2); flattenBezierWithoutInflections(bez2, p); } } else { flattenBezierWithoutInflections(tmpBez, p); } } else { QBezier bez = *this; flattenBezierWithoutInflections(bez, p); } p->append(QPointF(x4, y4)); } QT_END_NAMESPACE