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#include "qbezier_p.h"
#include <qdebug.h>
#include <qline.h>
#include <qpolygon.h>
#include <qvector.h>
#include <qlist.h>
#include <qmath.h>
#include <private/qnumeric_p.h>
#include <private/qmath_p.h>
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 (qFuzzyIsNull(normalized))
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 || qFuzzyIsNull(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 (qFuzzyIsNull(a)) {
if (qFuzzyIsNull(b))
return 0;
*x1 = *x2 = (-c / b);
return 1;
} else {
const qreal det = b * b - 4 * a * c;
if (qFuzzyIsNull(det)) {
*x1 = *x2 = -b / (2 * a);
return 1;
}
if (det > 0) {
if (qFuzzyIsNull(b)) {
*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 (!qFuzzyIsNull(a))
*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 ((qFuzzyIsNull(a) && qFuzzyIsNull(b)) ||
(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 (qFuzzyIsNull(r)) {
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 (qFuzzyIsNull(dist))
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 (qFuzzyIsNull(dist))
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 << '[' << bz.x1<< ", " << bz.y1 << "], "
<< '[' << bz.x2 <<", " << bz.y2 << "], "
<< '[' << bz.x3 <<", " << bz.y3 << "], "
<< '[' << bz.x4 <<", " << bz.y4 << ']';
return dbg;
}
#endif
static bool RecursivelyIntersect(const QBezier &a, qreal t0, qreal t1, int deptha,
const QBezier &b, qreal u0, qreal u1, int depthb,
QVector<QPair<qreal, qreal> > *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)"<<A[0];
//qDebug()<<"\t2)"<<A[1];
deptha--;
if (depthb > 0) {
QBezier B[2];
b.split(&B[0], &B[1]);
//qDebug()<<"\t3)"<<B[0];
//qDebug()<<"\t4)"<<B[1];
qreal umid = (u0+u1)*0.5;
depthb--;
if (IntersectBB(A[0], B[0])) {
//fprintf(stderr, "\t 1 from %d\n", currentD);
if (RecursivelyIntersect(A[0], t0, tmid, deptha,
B[0], u0, umid, depthb,
t) && !t)
return true;
}
if (IntersectBB(A[1], B[0])) {
//fprintf(stderr, "\t 2 from %d\n", currentD);
if (RecursivelyIntersect(A[1], tmid, t1, deptha,
B[0], u0, umid, depthb,
t) && !t)
return true;
}
if (IntersectBB(A[0], B[1])) {
//fprintf(stderr, "\t 3 from %d\n", currentD);
if (RecursivelyIntersect(A[0], t0, tmid, deptha,
B[1], umid, u1, depthb,
t) && !t)
return true;
}
if (IntersectBB(A[1], B[1])) {
//fprintf(stderr, "\t 4 from %d\n", currentD);
if (RecursivelyIntersect(A[1], tmid, t1, deptha,
B[1], umid, u1, depthb,
t) && !t)
return true;
}
return t ? !t->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<qreal, qreal> > QBezier::findIntersections(const QBezier &a, const QBezier &b)
{
QVector< QPair<qreal, qreal> > v(2);
findIntersections(a, b, &v);
return v;
}
bool QBezier::findIntersections(const QBezier &a, const QBezier &b,
QVector<QPair<qreal, qreal> > *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<qreal>());
//qSort(parameters[1].begin(), parameters[1].end(), qLess<qreal>());
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> > QBezier::splitAtIntersections(QBezier &b)
{
QVector< QList<QBezier> > curves(2);
QVector< QPair<qreal, qreal> > allInters = findIntersections(*this, b);
QList<qreal> inters1;
QList<qreal> inters2;
for (int i = 0; i < allInters.size(); ++i) {
inters1 << allInters[i].first;
inters2 << allInters[i].second;
}
qSort(inters1.begin(), inters1.end(), qLess<qreal>());
qSort(inters2.begin(), inters2.end(), qLess<qreal>());
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<qreal> result;
if (qFuzzyIsNull(reciprocal)) {
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 "<<l<<len;
qreal lastBigger = 1.;
while (1) {
//qDebug()<<"\tt is "<<t;
QBezier right = *this;
QBezier left;
right.parameterSplitLeft(t, &left);
qreal lLen = left.length();
if (qAbs(lLen - l) < error)
break;
if (lLen < l) {
t += (lastBigger - t)*.5;
} else {
lastBigger = t;
t -= t*.5;
}
//++iters;
}
//qDebug()<<"number of iters is "<<iters;
return t;
}
QBezier QBezier::bezierOnInterval(qreal t0, qreal t1) const
{
if (t0 == 0 && t1 == 1)
return *this;
QBezier bezier = *this;
QBezier result;
bezier.parameterSplitLeft(t0, &result);
qreal trueT = (t1-t0)/(1-t0);
bezier.parameterSplitLeft(trueT, &result);
return result;
}
static inline void bindInflectionPoint(const QBezier &bez, const qreal t,
qreal *tMinus , qreal *tPlus)
{
if (t <= 0) {
*tMinus = *tPlus = -1;
return;
} else if (t >= 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