qbezier.cpp

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// Copyright (C) 2016 The Qt Company Ltd.
// SPDX-License-Identifier: LicenseRef-Qt-Commercial OR LGPL-3.0-only OR GPL-2.0-only OR GPL-3.0-only
 
#include "qbezier_p.h"
#include <qdebug.h>
#include <qline.h>
#include <qmath.h>
#include <qpolygon.h>
 
#include <private/qnumeric_p.h>
 
#include <tuple> // for std::tie()
 
QT_BEGIN_NAMESPACE
 
//#define QDEBUG_BEZIER
 
QPolygonF QBezier::toPolygon(qreal bezier_flattening_threshold) 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, bezier_flattening_threshold);
    return polygon;
}
 
QBezier QBezier::mapBy(const QTransform &transform) const
{
    return QBezier::fromPoints(transform.map(pt1()), transform.map(pt2()), transform.map(pt3()), transform.map(pt4()));
}
 
QBezier QBezier::getSubRange(qreal t0, qreal t1) const
{
    QBezier result;
    QBezier temp;
 
    // cut at t1
    if (qFuzzyIsNull(t1 - qreal(1.))) {
        result = *this;
    } else {
        temp = *this;
        temp.parameterSplitLeft(t1, &result);
    }
 
    // cut at t0
    if (!qFuzzyIsNull(t0))
        result.parameterSplitLeft(t0 / t1, &temp);
 
    return result;
}
 
void QBezier::addToPolygon(QPolygonF *polygon, qreal bezier_flattening_threshold) const
{
    QBezier beziers[10];
    int levels[10];
    beziers[0] = *this;
    levels[0] = 9;
    int top = 0;
 
    while (top >= 0) {
        QBezier *b = &beziers[top];
        // 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 < bezier_flattening_threshold * l || levels[top] == 0) {
            // good enough, we pop it off and add the endpoint
            polygon->append(QPointF(b->x4, b->y4));
            --top;
        } else {
            // split, second half of the polygon goes lower into the stack
            std::tie(b[1], b[0]) = b->split();
            levels[top + 1] = --levels[top];
            ++top;
        }
    }
}
 
void QBezier::addToPolygon(QDataBuffer<QPointF> &polygon, qreal bezier_flattening_threshold) const
{
    QBezier beziers[10];
    int levels[10];
    beziers[0] = *this;
    levels[0] = 9;
    int top = 0;
 
    while (top >= 0) {
        QBezier *b = &beziers[top];
        // 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 < bezier_flattening_threshold * l || levels[top] == 0) {
            // good enough, we pop it off and add the endpoint
            polygon.add(QPointF(b->x4, b->y4));
            --top;
        } else {
            // split, second half of the polygon goes lower into the stack
            std::tie(b[1], b[0]) = b->split();
            levels[top + 1] = --levels[top];
            ++top;
        }
    }
}
 
QPolygonF QBezier::toQuadratics(qreal errorLimit) const
{
    qreal infPoints[2];
    int numInfPoints = inflectionPoints(infPoints);
    QPolygonF res;
    res.reserve((numInfPoints + 1) * 3 * 2);
    res.append(pt1());
    qreal t0 = 0;
    for (int i = 0; i < numInfPoints + 1; i++) { // #segments == #inflectionpoints + 1
        qreal t1 = (i < numInfPoints) ? infPoints[i] : 1;
        QBezier segment = bezierOnInterval(t0, t1);
        segment.addToQuadratics(&res, t1 - t0, errorLimit);
        t0 = t1;
    }
    return res;
}
 
static inline qreal scoreQuadratic(const QBezier &b, QPointF qcp)
{
    // Construct a cubic from the quadratic, and compare its control points to the originals'
    const QRectF bounds = b.bounds();
    qreal dim = QLineF(bounds.topLeft(), bounds.bottomRight()).length();
    if (qFuzzyIsNull(dim))
        return 1;
    const qreal f = 2.0 / 3;
    const QPointF cp1 = b.pt1() + f * (qcp - b.pt1());
    const QPointF cp2 = b.pt4() + f * (qcp - b.pt4());
    const QLineF d1(b.pt2(), cp1);
    const QLineF d2(b.pt3(), cp2);
    return qMax(d1.length(), d2.length()) / dim;
}
 
static inline QPointF quadraticForCubic(const QBezier &b)
{
    QPointF qcp;
    const QLineF st = b.startTangent();
    const QLineF et = b.endTangent();
    if (st.intersects(et, &qcp) == QLineF::NoIntersection)
        qcp = b.midPoint();
    return qcp;
}
 
void QBezier::addToQuadratics(QPolygonF *p, qreal tspan, qreal errorLimit) const
{
    Q_ASSERT((tspan > 0) && !(tspan > 1));
    static constexpr qreal MinimumTSpan = 0.1;
 
    QPointF qcp = quadraticForCubic(*this);
    if (tspan < MinimumTSpan || scoreQuadratic(*this, qcp) < errorLimit) {
        p->append(qcp);
        p->append(pt4());
    } else {
        std::pair<QBezier, QBezier> halves = split();
        halves.first.addToQuadratics(p, tspan / 2, errorLimit);
        halves.second.addToQuadratics(p, tspan / 2, errorLimit);
    }
}
 
int QBezier::inflectionPoints(qreal *tpoints) const
{
    auto isValidRoot = [](qreal r) {
        return qIsFinite(r) && (r > 0) && (!qFuzzyIsNull(float(r))) && (r < 1)
                && (!qFuzzyIsNull(float(r - 1)));
    };
 
    // normalize so pt1.x,pt1.y,pt4.y == 0
    QTransform xf;
    const QLineF l(pt1(), pt4());
    xf.rotate(l.angle());
    xf.translate(-pt1().x(), -pt1().y());
    const QBezier n = mapBy(xf);
    Q_ASSERT(n.pt1() == QPoint() && qFuzzyIsNull(float(n.pt4().y())));
 
    const qreal p = n.pt3().x() * n.pt2().y();
    const qreal q = n.pt4().x() * n.pt2().y();
    const qreal r = n.pt2().x() * n.pt3().y();
    const qreal s = n.pt4().x() * n.pt3().y();
 
    const qreal a = 36 * ((-3 * p) + (2 * q) + (3 * r) - s);
    if (!a)
        return 0;
    const qreal b = -18 * (((3 * p) - q) - (3 * r));
    const qreal c = 18 * (r - p);
    const qreal rad = (b * b) - (2 * a * c);
    if (rad < 0)
        return 0;
    const qreal sqr = qSqrt(rad);
    const qreal root1 = (b + sqr) / a;
    const qreal root2 = (b - sqr) / a;
 
    int res = 0;
    if (isValidRoot(root1))
        tpoints[res++] = root1;
    if (!qFuzzyCompare(root2, root1) && isValidRoot(root2))
        tpoints[res++] = root2;
 
    if (res == 2 && tpoints[0] > tpoints[1])
        qSwap(tpoints[0], tpoints[1]);
 
    return res;
}
 
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 int divisions = 4;
    const qreal spacing = qreal(1.0) / divisions;
    qreal t = spacing;
    for (int i = 1; i < divisions; ++i, t += spacing) {
        QPointF p1 = b1->pointAt(t);
        QPointF p2 = b2->pointAt(t);
        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(t);
        qreal l = qAbs(normalPoint.x()) + qAbs(normalPoint.y());
        if (l != qreal(0.0)) {
            d = qAbs( normalPoint.x()*(p1.y() - p2.y()) - normalPoint.y()*(p1.x() - p2.x()) ) / l;
            if (d > max_dist_normal)
                return Split;
        }
    }
    return Ok;
}
 
QT_WARNING_DISABLE_FLOAT_COMPARE
 
static ShiftResult shift(const QBezier *orig, QBezier *shifted, qreal offset, qreal threshold)
{
    int map[4];
    bool p1_p2_equal = qFuzzyCompare(orig->x1, orig->x2) && qFuzzyCompare(orig->y1, orig->y2);
    bool p2_p3_equal = qFuzzyCompare(orig->x2, orig->x3) && qFuzzyCompare(orig->y2, orig->y3);
    bool p3_p4_equal = qFuzzyCompare(orig->x3, orig->x4) && qFuzzyCompare(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;
 
    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->y2) *
                  (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]);
    if (!prev.length())
        return Discard;
    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 = qreal(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]]);
 
    if (np > 2)
        return good_offset(orig, shifted, offset, threshold);
    return Ok;
}
 
// 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 qreal(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] = qAcos(cos_a) * qreal(M_1_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(qreal(0.5)*(b->x1 + b->x4), qreal(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 = qreal(2.0) * 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 (qFuzzyCompare(x1, x2) && qFuzzyCompare(x1, x3) && qFuzzyCompare(x1, x4) &&
        qFuzzyCompare(y1, y2) && qFuzzyCompare(y1, y3) && qFuzzyCompare(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 *= qreal(1.5);
            if (threshold > qreal(2.0))
                goto give_up;
            goto redo;
        }
        ShiftResult res = shift(b, o, offset, threshold);
        if (res == Discard) {
            --b;
        } else if (res == Ok) {
            ++o;
            --b;
        } else if (res == Circle && maxSegments - (o - curveSegments) >= 2) {
            // add semi circle
            if (addCircle(b, offset, o))
                o += 2;
            --b;
        } else {
            std::tie(b[1], b[0]) = b->split();
            ++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;
}
 
#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
 
qreal QBezier::length(qreal error) const
{
    qreal length = qreal(0.0);
 
    addIfClose(&length, error);
 
    return length;
}
 
void QBezier::addIfClose(qreal *length, qreal error) const
{
    qreal len = qreal(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) {
        const auto halves = split();                  /* split in two */
        halves.first.addIfClose(length, error);       /* try left side */
        halves.second.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 = qreal(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) > qreal(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;
 
    if (qFuzzyIsNull(a)) {
        if (qFuzzyIsNull(b))
            return 0;
 
        t0 = -c / b;
        return t0 > 0 && t0 < 1;
    }
 
    qreal reciprocal = b * b - 4 * a * c;
 
    if (qFuzzyIsNull(reciprocal)) {
        t0 = -b / (2 * a);
        return t0 > 0 && t0 < 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   = qreal(1.0);
    const qreal error = qreal(0.01);
    if (l > len || qFuzzyCompare(l, len))
        return t;
 
    t *= qreal(0.5);
    //int iters = 0;
    //qDebug()<<"LEN is "<<l<<len;
    qreal lastBigger = qreal(1.0);
    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) * qreal(0.5);
        } else {
            lastBigger = t;
            t -= t * qreal(0.5);
        }
        //++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;
}
 
QT_END_NAMESPACE