source: trunk/src/gui/painting/qbezier.cpp

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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(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;
}

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, qreal bezier_flattening_threshold) 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 < bezier_flattening_threshold*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 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;

    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] = qAcos(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;
}

#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 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;
}

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;

    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   = 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;
}

QT_END_NAMESPACE