kis_algebra_2d.cpp

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/*
 *  SPDX-FileCopyrightText: 2014 Dmitry Kazakov <dimula73@gmail.com>
 *
 *  SPDX-License-Identifier: GPL-2.0-or-later
 */
 
#include "kis_algebra_2d.h"
 
#include <QTransform>
#include <QPainterPath>
#include <kis_debug.h>
 
#include <boost/accumulators/accumulators.hpp>
#include <boost/accumulators/statistics/stats.hpp>
#include <boost/accumulators/statistics/min.hpp>
#include <boost/accumulators/statistics/max.hpp>
 
#include <array>
#include <QVector2D>
#include <QVector3D>
 
#include <QtMath>
 
#include <config-gsl.h>
 
#ifdef HAVE_GSL
#include <gsl/gsl_multimin.h>
#endif /*HAVE_GSL*/
 
#include <Eigen/Eigenvalues>
 
#define SANITY_CHECKS
 
namespace KisAlgebra2D {
 
void adjustIfOnPolygonBoundary(const QPolygonF &poly, int polygonDirection, QPointF *pt)
{
    const int numPoints = poly.size();
    for (int i = 0; i < numPoints; i++) {
        int nextI = i + 1;
        if (nextI >= numPoints) {
            nextI = 0;
        }
 
        const QPointF &p0 = poly[i];
        const QPointF &p1 = poly[nextI];
 
        QPointF edge = p1 - p0;
 
        qreal cross = crossProduct(edge, *pt - p0)
            / (0.5 * edge.manhattanLength());
 
        if (cross < 1.0 &&
            isInRange(pt->x(), p0.x(), p1.x()) &&
            isInRange(pt->y(), p0.y(), p1.y())) {
 
            QPointF salt = 1.0e-3 * inwardUnitNormal(edge, polygonDirection);
 
            QPointF adjustedPoint = *pt + salt;
 
            // in case the polygon is self-intersecting, polygon direction
            // might not help
            if (kisDistanceToLine(adjustedPoint, QLineF(p0, p1)) < 1e-4) {
                adjustedPoint = *pt - salt;
 
#ifdef SANITY_CHECKS
                if (kisDistanceToLine(adjustedPoint, QLineF(p0, p1)) < 1e-4) {
                    dbgKrita << ppVar(*pt);
                    dbgKrita << ppVar(adjustedPoint);
                    dbgKrita << ppVar(QLineF(p0, p1));
                    dbgKrita << ppVar(salt);
 
                    dbgKrita << ppVar(poly.containsPoint(*pt, Qt::OddEvenFill));
 
                    dbgKrita << ppVar(kisDistanceToLine(*pt, QLineF(p0, p1)));
                    dbgKrita << ppVar(kisDistanceToLine(adjustedPoint, QLineF(p0, p1)));
                }
 
                *pt = adjustedPoint;
 
                KIS_ASSERT_RECOVER_NOOP(kisDistanceToLine(*pt, QLineF(p0, p1)) > 1e-4);
#endif /* SANITY_CHECKS */
            }
        }
    }
}
 
QPointF transformAsBase(const QPointF &pt, const QPointF &base1, const QPointF &base2) {
    qreal len1 = norm(base1);
    if (len1 < 1e-5) return pt;
    qreal sin1 = base1.y() / len1;
    qreal cos1 = base1.x() / len1;
 
    qreal len2 = norm(base2);
    if (len2 < 1e-5) return QPointF();
    qreal sin2 = base2.y() / len2;
    qreal cos2 = base2.x() / len2;
 
    qreal sinD = sin2 * cos1 - cos2 * sin1;
    qreal cosD = cos1 * cos2 + sin1 * sin2;
    qreal scaleD = len2 / len1;
 
    QPointF result;
    result.rx() = scaleD * (pt.x() * cosD - pt.y() * sinD);
    result.ry() = scaleD * (pt.x() * sinD + pt.y() * cosD);
 
    return result;
}
 
qreal angleBetweenVectors(const QPointF &v1, const QPointF &v2)
{
    qreal a1 = std::atan2(v1.y(), v1.x());
    qreal a2 = std::atan2(v2.y(), v2.x());
 
    return a2 - a1;
}
 
qreal directionBetweenPoints(const QPointF &p1, const QPointF &p2, qreal defaultAngle)
{
    if (fuzzyPointCompare(p1, p2)) {
        return defaultAngle;
    }
 
    const QVector2D diff(p2 - p1);
    return std::atan2(diff.y(), diff.x());
}
 
QPainterPath smallArrow()
{
    QPainterPath p;
 
    p.moveTo(5, 2);
    p.lineTo(-3, 8);
    p.lineTo(-5, 5);
    p.lineTo( 2, 0);
    p.lineTo(-5,-5);
    p.lineTo(-3,-8);
    p.lineTo( 5,-2);
    p.arcTo(QRectF(3, -2, 4, 4), 90, -180);
 
    return p;
}
 
template <class Point, class Rect>
inline Point ensureInRectImpl(Point pt, const Rect &bounds)
{
    if (pt.x() > bounds.right()) {
        pt.rx() = bounds.right();
    } else if (pt.x() < bounds.left()) {
        pt.rx() = bounds.left();
    }
 
    if (pt.y() > bounds.bottom()) {
        pt.ry() = bounds.bottom();
    } else if (pt.y() < bounds.top()) {
        pt.ry() = bounds.top();
    }
 
    return pt;
}
 
QPoint ensureInRect(QPoint pt, const QRect &bounds)
{
    return ensureInRectImpl(pt, bounds);
}
 
QPointF ensureInRect(QPointF pt, const QRectF &bounds)
{
    return ensureInRectImpl(pt, bounds);
}
 
bool intersectLineRect(QLineF &line, const QRect rect, bool extend)
{
    return intersectLineRect(line, rect, extend, extend);
}
 
bool intersectLineRect(QLineF &line, const QRect rect, bool extendFirst, bool extendSecond)
{
    // using Liang-Barsky algorithm
    // using parametric equation for a line:
    // X = x1 + t(x2-x1) = x1 + t*p2
    // Y = y1 + t(y2-y1) = y1 + t*p4
    // note that we have a line *segment* here, not a line
    // t1 = 0, t2 = 1, no matter what points we got
 
    double p1 = -(line.x2() - line.x1());
    double p2 = -p1;
    double p3 = -(line.y2() - line.y1());
    double p4 = -p3;
 
    QVector<double> p = QVector<double>({p1, p2, p3, p4});
    QVector<double> q = QVector<double>({line.x1() - rect.x(), rect.x() + rect.width() - line.x1(), line.y1() - rect.y(), rect.y() + rect.height() - line.y1()});
 
    float tmin = extendFirst ? -std::numeric_limits<float>::infinity() : 0; // line.p1() - first point, or infinity
    float tmax = extendSecond ? std::numeric_limits<float>::infinity() : 1; // line.p2() - second point, or infinity
 
    for (int i = 0; i < p.length(); i++) {
        // now clipping
        if (p[i] == 0 && q[i] < 0) {
            // line is parallel to the boundary and outside of it
            return false;
        } else if (p[i] < 0) {
            // line entry point (should be tmin)
            double t = q[i]/p[i];
            if (t > tmax) {
                if (extendSecond) {
                    tmax = t;
                } else {
                    return false;
                }
            }
            if (t > tmin) {
                tmin = t;
            }
 
 
        } else if (p[i] > 0) {
            // line leaving point (should be tmax)
            double t = q[i]/p[i];
            if (t < tmin) {
                if (extendFirst) {
                    tmin = t;
                } else {
                    return false;
                }
            }
            if (t < tmax) {
                tmax = t;
            }
        }
    }
 
    QPointF pt1 = line.p1();
    QPointF pt2 = line.p2();
 
    if (extendFirst || tmin > 0) {
        pt1 = QPointF(line.x1() + tmin*(line.x2() - line.x1()), line.y1() + tmin*(line.y2() - line.y1()));
    }
    if (extendSecond || tmax < 1) {
        pt2 = QPointF(line.x1() + tmax*(line.x2() - line.x1()), line.y1() + tmax*(line.y2() - line.y1()));
    }
 
    line.setP1(pt1);
    line.setP2(pt2);
 
    return true;
 
}
 
bool intersectLineConvexPolygon(QLineF &line, const QPolygonF polygon, bool extendFirst, bool extendSecond)
{
    qreal epsilon = 1e-07;
 
    if (polygon.size() == 0) {
        return false;
    }
 
    // trivial case: no extends, all points inside the polygon
    if (!extendFirst && !extendSecond && polygon.containsPoint(line.p1(), Qt::WindingFill) && polygon.containsPoint(line.p2(), Qt::WindingFill)) {
        return true;
    }
 
    // Cyrus-Beck algorithm: https://en.wikipedia.org/wiki/Cyrus%E2%80%93Beck_algorithm
 
    // parametric equation for the line:
    // p(t) = t*P1 + (1-t)*P2
 
    // we can't use infinity here, because the points would all end up as (+/- inf, +/- inf)
    // which would be useless if we have a very specific direction
    // hence we use just "a big number"
    float aBigNumber = 100000; // that will keep t*Px still on the line // this is LIMITATION of the algorithm
    float tmin = extendFirst ? -aBigNumber : 0; // line.p1() - first point, or infinity
    float tmax = extendSecond ? aBigNumber : 1; // line.p2() - second point, or infinity
 
 
 
    QList<QLineF> clippingLines;
    QList<QPointF> normals;
 
    bool clockwise = false;
    {
        // only temporary values to check whether the polygon is clockwise or counterclockwise
        QPointF vec1 = polygon[1] - polygon[0];
        QPointF vec2 = polygon[2] - polygon[0];
 
        QLineF line1(QPointF(0, 0), vec1);
        QLineF line2(QPointF(0, 0), vec2);
 
        qreal angle = line1.angleTo(line2); // range: <0, 360) // <0, 180) means counter-clockwise
        if (angle >= 180) {
            clockwise = true;
        }
    }
 
 
    for (int i = 0; i < polygon.size() - 1; i++) {
        clippingLines.append(QLineF(polygon[i], polygon[i + 1]));
        float xdiff = polygon[i].x() - polygon[i + 1].x();
        float ydiff = polygon[i].y() - polygon[i + 1].y();
 
        if (!clockwise) {
            normals.append(QPointF(ydiff, -xdiff));
        } else {
            normals.append(QPointF(-ydiff, xdiff));
        }
 
    }
 
    if (!polygon.isClosed()) {
        int i = polygon.size() - 1;
        clippingLines.append(QLineF(polygon[i], polygon[0]));
        float xdiff = polygon[i].x() - polygon[0].x();
        float ydiff = polygon[i].y() - polygon[0].y();
 
        if (!clockwise) {
            normals.append(QPointF(ydiff, -xdiff));
        } else {
            normals.append(QPointF(-ydiff, xdiff));
        }
 
    }
 
    QPointF lineVec = line.p2() - line.p1();
    // ax + by + c = 0
    // c = -ax - by
//    qreal cFromStandardEquation = -lineVec.x()*line.p1().x() - lineVec.y()*line.p1().y();
 
 
    QPointF p1 = line.p1();
    QPointF p2 = line.p2();
 
    qreal pA, pB, pC; // standard equation for the line: ax + by + c = 0
    if (qAbs(lineVec.x()) < epsilon) {
        // x1 ~= x2, line looks like: x = -pC
        pB = 0;
        pA = 1;
        pC = -p1.x();
    } else {
        pB = 1; // let b = 1 to simplify
        qreal tmp = (p1.y() - p2.y())/(p1.x() - p2.x());
        pA = -tmp;
        pC = tmp * p1.x() - p1.y();
    }
 
 
    int pEFound = 0;
 
 
    for (int i = 0; i < clippingLines.size(); i++) {
 
        // is the clipping line parallel to the line?
        QPointF clipVec = clippingLines[i].p2() - clippingLines[i].p1();
 
        if (qFuzzyCompare(clipVec.x()*lineVec.y(), clipVec.y()*lineVec.x())) {
 
            // vectors are parallel
            // let's check if one of the clipping points is on the line (extended) to see if it's the same line; if not, proceed normally
 
            qreal distanceBetweenLines = qAbs(pA*clippingLines[i].p1().x() + pB*clippingLines[i].p1().y() + pC)
                    /(sqrt(pA*pA + pB*pB));
 
 
            if (qAbs(distanceBetweenLines) < epsilon) {
                // they are the same line
 
                qreal t1;
                qreal t2;
 
                if (qAbs(p2.x() - p1.x()) > epsilon) {
                    t1 = (clippingLines[i].p1().x() - p1.x())/(p2.x() - p1.x());
                    t2 = (clippingLines[i].p2().x() - p1.x())/(p2.x() - p1.x());
                } else {
                    t1 = (clippingLines[i].p1().y() - p1.y())/(p2.y() - p1.y());
                    t2 = (clippingLines[i].p2().y() - p1.y())/(p2.y() - p1.y());
                }
 
                qreal tmin1 = qMin(t1, t2);
                qreal tmax2 = qMax(t1, t2);
 
 
 
                if (tmin < tmin1) {
                    tmin = tmin1;
                }
                if (tmax > tmax2) {
                    tmax = tmax2;
                }
                continue;
            }
            else {
                // if clipping line points are P1 and P2, and some other point is P(t) (given by parametric equation),
                // and N is the normal vector
                // and one of the points of the line we're intersecting is K,
                // then we have a following equation:
                // K = P(t) + a * N
                // where t and a are unknown.
                // after simplifying we get:
                // t*(p2 - p1) + a*(n) = k - p1
                // and since we have two axis, we get a linear system of two equations
                // A * [t; a] = B
 
                Eigen::Matrix2f A;
                Eigen::Vector2f b;
                A << p2.x() - p1.x(), normals[i].x(),
                     p2.y() - p1.y(), normals[i].y();
                b << clippingLines[i].p1().x() - p1.x(),
                     clippingLines[i].p1().y() - p1.y();
 
                Eigen::Vector2f ta = A.colPivHouseholderQr().solve(b);
 
                qreal tt = ta(1);
                if (tt < 0) {
                    return false;
                }
            }
 
        }
 
 
        boost::optional<QPointF> pEOptional = intersectLines(clippingLines[i], line); // bounded, unbounded
 
 
        if (pEOptional) {
            pEFound++;
 
            QPointF pE = pEOptional.value();
            QPointF n = normals[i];
 
            QPointF A = line.p2() - line.p1();
            QPointF B = line.p1() - pE;
 
            qreal over = (n.x()*B.x() + n.y()*B.y());
            qreal under = (n.x()*A.x() + n.y()*A.y());
            qreal t = -over/under;
 
//            if (pE.x() != p2.x()) {
//                qreal maybet = (pE.x() - p1.x())/(p2.x() - p1.x());
//            }
 
            qreal tminvalue = tmin * under + over;
            qreal tmaxvalue = tmax * under + over;
 
            if (tminvalue > 0 && tmaxvalue > 0) {
                // both outside of the edge
                return false;
            }
            if (tminvalue <= 0 && tmaxvalue <= 0) {
                // noop, both inside
            }
            if (tminvalue*tmaxvalue < 0) {
 
                // on two different sides
                if (tminvalue > 0) {
                    // tmin outside, tmax inside
                    if (t > tmin) {
                        tmin = t;
                    }
                } else {
                    if (t < tmax) {
                        tmax = t;
                    }
                }
            }
 
 
 
            if (tmax < tmin) {
                return false;
            }
        }
        else {
        }
 
    }
 
    if (pEFound == 0) {
        return false;
    }
 
    QLineF response = QLineF(tmin*line.p2() + (1-tmin)*line.p1(), tmax*line.p2() + (1-tmax)*line.p1());
    line = response;
    return true;
}
 
    template <class Rect, class Point>
    QVector<Point> sampleRectWithPoints(const Rect &rect)
    {
        QVector<Point> points;
 
        Point m1 = 0.5 * (rect.topLeft() + rect.topRight());
        Point m2 = 0.5 * (rect.bottomLeft() + rect.bottomRight());
 
        points << rect.topLeft();
        points << m1;
        points << rect.topRight();
 
        points << 0.5 * (rect.topLeft() + rect.bottomLeft());
        points << 0.5 * (m1 + m2);
        points << 0.5 * (rect.topRight() + rect.bottomRight());
 
        points << rect.bottomLeft();
        points << m2;
        points << rect.bottomRight();
 
        return points;
    }
 
    QVector<QPoint> sampleRectWithPoints(const QRect &rect)
    {
        return sampleRectWithPoints<QRect, QPoint>(rect);
    }
 
    QVector<QPointF> sampleRectWithPoints(const QRectF &rect)
    {
        return sampleRectWithPoints<QRectF, QPointF>(rect);
    }
 
 
    template <class Rect, class Point, bool alignPixels>
    Rect approximateRectFromPointsImpl(const QVector<Point> &points)
    {
        using namespace boost::accumulators;
        accumulator_set<qreal, stats<tag::min, tag::max > > accX;
        accumulator_set<qreal, stats<tag::min, tag::max > > accY;
 
        Q_FOREACH (const Point &pt, points) {
            accX(pt.x());
            accY(pt.y());
        }
 
        Rect resultRect;
 
        if (alignPixels) {
            resultRect.setCoords(std::floor(min(accX)), std::floor(min(accY)),
                                 std::ceil(max(accX)), std::ceil(max(accY)));
        } else {
            resultRect.setCoords(min(accX), min(accY),
                                 max(accX), max(accY));
        }
 
        return resultRect;
    }
 
    QRect approximateRectFromPoints(const QVector<QPoint> &points)
    {
        return approximateRectFromPointsImpl<QRect, QPoint, true>(points);
    }
 
    QRectF approximateRectFromPoints(const QVector<QPointF> &points)
    {
        return approximateRectFromPointsImpl<QRectF, QPointF, false>(points);
    }
 
    QRect approximateRectWithPointTransform(const QRect &rect, std::function<QPointF(QPointF)> func)
    {
        QVector<QPoint> points = sampleRectWithPoints(rect);
 
        using namespace boost::accumulators;
        accumulator_set<qreal, stats<tag::min, tag::max > > accX;
        accumulator_set<qreal, stats<tag::min, tag::max > > accY;
 
        Q_FOREACH (const QPoint &pt, points) {
            QPointF dstPt = func(pt);
 
            accX(dstPt.x());
            accY(dstPt.y());
        }
 
        QRect resultRect;
        resultRect.setCoords(std::floor(min(accX)), std::floor(min(accY)),
                             std::ceil(max(accX)), std::ceil(max(accY)));
 
        return resultRect;
    }
 
 
QRectF cutOffRect(const QRectF &rc, const KisAlgebra2D::RightHalfPlane &p)
{
    QVector<QPointF> points;
 
    const QLineF cutLine = p.getLine();
 
    points << rc.topLeft();
    points << rc.topRight();
    points << rc.bottomRight();
    points << rc.bottomLeft();
 
    QPointF p1 = points[3];
    bool p1Valid = p.pos(p1) >= 0;
 
    QVector<QPointF> resultPoints;
 
    for (int i = 0; i < 4; i++) {
        const QPointF p2 = points[i];
        const bool p2Valid = p.pos(p2) >= 0;
 
        if (p1Valid != p2Valid) {
            QPointF intersection;
            cutLine.intersects(QLineF(p1, p2), &intersection);
            resultPoints << intersection;
        }
 
        if (p2Valid) {
            resultPoints << p2;
        }
 
        p1 = p2;
        p1Valid = p2Valid;
    }
 
    return approximateRectFromPoints(resultPoints);
}
 
int quadraticEquation(qreal a, qreal b, qreal c, qreal *x1, qreal *x2)
{
    int numSolutions = 0;
 
    const qreal D = pow2(b) - 4 * a * c;
    const qreal eps = 1e-14;
 
    if (qAbs(D) <= eps) {
        *x1 = -b / (2 * a);
        numSolutions = 1;
    } else if (D < 0) {
        return 0;
    } else {
        const qreal sqrt_D = std::sqrt(D);
 
        *x1 = (-b + sqrt_D) / (2 * a);
        *x2 = (-b - sqrt_D) / (2 * a);
        numSolutions = 2;
    }
 
    return numSolutions;
}
 
QVector<QPointF> intersectTwoCircles(const QPointF &center1, qreal r1,
                                     const QPointF &center2, qreal r2)
{
    QVector<QPointF> points;
 
    const QPointF diff = (center2 - center1);
    const QPointF c1;
    const QPointF c2 = diff;
 
    const qreal centerDistance = norm(diff);
 
    if (centerDistance > r1 + r2) return points;
    if (centerDistance < qAbs(r1 - r2)) return points;
 
    if (centerDistance < qAbs(r1 - r2) + 0.001) {
        dbgKrita << "Skipping intersection" << ppVar(center1) << ppVar(center2) << ppVar(r1) << ppVar(r2) << ppVar(centerDistance) << ppVar(qAbs(r1-r2));
        return points;
    }
 
    const qreal x_kp1 = diff.x();
    const qreal y_kp1 = diff.y();
 
    const qreal F2 =
        0.5 * (pow2(x_kp1) +
               pow2(y_kp1) + pow2(r1) - pow2(r2));
 
    const qreal eps = 1e-6;
 
    if (qAbs(diff.y()) < eps) {
        qreal x = F2 / diff.x();
        qreal y1, y2;
        int result = KisAlgebra2D::quadraticEquation(
            1, 0,
            pow2(x) - pow2(r2),
            &y1, &y2);
 
        KIS_SAFE_ASSERT_RECOVER(result > 0) { return points; }
 
        if (result == 1) {
            points << QPointF(x, y1);
        } else if (result == 2) {
            KisAlgebra2D::RightHalfPlane p(c1, c2);
 
            QPointF p1(x, y1);
            QPointF p2(x, y2);
 
            if (p.pos(p1) >= 0) {
                points << p1;
                points << p2;
            } else {
                points << p2;
                points << p1;
            }
        }
    } else {
        const qreal A = diff.x() / diff.y();
        const qreal C = F2 / diff.y();
 
        qreal x1, x2;
        int result = KisAlgebra2D::quadraticEquation(
            1 + pow2(A), -2 * A * C,
            pow2(C) - pow2(r1),
            &x1, &x2);
 
        KIS_SAFE_ASSERT_RECOVER(result > 0) { return points; }
 
        if (result == 1) {
            points << QPointF(x1, C - x1 * A);
        } else if (result == 2) {
            KisAlgebra2D::RightHalfPlane p(c1, c2);
 
            QPointF p1(x1, C - x1 * A);
            QPointF p2(x2, C - x2 * A);
 
            if (p.pos(p1) >= 0) {
                points << p1;
                points << p2;
            } else {
                points << p2;
                points << p1;
            }
        }
    }
 
    for (int i = 0; i < points.size(); i++) {
        points[i] = center1 + points[i];
    }
 
    return points;
}
 
QTransform mapToRect(const QRectF &rect)
{
    return
        QTransform(rect.width(), 0, 0, rect.height(),
                   rect.x(), rect.y());
}
 
QTransform mapToRectInverse(const QRectF &rect)
{
    return
        QTransform::fromTranslate(-rect.x(), -rect.y()) *
        QTransform::fromScale(rect.width() != 0 ? 1.0 / rect.width() : 0.0,
                              rect.height() != 0 ? 1.0 / rect.height() : 0.0);
}
 
bool fuzzyMatrixCompare(const QTransform &t1, const QTransform &t2, qreal delta) {
    return
            qAbs(t1.m11() - t2.m11()) < delta &&
            qAbs(t1.m12() - t2.m12()) < delta &&
            qAbs(t1.m13() - t2.m13()) < delta &&
            qAbs(t1.m21() - t2.m21()) < delta &&
            qAbs(t1.m22() - t2.m22()) < delta &&
            qAbs(t1.m23() - t2.m23()) < delta &&
            qAbs(t1.m31() - t2.m31()) < delta &&
            qAbs(t1.m32() - t2.m32()) < delta &&
            qAbs(t1.m33() - t2.m33()) < delta;
}
 
bool fuzzyPointCompare(const QPointF &p1, const QPointF &p2)
{
    return qFuzzyCompare(p1.x(), p2.x()) && qFuzzyCompare(p1.y(), p2.y());
}
 
 
bool fuzzyPointCompare(const QPointF &p1, const QPointF &p2, qreal delta)
{
    return qAbs(p1.x() - p2.x()) < delta && qAbs(p1.y() - p2.y()) < delta;
}
 
 
inline QTransform toQTransformStraight(const Eigen::Matrix3d &m)
{
    return QTransform(m(0,0), m(0,1), m(0,2),
                      m(1,0), m(1,1), m(1,2),
                      m(2,0), m(2,1), m(2,2));
}
 
inline Eigen::Matrix3d fromQTransformStraight(const QTransform &t)
{
    Eigen::Matrix3d m;
 
    m << t.m11() , t.m12() , t.m13()
        ,t.m21() , t.m22() , t.m23()
        ,t.m31() , t.m32() , t.m33();
 
    return m;
}
 
/********************************************************/
/*             DecomposedMatrix                         */
/********************************************************/
 
DecomposedMatrix::DecomposedMatrix()
{
}
 
DecomposedMatrix::DecomposedMatrix(const QTransform &t0)
{
    QTransform t(t0);
 
    QTransform projMatrix;
 
    if (t.m33() == 0.0 || t0.determinant() == 0.0) {
        qWarning() << "Cannot decompose matrix!" << t;
        valid = false;
        return;
    }
 
    if (t.type() == QTransform::TxProject) {
        QTransform affineTransform(
            t.m11(), t.m12(), 0,
            t.m21(), t.m22(), 0,
            t.m31(), t.m32(), 1
        );
        projMatrix = affineTransform.inverted() * t;
 
        t = affineTransform;
        proj[0] = projMatrix.m13();
        proj[1] = projMatrix.m23();
        proj[2] = projMatrix.m33();
    }
 
    // can't use QVector3D, because they have too little accuracy for ellipse in perspective calculations
    std::array<Eigen::Vector3d, 3> rows;
 
 
    //  << t.m11() << t.m12() << t.m13())
    rows[0] = Eigen::Vector3d(t.m11(), t.m12(), t.m13());
    rows[1] = Eigen::Vector3d(t.m21(), t.m22(), t.m23());
    rows[2] = Eigen::Vector3d(t.m31(), t.m32(), t.m33());
 
    if (!qFuzzyCompare(t.m33(), 1.0)) {
        const qreal invM33 = 1.0 / t.m33();
 
        for (auto &row : rows) {
            row *= invM33;
        }
    }
 
    dx = rows[2].x();
    dy = rows[2].y();
 
    rows[2] = Eigen::Vector3d(0,0,1);
 
 
    scaleX = rows[0].norm();
    rows[0] *= 1.0 / scaleX;
 
    shearXY = rows[0].dot(rows[1]);
    rows[1] = rows[1] - shearXY * rows[0];
 
    scaleY = rows[1].norm();
    rows[1] *= 1.0 / scaleY;
    shearXY *= 1.0 / scaleY;
 
    // If determinant is negative, one axis was flipped.
    qreal determinant = rows[0].x() * rows[1].y() - rows[0].y() * rows[1].x();
    if (determinant < 0) {
        // Flip axis with minimum unit vector dot product.
        if (rows[0].x() < rows[1].y()) {
            scaleX = -scaleX;
            rows[0] = -rows[0];
        } else {
            scaleY = -scaleY;
            rows[1] = -rows[1];
        }
        shearXY = - shearXY;
    }
 
    angle = kisRadiansToDegrees(std::atan2(rows[0].y(), rows[0].x()));
 
    if (angle != 0.0) {
        // Rotate(-angle) = [cos(angle), sin(angle), -sin(angle), cos(angle)]
        //                = [row0x, -row0y, row0y, row0x]
        // Thanks to the normalization above.
 
        qreal sn = -rows[0].y();
        qreal cs = rows[0].x();
        qreal m11 = rows[0].x();
        qreal m12 = rows[0].y();
        qreal m21 = rows[1].x();
        qreal m22 = rows[1].y();
        rows[0][0] = (cs * m11 + sn * m21);
        rows[0][1] = (cs * m12 + sn * m22);
        rows[1][0] = (-sn * m11 + cs * m21);
        rows[1][1] = (-sn * m12 + cs * m22);
    }
 
    QTransform leftOver(
                rows[0].x(), rows[0].y(), rows[0].z(),
            rows[1].x(), rows[1].y(), rows[1].z(),
            rows[2].x(), rows[2].y(), rows[2].z());
 
    if (/*true || */!fuzzyMatrixCompare(leftOver, QTransform(), 1e-4)) {
        // what's wrong?
        ENTER_FUNCTION() << "FAILING THE ASSERT BELOW!";
        ENTER_FUNCTION() << ppVar(leftOver);
        ENTER_FUNCTION() << "matrix to decompose was: " << ppVar(t0);
        ENTER_FUNCTION() << ppVar(t.m33()) << ppVar(t0.determinant());
        Eigen::Matrix3d mat1 = fromQTransformStraight(QTransform());
        Eigen::Matrix3d mat2 = fromQTransformStraight(leftOver);
        Eigen::Matrix3d mat3 = mat2 - mat1;
        ENTER_FUNCTION() << mat3(0, 0) << mat3(0, 1) << mat3(0, 2);
        ENTER_FUNCTION() << mat3(1, 0) << mat3(1, 1) << mat3(1, 2);
        ENTER_FUNCTION() << mat3(2, 0) << mat3(2, 1) << mat3(2, 2);
        //ENTER_FUNCTION() << ppVar(mat1 - mat2);
    }
 
 
    KIS_SAFE_ASSERT_RECOVER_NOOP(fuzzyMatrixCompare(leftOver, QTransform(), 1e-4));
    KIS_ASSERT(fuzzyMatrixCompare(leftOver, QTransform(), 1e-4));
}
 
 
std::pair<QPointF, QTransform> transformEllipse(const QPointF &axes, const QTransform &fullLocalToGlobal)
{
    KisAlgebra2D::DecomposedMatrix decomposed(fullLocalToGlobal);
    const QTransform localToGlobal =
            decomposed.scaleTransform() *
            decomposed.shearTransform() *
 
            /* decomposed.projectTransform() * */
            /*decomposed.translateTransform() * */
 
            decomposed.rotateTransform();
 
    const QTransform localEllipse = QTransform(1.0 / pow2(axes.x()), 0.0, 0.0,
                                               0.0, 1.0 / pow2(axes.y()), 0.0,
                                               0.0, 0.0, 1.0);
 
 
    const QTransform globalToLocal = localToGlobal.inverted();
 
    Eigen::Matrix3d eqM =
        fromQTransformStraight(globalToLocal *
                               localEllipse *
                               globalToLocal.transposed());
 
//    std::cout << "eqM:" << std::endl << eqM << std::endl;
 
    Eigen::EigenSolver<Eigen::Matrix3d> eigenSolver(eqM);
 
    const Eigen::Matrix3d T = eigenSolver.eigenvalues().real().asDiagonal();
    const Eigen::Matrix3d U = eigenSolver.eigenvectors().real();
 
    const Eigen::Matrix3d Ti = eigenSolver.eigenvalues().imag().asDiagonal();
    const Eigen::Matrix3d Ui = eigenSolver.eigenvectors().imag();
 
    KIS_SAFE_ASSERT_RECOVER_NOOP(Ti.isZero());
    KIS_SAFE_ASSERT_RECOVER_NOOP(Ui.isZero());
    KIS_SAFE_ASSERT_RECOVER_NOOP((U * U.transpose()).isIdentity());
 
//    std::cout << "T:" << std::endl << T << std::endl;
//    std::cout << "U:" << std::endl << U << std::endl;
 
//    std::cout << "Ti:" << std::endl << Ti << std::endl;
//    std::cout << "Ui:" << std::endl << Ui << std::endl;
 
//    std::cout << "UTU':" << std::endl << U * T * U.transpose() << std::endl;
 
    const qreal newA = 1.0 / std::sqrt(T(0,0) * T(2,2));
    const qreal newB = 1.0 / std::sqrt(T(1,1) * T(2,2));
 
    const QTransform newGlobalToLocal = toQTransformStraight(U);
    const QTransform newLocalToGlobal = QTransform::fromScale(-1,-1) *
            newGlobalToLocal.inverted() *
            decomposed.translateTransform();
 
    return std::make_pair(QPointF(newA, newB), newLocalToGlobal);
}
 
QPointF alignForZoom(const QPointF &pt, qreal zoom)
{
    return QPointF((pt * zoom).toPoint()) / zoom;
}
 
boost::optional<QPointF> intersectLines(const QLineF &boundedLine, const QLineF &unboundedLine)
{
    const QPointF B1 = unboundedLine.p1();
    const QPointF A1 = unboundedLine.p2() - unboundedLine.p1();
 
    const QPointF B2 = boundedLine.p1();
    const QPointF A2 = boundedLine.p2() - boundedLine.p1();
 
    Eigen::Matrix<float, 2, 2> A;
    A << A1.x(), -A2.x(),
         A1.y(), -A2.y();
 
    Eigen::Matrix<float, 2, 1> B;
    B << B2.x() - B1.x(),
         B2.y() - B1.y();
 
    if (qFuzzyIsNull(A.determinant())) {
        return boost::none;
    }
 
    Eigen::Matrix<float, 2, 1> T;
 
    T = A.inverse() * B;
 
    const qreal t2 = T(1,0);
    double epsilon = 1e-06; // to avoid precision issues
 
    if (t2 < 0.0 || t2 > 1.0) {
        if (qAbs(t2) > epsilon && qAbs(t2 - 1.0) > epsilon) {
            return boost::none;
        }
    }
 
    return t2 * A2 + B2;
}
 
boost::optional<QPointF> intersectLines(const QPointF &p1, const QPointF &p2,
                                        const QPointF &q1, const QPointF &q2)
{
    return intersectLines(QLineF(p1, p2), QLineF(q1, q2));
}
 
QVector<QPointF> findTrianglePoint(const QPointF &p1, const QPointF &p2, qreal a, qreal b)
{
    using KisAlgebra2D::norm;
    using KisAlgebra2D::dotProduct;
 
    QVector<QPointF> result;
 
    const QPointF p = p2 - p1;
 
    const qreal pSq = dotProduct(p, p);
 
    const qreal T =  0.5 * (-pow2(b) + pow2(a) + pSq);
 
    if (p.isNull()) return result;
 
    if (qAbs(p.x()) > qAbs(p.y())) {
        const qreal A = 1.0;
        const qreal B2 = -T * p.y() / pSq;
        const qreal C = pow2(T) / pSq - pow2(a * p.x()) / pSq;
 
        const qreal D4 = pow2(B2) - A * C;
 
        if (D4 > 0 || qFuzzyIsNull(D4)) {
            if (qFuzzyIsNull(D4)) {
                const qreal y = -B2 / A;
                const qreal x = (T - y * p.y()) / p.x();
                result << p1 + QPointF(x, y);
            } else {
                const qreal y1 = (-B2 + std::sqrt(D4)) / A;
                const qreal x1 = (T - y1 * p.y()) / p.x();
                result << p1 + QPointF(x1, y1);
 
                const qreal y2 = (-B2 - std::sqrt(D4)) / A;
                const qreal x2 = (T - y2 * p.y()) / p.x();
                result << p1 + QPointF(x2, y2);
            }
        }
    } else {
        const qreal A = 1.0;
        const qreal B2 = -T * p.x() / pSq;
        const qreal C = pow2(T) / pSq - pow2(a * p.y()) / pSq;
 
        const qreal D4 = pow2(B2) - A * C;
 
        if (D4 > 0 || qFuzzyIsNull(D4)) {
            if (qFuzzyIsNull(D4)) {
                const qreal x = -B2 / A;
                const qreal y = (T - x * p.x()) / p.y();
                result << p1 + QPointF(x, y);
            } else {
                const qreal x1 = (-B2 + std::sqrt(D4)) / A;
                const qreal y1 = (T - x1 * p.x()) / p.y();
                result << p1 + QPointF(x1, y1);
 
                const qreal x2 = (-B2 - std::sqrt(D4)) / A;
                const qreal y2 = (T - x2 * p.x()) / p.y();
                result << p1 + QPointF(x2, y2);
            }
        }
    }
 
    return result;
}
 
boost::optional<QPointF> findTrianglePointNearest(const QPointF &p1, const QPointF &p2, qreal a, qreal b, const QPointF &nearest)
{
    const QVector<QPointF> points = findTrianglePoint(p1, p2, a, b);
 
    boost::optional<QPointF> result;
 
    if (points.size() == 1) {
        result = points.first();
    } else if (points.size() > 1) {
        result = kisDistance(points.first(), nearest) < kisDistance(points.last(), nearest) ? points.first() : points.last();
    }
 
    return result;
}
 
QPointF moveElasticPoint(const QPointF &pt, const QPointF &base, const QPointF &newBase, const QPointF &wingA, const QPointF &wingB)
{
    using KisAlgebra2D::norm;
    using KisAlgebra2D::dotProduct;
    using KisAlgebra2D::crossProduct;
 
    const QPointF vecL = base - pt;
    const QPointF vecLa = wingA - pt;
    const QPointF vecLb = wingB - pt;
 
    const qreal L = norm(vecL);
    const qreal La = norm(vecLa);
    const qreal Lb = norm(vecLb);
 
    const qreal sinMuA = crossProduct(vecLa, vecL) / (La * L);
    const qreal sinMuB = crossProduct(vecL, vecLb) / (Lb * L);
 
    const qreal cosMuA = dotProduct(vecLa, vecL) / (La * L);
    const qreal cosMuB = dotProduct(vecLb, vecL) / (Lb * L);
 
    const qreal dL = dotProduct(newBase - base, -vecL) / L;
 
 
    const qreal divB = (cosMuB + La / Lb * sinMuB * cosMuA / sinMuA) / L +
            (cosMuB + sinMuB * cosMuA / sinMuA) / Lb;
    const qreal dLb = dL / ( L * divB);
 
    const qreal divA = (cosMuA + Lb / La * sinMuA * cosMuB / sinMuB) / L +
            (cosMuA + sinMuA * cosMuB / sinMuB) / La;
    const qreal dLa = dL / ( L * divA);
 
    boost::optional<QPointF> result =
        KisAlgebra2D::findTrianglePointNearest(wingA, wingB, La + dLa, Lb + dLb, pt);
 
    const QPointF resultPoint = result ? *result : pt;
 
    return resultPoint;
}
 
#ifdef HAVE_GSL
 
struct ElasticMotionData
{
    QPointF oldBasePos;
    QPointF newBasePos;
    QVector<QPointF> anchorPoints;
 
    QPointF oldResultPoint;
};
 
double elasticMotionError(const gsl_vector * x, void *paramsPtr)
{
    using KisAlgebra2D::norm;
    using KisAlgebra2D::dotProduct;
    using KisAlgebra2D::crossProduct;
 
    const QPointF newResultPoint(gsl_vector_get(x, 0), gsl_vector_get(x, 1));
 
    const ElasticMotionData *p = static_cast<const ElasticMotionData*>(paramsPtr);
 
    const QPointF vecL = newResultPoint - p->newBasePos;
    const qreal L = norm(vecL);
 
    const qreal deltaL = L - kisDistance(p->oldBasePos, p->oldResultPoint);
 
    QVector<qreal> deltaLi;
    QVector<qreal> Li;
    QVector<qreal> cosMuI;
    QVector<qreal> sinMuI;
    Q_FOREACH (const QPointF &anchorPoint, p->anchorPoints) {
        const QPointF vecLi = newResultPoint - anchorPoint;
        const qreal _Li = norm(vecLi);
 
        Li << _Li;
        deltaLi << _Li - kisDistance(p->oldResultPoint, anchorPoint);
        cosMuI << dotProduct(vecLi, vecL) / (_Li * L);
        sinMuI << crossProduct(vecL, vecLi) / (_Li * L);
    }
 
    qreal finalError = 0;
 
    qreal tangentialForceSum = 0;
    for (int i = 0; i < p->anchorPoints.size(); i++) {
        const qreal sum = deltaLi[i] * sinMuI[i] / Li[i];
        tangentialForceSum += sum;
    }
 
    finalError += pow2(tangentialForceSum);
 
    qreal normalForceSum = 0;
    {
        qreal sum = 0;
        for (int i = 0; i < p->anchorPoints.size(); i++) {
            sum += deltaLi[i] * cosMuI[i] / Li[i];
        }
        normalForceSum = (-deltaL) / L - sum;
    }
 
    finalError += pow2(normalForceSum);
 
    return finalError;
}
 
 
#endif /* HAVE_GSL */
 
QPointF moveElasticPoint(const QPointF &pt,
                         const QPointF &base, const QPointF &newBase,
                         const QVector<QPointF> &anchorPoints)
{
    const QPointF offset = newBase - base;
 
    QPointF newResultPoint = pt + offset;
 
#ifdef HAVE_GSL
 
    ElasticMotionData data;
    data.newBasePos = newBase;
    data.oldBasePos = base;
    data.anchorPoints = anchorPoints;
    data.oldResultPoint = pt;
 
    const gsl_multimin_fminimizer_type *T =
        gsl_multimin_fminimizer_nmsimplex2;
    gsl_multimin_fminimizer *s = 0;
    gsl_vector *ss, *x;
    gsl_multimin_function minex_func;
 
    size_t iter = 0;
    int status;
    double size;
 
    /* Starting point */
    x = gsl_vector_alloc (2);
    gsl_vector_set (x, 0, newResultPoint.x());
    gsl_vector_set (x, 1, newResultPoint.y());
 
    /* Set initial step sizes to 0.1 */
    ss = gsl_vector_alloc (2);
    gsl_vector_set (ss, 0, 0.1 * offset.x());
    gsl_vector_set (ss, 1, 0.1 * offset.y());
 
    /* Initialize method and iterate */
    minex_func.n = 2;
    minex_func.f = elasticMotionError;
    minex_func.params = (void*)&data;
 
    s = gsl_multimin_fminimizer_alloc (T, 2);
    gsl_multimin_fminimizer_set (s, &minex_func, x, ss);
 
    do
    {
        iter++;
        status = gsl_multimin_fminimizer_iterate(s);
 
        if (status)
            break;
 
        size = gsl_multimin_fminimizer_size (s);
        status = gsl_multimin_test_size (size, 1e-6);
 
        if (status == GSL_SUCCESS && elasticMotionError(s->x, &data) > 0.5) {
            status = GSL_CONTINUE;
        }
 
        if (status == GSL_SUCCESS)
        {
//             qDebug() << "*******Converged to minimum";
//             qDebug() << gsl_vector_get (s->x, 0)
//                      << gsl_vector_get (s->x, 1)
//                      << "|" << s->fval << size;
//             qDebug() << ppVar(iter);
 
             newResultPoint.rx() = gsl_vector_get (s->x, 0);
             newResultPoint.ry() = gsl_vector_get (s->x, 1);
        }
    }
    while (status == GSL_CONTINUE && iter < 10000);
 
    if (status != GSL_SUCCESS) {
        ENTER_FUNCTION() << "failed to find point" << ppVar(pt) << ppVar(base) << ppVar(newBase);
    }
 
    gsl_vector_free(x);
    gsl_vector_free(ss);
    gsl_multimin_fminimizer_free (s);
#endif
 
    return newResultPoint;
}
 
void cropLineToRect(QLineF &line, const QRect rect, bool extendFirst, bool extendSecond)
{
    bool intersects = intersectLineRect(line, rect, extendFirst, extendSecond);
    if (!intersects) {
        line = QLineF(); // empty line to help with drawing
    }
}
 
void cropLineToConvexPolygon(QLineF &line, const QPolygonF polygon, bool extendFirst, bool extendSecond)
{
    bool intersects = intersectLineConvexPolygon(line, polygon, extendFirst, extendSecond);
    if (!intersects) {
        line = QLineF(); // empty line to help with drawing
    }
}
 
 
qreal findMinimumGoldenSection(std::function<qreal(qreal)> f, qreal xA, qreal xB, qreal eps, int maxIter = 100)
{
    // requirements:
    // only one local minimum between xA and xB
 
    int i = 0;
    const double phi = 0.618033988749894; // 1/(golden ratio)
    qreal a = xA;
    qreal b = xB;
    qreal c = b - (b - a)*phi;
    qreal d = a + (b - a)*phi;
 
    while (qAbs(b - a) > eps) {
        if (f(c) < f(d)) {
            b = d;
        } else {
            a = c;
        }
 
        // Wikipedia says to recompute both c and d here to avoid loss of precision which may lead to incorrect results or infinite loop
        c = b - (b - a) * phi;
        d = a + (b - a) * phi;
 
        i++;
        if (i > maxIter) {
            break;
        }
 
    }
 
    return (b + a)/2;
}
 
qreal findMinimumTernarySection(std::function<qreal(qreal)> f, qreal xA, qreal xB, qreal eps, int maxIter = 100)
{
    // requirements:
    // only one local minimum between xA and xB
 
    int i = 0;
    qreal l = qMin(xA, xB);
    qreal r = qMax(xA, xB);
 
 
    qreal m1 = l + (r - l)/3;
    qreal m2 = r - (r - l)/3;
 
    while ((r - l) > eps) {
        qreal f1 = f(m1);
        qreal f2 = f(m2);
 
        if (f1 > f2) {
            l = m1;
        } else {
            r = m2;
        }
 
        // In Golden Section, Wikipedia says to recompute both c and d here to avoid loss of precision which may lead to incorrect results or infinite loop
        // so it's probably a good idea to do it here too
        m1 = l + (r - l)/3;
        m2 = r - (r - l)/3;
 
        i++;
 
        if (i > maxIter) {
            break;
        }
    }
 
    return (l + r)/2;
}
 
qreal pointToLineDistSquared(const QPointF &pt, const QLineF &line)
{
    // distance = |(p2 - p1) x (p1 - pt)| / |p2 - p1|
 
    // magnitude of (p2 - p1) x (p1 - pt)
    const qreal cross = (line.dx() * (line.y1() - pt.y()) - line.dy() * (line.x1() - pt.x()));
 
    return cross * cross / (line.dx() * line.dx() + line.dy() * line.dy());
}
 
}