qtransform.cpp

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// Copyright (C) 2021 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 "qtransform.h"
 
#include "qdatastream.h"
#include "qdebug.h"
#include "qhashfunctions.h"
#include "qregion.h"
#include "qpainterpath.h"
#include "qpainterpath_p.h"
#include "qvariant.h"
#include "qmath_p.h"
#include <qnumeric.h>
 
#include <private/qbezier_p.h>
 
QT_BEGIN_NAMESPACE
 
#ifndef QT_NO_DEBUG
Q_NEVER_INLINE
static void nanWarning(const char *func)
{
    qWarning("QTransform::%s with NaN called", func);
}
#endif // QT_NO_DEBUG
 
#define Q_NEAR_CLIP (sizeof(qreal) == sizeof(double) ? 0.000001 : 0.0001)
 
void QTransform::do_map(qreal x, qreal y, qreal &nx, qreal &ny) const
{
    const TransformationType t = inline_type();
    switch (t) {
    case QTransform::TxNone:
        nx = x;
        ny = y;
        return;
    case QTransform::TxTranslate:
        nx = x + m_matrix[2][0];
        ny = y + m_matrix[2][1];
        return;
    case QTransform::TxScale:
        nx = m_matrix[0][0] * x + m_matrix[2][0];
        ny = m_matrix[1][1] * y + m_matrix[2][1];
        return;
    case QTransform::TxRotate:
    case QTransform::TxShear:
    case QTransform::TxProject:
        nx = m_matrix[0][0] * x + m_matrix[1][0] * y + m_matrix[2][0];
        ny = m_matrix[0][1] * x + m_matrix[1][1] * y + m_matrix[2][1];
        if (t == QTransform::TxProject) {
            qreal w = (m_matrix[0][2] * x + m_matrix[1][2] * y + m_matrix[2][2]);
            if (w < qreal(Q_NEAR_CLIP)) w = qreal(Q_NEAR_CLIP);
            w = qreal(1.)/w;
            nx *= w;
            ny *= w;
        }
        return;
    }
    Q_UNREACHABLE_RETURN();
}
 
/*!
    \class QTransform
    \brief The QTransform class specifies 2D transformations of a coordinate system.
    \since 4.3
    \ingroup painting
    \inmodule QtGui
 
    A transformation specifies how to translate, scale, shear, rotate
    or project the coordinate system, and is typically used when
    rendering graphics.
 
    A QTransform object can be built using the setMatrix(), scale(),
    rotate(), translate() and shear() functions.  Alternatively, it
    can be built by applying \l {QTransform#Basic Matrix
    Operations}{basic matrix operations}. The matrix can also be
    defined when constructed, and it can be reset to the identity
    matrix (the default) using the reset() function.
 
    The QTransform class supports mapping of graphic primitives: A given
    point, line, polygon, region, or painter path can be mapped to the
    coordinate system defined by \e this matrix using the map()
    function. In case of a rectangle, its coordinates can be
    transformed using the mapRect() function. A rectangle can also be
    transformed into a \e polygon (mapped to the coordinate system
    defined by \e this matrix), using the mapToPolygon() function.
 
    QTransform provides the isIdentity() function which returns \c true if
    the matrix is the identity matrix, and the isInvertible() function
    which returns \c true if the matrix is non-singular (i.e. AB = BA =
    I). The inverted() function returns an inverted copy of \e this
    matrix if it is invertible (otherwise it returns the identity
    matrix), and adjoint() returns the matrix's classical adjoint.
    In addition, QTransform provides the determinant() function which
    returns the matrix's determinant.
 
    Finally, the QTransform class supports matrix multiplication, addition
    and subtraction, and objects of the class can be streamed as well
    as compared.
 
    \section1 Rendering Graphics
 
    When rendering graphics, the matrix defines the transformations
    but the actual transformation is performed by the drawing routines
    in QPainter.
 
    By default, QPainter operates on the associated device's own
    coordinate system.  The standard coordinate system of a
    QPaintDevice has its origin located at the top-left position. The
    \e x values increase to the right; \e y values increase
    downward. For a complete description, see the \l {Coordinate
    System} {coordinate system} documentation.
 
    QPainter has functions to translate, scale, shear and rotate the
    coordinate system without using a QTransform. For example:
 
    \table 100%
    \row
    \li \inlineimage qtransform-simpletransformation.png
    \li
    \snippet transform/main.cpp 0
    \endtable
 
    Although these functions are very convenient, it can be more
    efficient to build a QTransform and call QPainter::setTransform() if you
    want to perform more than a single transform operation. For
    example:
 
    \table 100%
    \row
    \li \inlineimage qtransform-combinedtransformation.png
    \li
    \snippet transform/main.cpp 1
    \endtable
 
    \section1 Basic Matrix Operations
 
    \image qtransform-representation.png {Illustration showing an object that
           contains a 3x3 matrix where every row is a different element}
 
    A QTransform object contains a 3 x 3 matrix.  The \c m31 (\c dx) and
    \c m32 (\c dy) elements specify horizontal and vertical translation.
    The \c m11 and \c m22 elements specify horizontal and vertical scaling.
    The \c m21 and \c m12 elements specify horizontal and vertical \e shearing.
    And finally, the \c m13 and \c m23 elements specify horizontal and vertical
    projection, with \c m33 as an additional projection factor.
 
    QTransform transforms a point in the plane to another point using the
    following formulas:
 
    \snippet code/src_gui_painting_qtransform.cpp 0
 
    The point \e (x, y) is the original point, and \e (x', y') is the
    transformed point. \e (x', y') can be transformed back to \e (x,
    y) by performing the same operation on the inverted() matrix.
 
    The various matrix elements can be set when constructing the
    matrix, or by using the setMatrix() function later on. They can also
    be manipulated using the translate(), rotate(), scale() and
    shear() convenience functions. The currently set values can be
    retrieved using the m11(), m12(), m13(), m21(), m22(), m23(),
    m31(), m32(), m33(), dx() and dy() functions.
 
    Translation is the simplest transformation. Setting \c dx and \c
    dy will move the coordinate system \c dx units along the X axis
    and \c dy units along the Y axis.  Scaling can be done by setting
    \c m11 and \c m22. For example, setting \c m11 to 2 and \c m22 to
    1.5 will double the height and increase the width by 50%.  The
    identity matrix has \c m11, \c m22, and \c m33 set to 1 (all others are set
    to 0) mapping a point to itself. Shearing is controlled by \c m12
    and \c m21. Setting these elements to values different from zero
    will twist the coordinate system. Rotation is achieved by
    setting both the shearing factors and the scaling factors. Perspective
    transformation is achieved by setting both the projection factors and
    the scaling factors.
 
    \section2 Combining Transforms
    Here's the combined transformations example using basic matrix
    operations:
 
    \table 100%
    \row
    \li \inlineimage qtransform-combinedtransformation2.png
    \li
    \snippet transform/main.cpp 2
    \endtable
 
    The combined transform first scales each operand, then rotates it, and
    finally translates it, just as in the order in which the product of its
    factors is written. This means the point to which the transforms are
    applied is implicitly multiplied on the left with the transform
    to its right.
 
    \section2 Relation to Matrix Notation
    The matrix notation in QTransform is the transpose of a commonly-taught
    convention which represents transforms and points as matrices and vectors.
    That convention multiplies its matrix on the left and column vector to the
    right. In other words, when several transforms are applied to a point, the
    right-most matrix acts directly on the vector first. Then the next matrix
    to the left acts on the result of the first operation - and so on. As a
    result, that convention multiplies the matrices that make up a composite
    transform in the reverse of the order in QTransform, as you can see in
    \l {Combining Transforms}. Transposing the matrices, and combining them to
    the right of a row vector that represents the point, lets the matrices of
    transforms appear, in their product, in the order in which we think of the
    transforms being applied to the point.
 
    \sa QPainter, {Coordinate System}, {painting/affine}{Affine
    Transformations Example}, {Transformations Example}
*/
 
/*!
    \enum QTransform::TransformationType
 
    \value TxNone
    \value TxTranslate
    \value TxScale
    \value TxRotate
    \value TxShear
    \value TxProject
*/
 
/*!
    \fn QTransform::QTransform(Qt::Initialization)
    \internal
*/
 
/*!
    \fn QTransform::QTransform()
 
    Constructs an identity matrix.
 
    All elements are set to zero except \c m11 and \c m22 (specifying
    the scale) and \c m33 which are set to 1.
 
    \sa reset()
*/
 
/*!
    \fn QTransform::QTransform(qreal m11, qreal m12, qreal m13, qreal m21, qreal m22, qreal m23, qreal m31, qreal m32, qreal m33)
 
    Constructs a matrix with the elements, \a m11, \a m12, \a m13,
    \a m21, \a m22, \a m23, \a m31, \a m32, \a m33.
 
    \sa setMatrix()
*/
 
/*!
    \fn QTransform::QTransform(qreal m11, qreal m12, qreal m21, qreal m22, qreal dx, qreal dy)
 
    Constructs a matrix with the elements, \a m11, \a m12, \a m21, \a m22, \a dx and \a dy.
 
    \sa setMatrix()
*/
 
/*!
    Returns the adjoint of this matrix.
*/
QTransform QTransform::adjoint() const
{
    qreal h11, h12, h13,
          h21, h22, h23,
          h31, h32, h33;
    h11 = m_matrix[1][1] * m_matrix[2][2] - m_matrix[1][2] * m_matrix[2][1];
    h21 = m_matrix[1][2] * m_matrix[2][0] - m_matrix[1][0] * m_matrix[2][2];
    h31 = m_matrix[1][0] * m_matrix[2][1] - m_matrix[1][1] * m_matrix[2][0];
    h12 = m_matrix[0][2] * m_matrix[2][1] - m_matrix[0][1] * m_matrix[2][2];
    h22 = m_matrix[0][0] * m_matrix[2][2] - m_matrix[0][2] * m_matrix[2][0];
    h32 = m_matrix[0][1] * m_matrix[2][0] - m_matrix[0][0] * m_matrix[2][1];
    h13 = m_matrix[0][1] * m_matrix[1][2] - m_matrix[0][2] * m_matrix[1][1];
    h23 = m_matrix[0][2] * m_matrix[1][0] - m_matrix[0][0] * m_matrix[1][2];
    h33 = m_matrix[0][0] * m_matrix[1][1] - m_matrix[0][1] * m_matrix[1][0];
 
    return QTransform(h11, h12, h13,
                      h21, h22, h23,
                      h31, h32, h33);
}
 
/*!
    Returns the transpose of this matrix.
*/
QTransform QTransform::transposed() const
{
    QTransform t(m_matrix[0][0], m_matrix[1][0], m_matrix[2][0],
                 m_matrix[0][1], m_matrix[1][1], m_matrix[2][1],
                 m_matrix[0][2], m_matrix[1][2], m_matrix[2][2]);
    return t;
}
 
/*!
    Returns an inverted copy of this matrix.
 
    If the matrix is singular (not invertible), the returned matrix is
    the identity matrix. If \a invertible is valid (i.e. not 0), its
    value is set to true if the matrix is invertible, otherwise it is
    set to false.
 
    \sa isInvertible()
*/
QTransform QTransform::inverted(bool *invertible) const
{
    QTransform invert;
    bool inv = true;
 
    switch(inline_type()) {
    case TxNone:
        break;
    case TxTranslate:
        invert.m_matrix[2][0] = -m_matrix[2][0];
        invert.m_matrix[2][1] = -m_matrix[2][1];
        break;
    case TxScale:
        inv = !qFuzzyIsNull(m_matrix[0][0]);
        inv &= !qFuzzyIsNull(m_matrix[1][1]);
        if (inv) {
            invert.m_matrix[0][0] = 1. / m_matrix[0][0];
            invert.m_matrix[1][1] = 1. / m_matrix[1][1];
            invert.m_matrix[2][0] = -m_matrix[2][0] * invert.m_matrix[0][0];
            invert.m_matrix[2][1] = -m_matrix[2][1] * invert.m_matrix[1][1];
        }
        break;
    default:
        // general case
        qreal det = determinant();
        inv = !qFuzzyIsNull(det);
        if (inv)
            invert = adjoint() / det;
        break;
    }
 
    if (invertible)
        *invertible = inv;
 
    if (inv) {
        // inverting doesn't change the type
        invert.m_type = m_type;
        invert.m_dirty = m_dirty;
    }
 
    return invert;
}
 
/*!
    Moves the coordinate system \a dx along the x axis and \a dy along
    the y axis, and returns a reference to the matrix.
 
    \sa setMatrix()
*/
QTransform &QTransform::translate(qreal dx, qreal dy)
{
    if (dx == 0 && dy == 0)
        return *this;
#ifndef QT_NO_DEBUG
    if (qIsNaN(dx) || qIsNaN(dy)) {
        nanWarning("translate");
        return *this;
    }
#endif
 
    switch(inline_type()) {
    case TxNone:
        m_matrix[2][0] = dx;
        m_matrix[2][1] = dy;
        break;
    case TxTranslate:
        m_matrix[2][0] += dx;
        m_matrix[2][1] += dy;
        break;
    case TxScale:
        m_matrix[2][0] += dx * m_matrix[0][0];
        m_matrix[2][1] += dy * m_matrix[1][1];
        break;
    case TxProject:
        m_matrix[2][2] += dx * m_matrix[0][2] + dy * m_matrix[1][2];
        Q_FALLTHROUGH();
    case TxShear:
    case TxRotate:
        m_matrix[2][0] += dx * m_matrix[0][0] + dy * m_matrix[1][0];
        m_matrix[2][1] += dy * m_matrix[1][1] + dx * m_matrix[0][1];
        break;
    }
    if (m_dirty < TxTranslate)
        m_dirty = TxTranslate;
    return *this;
}
 
/*!
    Creates a matrix which corresponds to a translation of \a dx along
    the x axis and \a dy along the y axis. This is the same as
    QTransform().translate(dx, dy) but slightly faster.
 
    \since 4.5
*/
QTransform QTransform::fromTranslate(qreal dx, qreal dy)
{
#ifndef QT_NO_DEBUG
    if (qIsNaN(dx) || qIsNaN(dy)) {
        nanWarning("fromTranslate");
        return QTransform();
}
#endif
    QTransform transform(1, 0, 0, 0, 1, 0, dx, dy, 1);
    if (dx == 0 && dy == 0)
        transform.m_type = TxNone;
    else
        transform.m_type = TxTranslate;
    transform.m_dirty = TxNone;
    return transform;
}
 
/*!
    Scales the coordinate system by \a sx horizontally and \a sy
    vertically, and returns a reference to the matrix.
 
    \sa setMatrix()
*/
QTransform & QTransform::scale(qreal sx, qreal sy)
{
    if (sx == 1 && sy == 1)
        return *this;
#ifndef QT_NO_DEBUG
    if (qIsNaN(sx) || qIsNaN(sy)) {
        nanWarning("scale");
        return *this;
    }
#endif
 
    switch(inline_type()) {
    case TxNone:
    case TxTranslate:
        m_matrix[0][0] = sx;
        m_matrix[1][1] = sy;
        break;
    case TxProject:
        m_matrix[0][2] *= sx;
        m_matrix[1][2] *= sy;
        Q_FALLTHROUGH();
    case TxRotate:
    case TxShear:
        m_matrix[0][1] *= sx;
        m_matrix[1][0] *= sy;
        Q_FALLTHROUGH();
    case TxScale:
        m_matrix[0][0] *= sx;
        m_matrix[1][1] *= sy;
        break;
    }
    if (m_dirty < TxScale)
        m_dirty = TxScale;
    return *this;
}
 
/*!
    Creates a matrix which corresponds to a scaling of
    \a sx horizontally and \a sy vertically.
    This is the same as QTransform().scale(sx, sy) but slightly faster.
 
    \since 4.5
*/
QTransform QTransform::fromScale(qreal sx, qreal sy)
{
#ifndef QT_NO_DEBUG
    if (qIsNaN(sx) || qIsNaN(sy)) {
        nanWarning("fromScale");
        return QTransform();
}
#endif
    QTransform transform(sx, 0, 0, 0, sy, 0, 0, 0, 1);
    if (sx == 1. && sy == 1.)
        transform.m_type = TxNone;
    else
        transform.m_type = TxScale;
    transform.m_dirty = TxNone;
    return transform;
}
 
/*!
    Shears the coordinate system by \a sh horizontally and \a sv
    vertically, and returns a reference to the matrix.
 
    \sa setMatrix()
*/
QTransform & QTransform::shear(qreal sh, qreal sv)
{
    if (sh == 0 && sv == 0)
        return *this;
#ifndef QT_NO_DEBUG
    if (qIsNaN(sh) || qIsNaN(sv)) {
        nanWarning("shear");
        return *this;
    }
#endif
 
    switch(inline_type()) {
    case TxNone:
    case TxTranslate:
        m_matrix[0][1] = sv;
        m_matrix[1][0] = sh;
        break;
    case TxScale:
        m_matrix[0][1] = sv*m_matrix[1][1];
        m_matrix[1][0] = sh*m_matrix[0][0];
        break;
    case TxProject: {
        qreal tm13 = sv * m_matrix[1][2];
        qreal tm23 = sh * m_matrix[0][2];
        m_matrix[0][2] += tm13;
        m_matrix[1][2] += tm23;
    }
        Q_FALLTHROUGH();
    case TxRotate:
    case TxShear: {
        qreal tm11 = sv * m_matrix[1][0];
        qreal tm22 = sh * m_matrix[0][1];
        qreal tm12 = sv * m_matrix[1][1];
        qreal tm21 = sh * m_matrix[0][0];
        m_matrix[0][0] += tm11;
        m_matrix[0][1] += tm12;
        m_matrix[1][0] += tm21;
        m_matrix[1][1] += tm22;
        break;
    }
    }
    if (m_dirty < TxShear)
        m_dirty = TxShear;
    return *this;
}
 
/*!
    \since 6.5
 
    Rotates the coordinate system counterclockwise by the given angle \a a
    about the specified \a axis at distance \a distanceToPlane from the
    screen and returns a reference to the matrix.
 
//! [transform-rotate-note]
    Note that if you apply a QTransform to a point defined in widget
    coordinates, the direction of the rotation will be clockwise
    because the y-axis points downwards.
 
    The angle is specified in degrees.
//! [transform-rotate-note]
 
    If \a distanceToPlane is zero, it will be ignored. This is suitable
    for implementing orthographic projections where the z coordinate should
    be dropped rather than projected.
 
    \sa setMatrix()
*/
QTransform & QTransform::rotate(qreal a, Qt::Axis axis, qreal distanceToPlane)
{
    if (a == 0)
        return *this;
#ifndef QT_NO_DEBUG
    if (qIsNaN(a) || qIsNaN(distanceToPlane)) {
        nanWarning("rotate");
        return *this;
    }
#endif
 
    qreal sina = 0;
    qreal cosa = 0;
    if (a == 90. || a == -270.)
        sina = 1.;
    else if (a == 270. || a == -90.)
        sina = -1.;
    else if (a == 180.)
        cosa = -1.;
    else{
        qreal b = qDegreesToRadians(a);
        sina = qSin(b);               // fast and convenient
        cosa = qCos(b);
    }
 
    if (axis == Qt::ZAxis) {
        switch(inline_type()) {
        case TxNone:
        case TxTranslate:
            m_matrix[0][0] = cosa;
            m_matrix[0][1] = sina;
            m_matrix[1][0] = -sina;
            m_matrix[1][1] = cosa;
            break;
        case TxScale: {
            qreal tm11 = cosa * m_matrix[0][0];
            qreal tm12 = sina * m_matrix[1][1];
            qreal tm21 = -sina * m_matrix[0][0];
            qreal tm22 = cosa * m_matrix[1][1];
            m_matrix[0][0] = tm11;
            m_matrix[0][1] = tm12;
            m_matrix[1][0] = tm21;
            m_matrix[1][1] = tm22;
            break;
        }
        case TxProject: {
            qreal tm13 = cosa * m_matrix[0][2] + sina * m_matrix[1][2];
            qreal tm23 = -sina * m_matrix[0][2] + cosa * m_matrix[1][2];
            m_matrix[0][2] = tm13;
            m_matrix[1][2] = tm23;
            Q_FALLTHROUGH();
        }
        case TxRotate:
        case TxShear: {
            qreal tm11 = cosa * m_matrix[0][0] + sina * m_matrix[1][0];
            qreal tm12 = cosa * m_matrix[0][1] + sina * m_matrix[1][1];
            qreal tm21 = -sina * m_matrix[0][0] + cosa * m_matrix[1][0];
            qreal tm22 = -sina * m_matrix[0][1] + cosa * m_matrix[1][1];
            m_matrix[0][0] = tm11;
            m_matrix[0][1] = tm12;
            m_matrix[1][0] = tm21;
            m_matrix[1][1] = tm22;
            break;
        }
        }
        if (m_dirty < TxRotate)
            m_dirty = TxRotate;
    } else {
        if (!qIsNull(distanceToPlane))
            sina /= distanceToPlane;
 
        QTransform result;
        if (axis == Qt::YAxis) {
            result.m_matrix[0][0] = cosa;
            result.m_matrix[0][2] = -sina;
        } else {
            result.m_matrix[1][1] = cosa;
            result.m_matrix[1][2] = -sina;
        }
        result.m_type = TxProject;
        *this = result * *this;
    }
 
    return *this;
}
 
#if QT_VERSION < QT_VERSION_CHECK(7, 0, 0)
/*!
    \overload
 
    Rotates the coordinate system counterclockwise by the given angle \a a
    about the specified \a axis at distance 1024.0 from the screen and
    returns a reference to the matrix.
 
    \include qtransform.cpp transform-rotate-note
 
    \sa setMatrix
*/
QTransform &QTransform::rotate(qreal a, Qt::Axis axis)
{
    return rotate(a, axis, 1024.0);
}
#endif
 
/*!
    \since 6.5
 
    Rotates the coordinate system counterclockwise by the given angle \a a
    about the specified \a axis at distance \a distanceToPlane from the
    screen and returns a reference to the matrix.
 
//! [transform-rotate-radians-note]
    Note that if you apply a QTransform to a point defined in widget
    coordinates, the direction of the rotation will be clockwise
    because the y-axis points downwards.
 
    The angle is specified in radians.
//! [transform-rotate-radians-note]
 
    If \a distanceToPlane is zero, it will be ignored. This is suitable
    for implementing orthographic projections where the z coordinate should
    be dropped rather than projected.
 
    \sa setMatrix()
*/
QTransform & QTransform::rotateRadians(qreal a, Qt::Axis axis, qreal distanceToPlane)
{
#ifndef QT_NO_DEBUG
    if (qIsNaN(a) || qIsNaN(distanceToPlane)) {
        nanWarning("rotateRadians");
        return *this;
    }
#endif
    qreal sina = qSin(a);
    qreal cosa = qCos(a);
 
    if (axis == Qt::ZAxis) {
        switch(inline_type()) {
        case TxNone:
        case TxTranslate:
            m_matrix[0][0] = cosa;
            m_matrix[0][1] = sina;
            m_matrix[1][0] = -sina;
            m_matrix[1][1] = cosa;
            break;
        case TxScale: {
            qreal tm11 = cosa * m_matrix[0][0];
            qreal tm12 = sina * m_matrix[1][1];
            qreal tm21 = -sina * m_matrix[0][0];
            qreal tm22 = cosa * m_matrix[1][1];
            m_matrix[0][0] = tm11;
            m_matrix[0][1] = tm12;
            m_matrix[1][0] = tm21;
            m_matrix[1][1] = tm22;
            break;
        }
        case TxProject: {
            qreal tm13 = cosa * m_matrix[0][2] + sina * m_matrix[1][2];
            qreal tm23 = -sina * m_matrix[0][2] + cosa * m_matrix[1][2];
            m_matrix[0][2] = tm13;
            m_matrix[1][2] = tm23;
            Q_FALLTHROUGH();
        }
        case TxRotate:
        case TxShear: {
            qreal tm11 = cosa * m_matrix[0][0] + sina * m_matrix[1][0];
            qreal tm12 = cosa * m_matrix[0][1] + sina * m_matrix[1][1];
            qreal tm21 = -sina * m_matrix[0][0] + cosa * m_matrix[1][0];
            qreal tm22 = -sina * m_matrix[0][1] + cosa * m_matrix[1][1];
            m_matrix[0][0] = tm11;
            m_matrix[0][1] = tm12;
            m_matrix[1][0] = tm21;
            m_matrix[1][1] = tm22;
            break;
        }
        }
        if (m_dirty < TxRotate)
            m_dirty = TxRotate;
    } else {
        if (!qIsNull(distanceToPlane))
            sina /= distanceToPlane;
 
        QTransform result;
        if (axis == Qt::YAxis) {
            result.m_matrix[0][0] = cosa;
            result.m_matrix[0][2] = -sina;
        } else {
            result.m_matrix[1][1] = cosa;
            result.m_matrix[1][2] = -sina;
        }
        result.m_type = TxProject;
        *this = result * *this;
    }
    return *this;
}
 
#if QT_VERSION < QT_VERSION_CHECK(7, 0, 0)
/*!
    \overload
 
    Rotates the coordinate system counterclockwise by the given angle \a a
    about the specified \a axis at distance 1024.0 from the screen and
    returns a reference to the matrix.
 
    \include qtransform.cpp transform-rotate-radians-note
 
    \sa setMatrix()
*/
QTransform &QTransform::rotateRadians(qreal a, Qt::Axis axis)
{
    return rotateRadians(a, axis, 1024.0);
}
#endif
 
/*!
    \fn bool QTransform::operator==(const QTransform &matrix) const
    Returns \c true if this matrix is equal to the given \a matrix,
    otherwise returns \c false.
*/
bool QTransform::operator==(const QTransform &o) const
{
    return m_matrix[0][0] == o.m_matrix[0][0] &&
           m_matrix[0][1] == o.m_matrix[0][1] &&
           m_matrix[1][0] == o.m_matrix[1][0] &&
           m_matrix[1][1] == o.m_matrix[1][1] &&
           m_matrix[2][0] == o.m_matrix[2][0] &&
           m_matrix[2][1] == o.m_matrix[2][1] &&
           m_matrix[0][2] == o.m_matrix[0][2] &&
           m_matrix[1][2] == o.m_matrix[1][2] &&
           m_matrix[2][2] == o.m_matrix[2][2];
}
 
/*!
    \since 5.6
    \qhashold{QTransform}
*/
size_t qHash(const QTransform &key, size_t seed) noexcept
{
    QtPrivate::QHashCombineWithSeed hash(seed);
    seed = hash(seed, key.m11());
    seed = hash(seed, key.m12());
    seed = hash(seed, key.m21());
    seed = hash(seed, key.m22());
    seed = hash(seed, key.dx());
    seed = hash(seed, key.dy());
    seed = hash(seed, key.m13());
    seed = hash(seed, key.m23());
    seed = hash(seed, key.m33());
    return seed;
}
 
 
/*!
    \fn bool QTransform::operator!=(const QTransform &matrix) const
    Returns \c true if this matrix is not equal to the given \a matrix,
    otherwise returns \c false.
*/
bool QTransform::operator!=(const QTransform &o) const
{
    return !operator==(o);
}
 
/*!
    \fn QTransform & QTransform::operator*=(const QTransform &matrix)
    \overload
 
    Returns the result of multiplying this matrix by the given \a
    matrix.
*/
QTransform & QTransform::operator*=(const QTransform &o)
{
    const TransformationType otherType = o.inline_type();
    if (otherType == TxNone)
        return *this;
 
    const TransformationType thisType = inline_type();
    if (thisType == TxNone)
        return operator=(o);
 
    TransformationType t = qMax(thisType, otherType);
    switch(t) {
    case TxNone:
        break;
    case TxTranslate:
        m_matrix[2][0] += o.m_matrix[2][0];
        m_matrix[2][1] += o.m_matrix[2][1];
        break;
    case TxScale:
    {
        qreal m11 = m_matrix[0][0] * o.m_matrix[0][0];
        qreal m22 = m_matrix[1][1] * o.m_matrix[1][1];
 
        qreal m31 = m_matrix[2][0] * o.m_matrix[0][0] + o.m_matrix[2][0];
        qreal m32 = m_matrix[2][1] * o.m_matrix[1][1] + o.m_matrix[2][1];
 
        m_matrix[0][0] = m11;
        m_matrix[1][1] = m22;
        m_matrix[2][0] = m31; m_matrix[2][1] = m32;
        break;
    }
    case TxRotate:
    case TxShear:
    {
        qreal m11 = m_matrix[0][0] * o.m_matrix[0][0] + m_matrix[0][1] * o.m_matrix[1][0];
        qreal m12 = m_matrix[0][0] * o.m_matrix[0][1] + m_matrix[0][1] * o.m_matrix[1][1];
 
        qreal m21 = m_matrix[1][0] * o.m_matrix[0][0] + m_matrix[1][1] * o.m_matrix[1][0];
        qreal m22 = m_matrix[1][0] * o.m_matrix[0][1] + m_matrix[1][1] * o.m_matrix[1][1];
 
        qreal m31 = m_matrix[2][0] * o.m_matrix[0][0] + m_matrix[2][1] * o.m_matrix[1][0] + o.m_matrix[2][0];
        qreal m32 = m_matrix[2][0] * o.m_matrix[0][1] + m_matrix[2][1] * o.m_matrix[1][1] + o.m_matrix[2][1];
 
        m_matrix[0][0] = m11;
        m_matrix[0][1] = m12;
        m_matrix[1][0] = m21;
        m_matrix[1][1] = m22;
        m_matrix[2][0] = m31;
        m_matrix[2][1] = m32;
        break;
    }
    case TxProject:
    {
        qreal m11 = m_matrix[0][0] * o.m_matrix[0][0] + m_matrix[0][1] * o.m_matrix[1][0] + m_matrix[0][2] * o.m_matrix[2][0];
        qreal m12 = m_matrix[0][0] * o.m_matrix[0][1] + m_matrix[0][1] * o.m_matrix[1][1] + m_matrix[0][2] * o.m_matrix[2][1];
        qreal m13 = m_matrix[0][0] * o.m_matrix[0][2] + m_matrix[0][1] * o.m_matrix[1][2] + m_matrix[0][2] * o.m_matrix[2][2];
 
        qreal m21 = m_matrix[1][0] * o.m_matrix[0][0] + m_matrix[1][1] * o.m_matrix[1][0] + m_matrix[1][2] * o.m_matrix[2][0];
        qreal m22 = m_matrix[1][0] * o.m_matrix[0][1] + m_matrix[1][1] * o.m_matrix[1][1] + m_matrix[1][2] * o.m_matrix[2][1];
        qreal m23 = m_matrix[1][0] * o.m_matrix[0][2] + m_matrix[1][1] * o.m_matrix[1][2] + m_matrix[1][2] * o.m_matrix[2][2];
 
        qreal m31 = m_matrix[2][0] * o.m_matrix[0][0] + m_matrix[2][1] * o.m_matrix[1][0] + m_matrix[2][2] * o.m_matrix[2][0];
        qreal m32 = m_matrix[2][0] * o.m_matrix[0][1] + m_matrix[2][1] * o.m_matrix[1][1] + m_matrix[2][2] * o.m_matrix[2][1];
        qreal m33 = m_matrix[2][0] * o.m_matrix[0][2] + m_matrix[2][1] * o.m_matrix[1][2] + m_matrix[2][2] * o.m_matrix[2][2];
 
        m_matrix[0][0] = m11; m_matrix[0][1] = m12; m_matrix[0][2] = m13;
        m_matrix[1][0] = m21; m_matrix[1][1] = m22; m_matrix[1][2] = m23;
        m_matrix[2][0] = m31; m_matrix[2][1] = m32; m_matrix[2][2] = m33;
    }
    }
 
    m_dirty = t;
    m_type = t;
 
    return *this;
}
 
/*!
    \fn QTransform QTransform::operator*(const QTransform &matrix) const
    Returns the result of multiplying this matrix by the given \a
    matrix.
 
    Note that matrix multiplication is not commutative, i.e. a*b !=
    b*a.
*/
QTransform QTransform::operator*(const QTransform &m) const
{
    const TransformationType otherType = m.inline_type();
    if (otherType == TxNone)
        return *this;
 
    const TransformationType thisType = inline_type();
    if (thisType == TxNone)
        return m;
 
    QTransform t;
    TransformationType type = qMax(thisType, otherType);
    switch(type) {
    case TxNone:
        break;
    case TxTranslate:
        t.m_matrix[2][0] = m_matrix[2][0] + m.m_matrix[2][0];
        t.m_matrix[2][1] = m_matrix[2][1] + m.m_matrix[2][1];
        break;
    case TxScale:
    {
        qreal m11 = m_matrix[0][0] * m.m_matrix[0][0];
        qreal m22 = m_matrix[1][1] * m.m_matrix[1][1];
 
        qreal m31 = m_matrix[2][0] * m.m_matrix[0][0] + m.m_matrix[2][0];
        qreal m32 = m_matrix[2][1] * m.m_matrix[1][1] + m.m_matrix[2][1];
 
        t.m_matrix[0][0] = m11;
        t.m_matrix[1][1] = m22;
        t.m_matrix[2][0] = m31;
        t.m_matrix[2][1] = m32;
        break;
    }
    case TxRotate:
    case TxShear:
    {
        qreal m11 = m_matrix[0][0] * m.m_matrix[0][0] + m_matrix[0][1] * m.m_matrix[1][0];
        qreal m12 = m_matrix[0][0] * m.m_matrix[0][1] + m_matrix[0][1] * m.m_matrix[1][1];
 
        qreal m21 = m_matrix[1][0] * m.m_matrix[0][0] + m_matrix[1][1] * m.m_matrix[1][0];
        qreal m22 = m_matrix[1][0] * m.m_matrix[0][1] + m_matrix[1][1] * m.m_matrix[1][1];
 
        qreal m31 = m_matrix[2][0] * m.m_matrix[0][0] + m_matrix[2][1] * m.m_matrix[1][0] + m.m_matrix[2][0];
        qreal m32 = m_matrix[2][0] * m.m_matrix[0][1] + m_matrix[2][1] * m.m_matrix[1][1] + m.m_matrix[2][1];
 
        t.m_matrix[0][0] = m11; t.m_matrix[0][1] = m12;
        t.m_matrix[1][0] = m21; t.m_matrix[1][1] = m22;
        t.m_matrix[2][0] = m31; t.m_matrix[2][1] = m32;
        break;
    }
    case TxProject:
    {
        qreal m11 = m_matrix[0][0] * m.m_matrix[0][0] + m_matrix[0][1] * m.m_matrix[1][0] + m_matrix[0][2] * m.m_matrix[2][0];
        qreal m12 = m_matrix[0][0] * m.m_matrix[0][1] + m_matrix[0][1] * m.m_matrix[1][1] + m_matrix[0][2] * m.m_matrix[2][1];
        qreal m13 = m_matrix[0][0] * m.m_matrix[0][2] + m_matrix[0][1] * m.m_matrix[1][2] + m_matrix[0][2] * m.m_matrix[2][2];
 
        qreal m21 = m_matrix[1][0] * m.m_matrix[0][0] + m_matrix[1][1] * m.m_matrix[1][0] + m_matrix[1][2] * m.m_matrix[2][0];
        qreal m22 = m_matrix[1][0] * m.m_matrix[0][1] + m_matrix[1][1] * m.m_matrix[1][1] + m_matrix[1][2] * m.m_matrix[2][1];
        qreal m23 = m_matrix[1][0] * m.m_matrix[0][2] + m_matrix[1][1] * m.m_matrix[1][2] + m_matrix[1][2] * m.m_matrix[2][2];
 
        qreal m31 = m_matrix[2][0] * m.m_matrix[0][0] + m_matrix[2][1] * m.m_matrix[1][0] + m_matrix[2][2] * m.m_matrix[2][0];
        qreal m32 = m_matrix[2][0] * m.m_matrix[0][1] + m_matrix[2][1] * m.m_matrix[1][1] + m_matrix[2][2] * m.m_matrix[2][1];
        qreal m33 = m_matrix[2][0] * m.m_matrix[0][2] + m_matrix[2][1] * m.m_matrix[1][2] + m_matrix[2][2] * m.m_matrix[2][2];
 
        t.m_matrix[0][0] = m11; t.m_matrix[0][1] = m12; t.m_matrix[0][2] = m13;
        t.m_matrix[1][0] = m21; t.m_matrix[1][1] = m22; t.m_matrix[1][2] = m23;
        t.m_matrix[2][0] = m31; t.m_matrix[2][1] = m32; t.m_matrix[2][2] = m33;
    }
    }
 
    t.m_dirty = type;
    t.m_type = type;
 
    return t;
}
 
/*!
    \fn QTransform & QTransform::operator*=(qreal scalar)
    \overload
 
    Returns the result of performing an element-wise multiplication of this
    matrix with the given \a scalar.
*/
 
/*!
    \fn QTransform & QTransform::operator/=(qreal scalar)
    \overload
 
    Returns the result of performing an element-wise division of this
    matrix by the given \a scalar.
*/
 
/*!
    \fn QTransform & QTransform::operator+=(qreal scalar)
    \overload
 
    Returns the matrix obtained by adding the given \a scalar to each
    element of this matrix.
*/
 
/*!
    \fn QTransform & QTransform::operator-=(qreal scalar)
    \overload
 
    Returns the matrix obtained by subtracting the given \a scalar from each
    element of this matrix.
*/
 
/*!
    \fn QTransform &QTransform::operator=(const QTransform &matrix) noexcept
 
    Assigns the given \a matrix's values to this matrix.
*/
 
/*!
    Resets the matrix to an identity matrix, i.e. all elements are set
    to zero, except \c m11 and \c m22 (specifying the scale) and \c m33
    which are set to 1.
 
    \sa QTransform(), isIdentity(), {QTransform#Basic Matrix
    Operations}{Basic Matrix Operations}
*/
void QTransform::reset()
{
    *this = QTransform();
}
 
#ifndef QT_NO_DATASTREAM
/*!
    \fn QDataStream &operator<<(QDataStream &stream, const QTransform &matrix)
    \since 4.3
    \relates QTransform
 
    Writes the given \a matrix to the given \a stream and returns a
    reference to the stream.
 
    \sa {Serializing Qt Data Types}
*/
QDataStream & operator<<(QDataStream &s, const QTransform &m)
{
    s << double(m.m11())
      << double(m.m12())
      << double(m.m13())
      << double(m.m21())
      << double(m.m22())
      << double(m.m23())
      << double(m.m31())
      << double(m.m32())
      << double(m.m33());
    return s;
}
 
/*!
    \fn QDataStream &operator>>(QDataStream &stream, QTransform &matrix)
    \since 4.3
    \relates QTransform
 
    Reads the given \a matrix from the given \a stream and returns a
    reference to the stream.
 
    \sa {Serializing Qt Data Types}
*/
QDataStream & operator>>(QDataStream &s, QTransform &t)
{
     double m11, m12, m13,
         m21, m22, m23,
         m31, m32, m33;
 
     s >> m11;
     s >> m12;
     s >> m13;
     s >> m21;
     s >> m22;
     s >> m23;
     s >> m31;
     s >> m32;
     s >> m33;
     t.setMatrix(m11, m12, m13,
                 m21, m22, m23,
                 m31, m32, m33);
     return s;
}
 
#endif // QT_NO_DATASTREAM
 
#ifndef QT_NO_DEBUG_STREAM
QDebug operator<<(QDebug dbg, const QTransform &m)
{
    static const char typeStr[][12] =
    {
        "TxNone",
        "TxTranslate",
        "TxScale",
        "",
        "TxRotate",
        "", "", "",
        "TxShear",
        "", "", "", "", "", "", "",
        "TxProject"
    };
 
    QDebugStateSaver saver(dbg);
    dbg.nospace() << "QTransform(type=" << typeStr[m.type()] << ','
                  << " 11=" << m.m11()
                  << " 12=" << m.m12()
                  << " 13=" << m.m13()
                  << " 21=" << m.m21()
                  << " 22=" << m.m22()
                  << " 23=" << m.m23()
                  << " 31=" << m.m31()
                  << " 32=" << m.m32()
                  << " 33=" << m.m33()
                  << ')';
 
    return dbg;
}
#endif
 
/*!
    \fn QPoint operator*(const QPoint &point, const QTransform &matrix)
    \relates QTransform
 
    This is the same as \a{matrix}.map(\a{point}).
 
    \sa QTransform::map()
*/
QPoint QTransform::map(const QPoint &p) const
{
    qreal fx = p.x();
    qreal fy = p.y();
 
    qreal x = 0, y = 0;
 
    do_map(fx, fy, x, y);
 
    return QPoint(qRound(x), qRound(y));
}
 
 
/*!
    \fn QPointF operator*(const QPointF &point, const QTransform &matrix)
    \relates QTransform
 
    Same as \a{matrix}.map(\a{point}).
 
    \sa QTransform::map()
*/
 
/*!
    \overload
 
    Creates and returns a QPointF object that is a copy of the given point,
    \a p, mapped into the coordinate system defined by this matrix.
*/
QPointF QTransform::map(const QPointF &p) const
{
    qreal fx = p.x();
    qreal fy = p.y();
 
    qreal x = 0, y = 0;
 
    do_map(fx, fy, x, y);
 
    return QPointF(x, y);
}
 
/*!
    \fn QPoint QTransform::map(const QPoint &point) const
    \overload
 
    Creates and returns a QPoint object that is a copy of the given \a
    point, mapped into the coordinate system defined by this
    matrix. Note that the transformed coordinates are rounded to the
    nearest integer.
*/
 
/*!
    \fn QLineF operator*(const QLineF &line, const QTransform &matrix)
    \relates QTransform
 
    This is the same as \a{matrix}.map(\a{line}).
 
    \sa QTransform::map()
*/
 
/*!
    \fn QLine operator*(const QLine &line, const QTransform &matrix)
    \relates QTransform
 
    This is the same as \a{matrix}.map(\a{line}).
 
    \sa QTransform::map()
*/
 
/*!
    \overload
 
    Creates and returns a QLineF object that is a copy of the given line,
    \a l, mapped into the coordinate system defined by this matrix.
*/
QLine QTransform::map(const QLine &l) const
{
    qreal fx1 = l.x1();
    qreal fy1 = l.y1();
    qreal fx2 = l.x2();
    qreal fy2 = l.y2();
 
    qreal x1 = 0, y1 = 0, x2 = 0, y2 = 0;
 
    do_map(fx1, fy1, x1, y1);
    do_map(fx2, fy2, x2, y2);
 
    return QLine(qRound(x1), qRound(y1), qRound(x2), qRound(y2));
}
 
/*!
    \overload
 
    \fn QLineF QTransform::map(const QLineF &line) const
 
    Creates and returns a QLine object that is a copy of the given \a
    line, mapped into the coordinate system defined by this matrix.
    Note that the transformed coordinates are rounded to the nearest
    integer.
*/
 
QLineF QTransform::map(const QLineF &l) const
{
    qreal fx1 = l.x1();
    qreal fy1 = l.y1();
    qreal fx2 = l.x2();
    qreal fy2 = l.y2();
 
    qreal x1 = 0, y1 = 0, x2 = 0, y2 = 0;
 
    do_map(fx1, fy1, x1, y1);
    do_map(fx2, fy2, x2, y2);
 
    return QLineF(x1, y1, x2, y2);
}
 
/*!
    \fn QPolygonF operator *(const QPolygonF &polygon, const QTransform &matrix)
    \since 4.3
    \relates QTransform
 
    This is the same as \a{matrix}.map(\a{polygon}).
 
    \sa QTransform::map()
*/
 
/*!
    \fn QPolygon operator*(const QPolygon &polygon, const QTransform &matrix)
    \relates QTransform
 
    This is the same as \a{matrix}.map(\a{polygon}).
 
    \sa QTransform::map()
*/
 
/*!
    \fn QPolygonF QTransform::map(const QPolygonF &polygon) const
    \overload
 
    Creates and returns a QPolygonF object that is a copy of the given
    \a polygon, mapped into the coordinate system defined by this
    matrix.
*/
QPolygonF QTransform::map(const QPolygonF &a) const
{
    TransformationType t = inline_type();
    if (t <= TxTranslate)
        return a.translated(m_matrix[2][0], m_matrix[2][1]);
 
    int size = a.size();
    int i;
    QPolygonF p(size);
    const QPointF *da = a.constData();
    QPointF *dp = p.data();
 
    for(i = 0; i < size; ++i) {
        do_map(da[i].x(), da[i].y(), dp[i].rx(), dp[i].ry());
    }
    return p;
}
 
/*!
    \fn QPolygon QTransform::map(const QPolygon &polygon) const
    \overload
 
    Creates and returns a QPolygon object that is a copy of the given
    \a polygon, mapped into the coordinate system defined by this
    matrix. Note that the transformed coordinates are rounded to the
    nearest integer.
*/
QPolygon QTransform::map(const QPolygon &a) const
{
    TransformationType t = inline_type();
    if (t <= TxTranslate)
        return a.translated(qRound(m_matrix[2][0]), qRound(m_matrix[2][1]));
 
    int size = a.size();
    int i;
    QPolygon p(size);
    const QPoint *da = a.constData();
    QPoint *dp = p.data();
 
    for(i = 0; i < size; ++i) {
        qreal nx = 0, ny = 0;
        do_map(da[i].x(), da[i].y(), nx, ny);
        dp[i].rx() = qRound(nx);
        dp[i].ry() = qRound(ny);
    }
    return p;
}
 
/*!
    \fn QRegion operator*(const QRegion &region, const QTransform &matrix)
    \relates QTransform
 
    This is the same as \a{matrix}.map(\a{region}).
 
    \sa QTransform::map()
*/
 
Q_GUI_EXPORT extern QPainterPath qt_regionToPath(const QRegion &region);
 
/*!
    \fn QRegion QTransform::map(const QRegion &region) const
    \overload
 
    Creates and returns a QRegion object that is a copy of the given
    \a region, mapped into the coordinate system defined by this matrix.
 
    Calling this method can be rather expensive if rotations or
    shearing are used.
*/
QRegion QTransform::map(const QRegion &r) const
{
    TransformationType t = inline_type();
    if (t == TxNone)
        return r;
 
    if (t == TxTranslate) {
        QRegion copy(r);
        copy.translate(qRound(m_matrix[2][0]), qRound(m_matrix[2][1]));
        return copy;
    }
 
    if (t == TxScale) {
        QRegion res;
        if (m11() < 0 || m22() < 0) {
            for (const QRect &rect : r)
                res += qt_mapFillRect(QRectF(rect), *this);
        } else {
            QVarLengthArray<QRect, 32> rects;
            rects.reserve(r.rectCount());
            for (const QRect &rect : r) {
                QRect nr = qt_mapFillRect(QRectF(rect), *this);
                if (!nr.isEmpty())
                    rects.append(nr);
            }
            res.setRects(rects.constData(), rects.size());
        }
        return res;
    }
 
    QPainterPath p = map(qt_regionToPath(r));
    return p.toFillPolygon().toPolygon();
}
 
struct QHomogeneousCoordinate
{
    qreal x;
    qreal y;
    qreal w;
 
    QHomogeneousCoordinate() {}
    QHomogeneousCoordinate(qreal x_, qreal y_, qreal w_) : x(x_), y(y_), w(w_) {}
 
    const QPointF toPoint() const {
        qreal iw = 1. / w;
        return QPointF(x * iw, y * iw);
    }
};
 
static inline QHomogeneousCoordinate mapHomogeneous(const QTransform &transform, const QPointF &p)
{
    QHomogeneousCoordinate c;
    c.x = transform.m11() * p.x() + transform.m21() * p.y() + transform.m31();
    c.y = transform.m12() * p.x() + transform.m22() * p.y() + transform.m32();
    c.w = transform.m13() * p.x() + transform.m23() * p.y() + transform.m33();
    return c;
}
 
static inline bool lineTo_clipped(QPainterPath &path, const QTransform &transform, const QPointF &a, const QPointF &b,
                                  bool needsMoveTo, bool needsLineTo = true)
{
    QHomogeneousCoordinate ha = mapHomogeneous(transform, a);
    QHomogeneousCoordinate hb = mapHomogeneous(transform, b);
 
    if (ha.w < Q_NEAR_CLIP && hb.w < Q_NEAR_CLIP)
        return false;
 
    if (hb.w < Q_NEAR_CLIP) {
        const qreal t = (Q_NEAR_CLIP - hb.w) / (ha.w - hb.w);
 
        hb.x += (ha.x - hb.x) * t;
        hb.y += (ha.y - hb.y) * t;
        hb.w = qreal(Q_NEAR_CLIP);
    } else if (ha.w < Q_NEAR_CLIP) {
        const qreal t = (Q_NEAR_CLIP - ha.w) / (hb.w - ha.w);
 
        ha.x += (hb.x - ha.x) * t;
        ha.y += (hb.y - ha.y) * t;
        ha.w = qreal(Q_NEAR_CLIP);
 
        const QPointF p = ha.toPoint();
        if (needsMoveTo) {
            path.moveTo(p);
            needsMoveTo = false;
        } else {
            path.lineTo(p);
        }
    }
 
    if (needsMoveTo)
        path.moveTo(ha.toPoint());
 
    if (needsLineTo)
        path.lineTo(hb.toPoint());
 
    return true;
}
Q_GUI_EXPORT bool qt_scaleForTransform(const QTransform &transform, qreal *scale);
 
static inline bool cubicTo_clipped(QPainterPath &path, const QTransform &transform, const QPointF &a, const QPointF &b, const QPointF &c, const QPointF &d, bool needsMoveTo)
{
    // Convert projective xformed curves to line
    // segments so they can be transformed more accurately
 
    qreal scale;
    qt_scaleForTransform(transform, &scale);
 
    qreal curveThreshold = scale == 0 ? qreal(0.25) : (qreal(0.25) / scale);
 
    QPolygonF segment = QBezier::fromPoints(a, b, c, d).toPolygon(curveThreshold);
 
    for (int i = 0; i < segment.size() - 1; ++i)
        if (lineTo_clipped(path, transform, segment.at(i), segment.at(i+1), needsMoveTo))
            needsMoveTo = false;
 
    return !needsMoveTo;
}
 
static QPainterPath mapProjective(const QTransform &transform, const QPainterPath &path)
{
    QPainterPath result;
 
    QPointF last;
    QPointF lastMoveTo;
    bool needsMoveTo = true;
    for (int i = 0; i < path.elementCount(); ++i) {
        switch (path.elementAt(i).type) {
        case QPainterPath::MoveToElement:
            if (i > 0 && lastMoveTo != last)
                lineTo_clipped(result, transform, last, lastMoveTo, needsMoveTo);
 
            lastMoveTo = path.elementAt(i);
            last = path.elementAt(i);
            needsMoveTo = true;
            break;
        case QPainterPath::LineToElement:
            if (lineTo_clipped(result, transform, last, path.elementAt(i), needsMoveTo))
                needsMoveTo = false;
            last = path.elementAt(i);
            break;
        case QPainterPath::CurveToElement:
            if (cubicTo_clipped(result, transform, last, path.elementAt(i), path.elementAt(i+1), path.elementAt(i+2), needsMoveTo))
                needsMoveTo = false;
            i += 2;
            last = path.elementAt(i);
            break;
        default:
            Q_ASSERT(false);
        }
    }
 
    if (path.elementCount() > 0 && lastMoveTo != last)
        lineTo_clipped(result, transform, last, lastMoveTo, needsMoveTo, false);
 
    result.setFillRule(path.fillRule());
    return result;
}
 
/*!
    \fn QPainterPath operator *(const QPainterPath &path, const QTransform &matrix)
    \since 4.3
    \relates QTransform
 
    This is the same as \a{matrix}.map(\a{path}).
 
    \sa QTransform::map()
*/
 
/*!
    \overload
 
    Creates and returns a QPainterPath object that is a copy of the
    given \a path, mapped into the coordinate system defined by this
    matrix.
*/
QPainterPath QTransform::map(const QPainterPath &path) const
{
    TransformationType t = inline_type();
    if (t == TxNone || path.elementCount() == 0)
        return path;
 
    if (t >= TxProject)
        return mapProjective(*this, path);
 
    QPainterPath copy = path;
 
    if (t == TxTranslate) {
        copy.translate(m_matrix[2][0], m_matrix[2][1]);
    } else {
        copy.setDirty(true);
        // Full xform
        for (int i=0; i<path.elementCount(); ++i) {
            QPainterPath::Element &e = copy.d_ptr->elements[i];
            do_map(e.x, e.y, e.x, e.y);
        }
    }
 
    return copy;
}
 
/*!
    \fn QPolygon QTransform::mapToPolygon(const QRect &rectangle) const
 
    Creates and returns a QPolygon representation of the given \a
    rectangle, mapped into the coordinate system defined by this
    matrix.
 
    The rectangle's coordinates are transformed using the following
    formulas:
 
    \snippet code/src_gui_painting_qtransform.cpp 1
 
    Polygons and rectangles behave slightly differently when
    transformed (due to integer rounding), so
    \c{matrix.map(QPolygon(rectangle))} is not always the same as
    \c{matrix.mapToPolygon(rectangle)}.
 
    \sa mapRect(), {QTransform#Basic Matrix Operations}{Basic Matrix
    Operations}
*/
QPolygon QTransform::mapToPolygon(const QRect &rect) const
{
    TransformationType t = inline_type();
 
    QPolygon a(4);
    qreal x[4] = { 0, 0, 0, 0 }, y[4] = { 0, 0, 0, 0 };
    if (t <= TxScale) {
        x[0] = m_matrix[0][0]*rect.x() + m_matrix[2][0];
        y[0] = m_matrix[1][1]*rect.y() + m_matrix[2][1];
        qreal w = m_matrix[0][0]*rect.width();
        qreal h = m_matrix[1][1]*rect.height();
        if (w < 0) {
            w = -w;
            x[0] -= w;
        }
        if (h < 0) {
            h = -h;
            y[0] -= h;
        }
        x[1] = x[0]+w;
        x[2] = x[1];
        x[3] = x[0];
        y[1] = y[0];
        y[2] = y[0]+h;
        y[3] = y[2];
    } else {
        auto right = rect.x() + rect.width();
        auto bottom = rect.y() + rect.height();
        do_map(rect.x(), rect.y(), x[0], y[0]);
        do_map(right, rect.y(), x[1], y[1]);
        do_map(right, bottom, x[2], y[2]);
        do_map(rect.x(), bottom, x[3], y[3]);
    }
 
    // all coordinates are correctly, transform to a pointarray
    // (rounding to the next integer)
    a.setPoints(4, qRound(x[0]), qRound(y[0]),
                qRound(x[1]), qRound(y[1]),
                qRound(x[2]), qRound(y[2]),
                qRound(x[3]), qRound(y[3]));
    return a;
}
 
/*!
    Creates a transformation matrix, \a trans, that maps a unit square
    to a four-sided polygon, \a quad. Returns \c true if the transformation
    is constructed or false if such a transformation does not exist.
 
    \sa quadToSquare(), quadToQuad()
*/
bool QTransform::squareToQuad(const QPolygonF &quad, QTransform &trans)
{
    if (quad.size() != (quad.isClosed() ? 5 : 4))
        return false;
 
    qreal dx0 = quad[0].x();
    qreal dx1 = quad[1].x();
    qreal dx2 = quad[2].x();
    qreal dx3 = quad[3].x();
 
    qreal dy0 = quad[0].y();
    qreal dy1 = quad[1].y();
    qreal dy2 = quad[2].y();
    qreal dy3 = quad[3].y();
 
    double ax  = dx0 - dx1 + dx2 - dx3;
    double ay  = dy0 - dy1 + dy2 - dy3;
 
    if (!ax && !ay) { //afine transform
        trans.setMatrix(dx1 - dx0, dy1 - dy0,  0,
                        dx2 - dx1, dy2 - dy1,  0,
                        dx0,       dy0,  1);
    } else {
        double ax1 = dx1 - dx2;
        double ax2 = dx3 - dx2;
        double ay1 = dy1 - dy2;
        double ay2 = dy3 - dy2;
 
        /*determinants */
        double gtop    =  ax  * ay2 - ax2 * ay;
        double htop    =  ax1 * ay  - ax  * ay1;
        double bottom  =  ax1 * ay2 - ax2 * ay1;
 
        double a, b, c, d, e, f, g, h;  /*i is always 1*/
 
        if (!bottom)
            return false;
 
        g = gtop/bottom;
        h = htop/bottom;
 
        a = dx1 - dx0 + g * dx1;
        b = dx3 - dx0 + h * dx3;
        c = dx0;
        d = dy1 - dy0 + g * dy1;
        e = dy3 - dy0 + h * dy3;
        f = dy0;
 
        trans.setMatrix(a, d, g,
                        b, e, h,
                        c, f, 1.0);
    }
 
    return true;
}
 
/*!
    \fn bool QTransform::quadToSquare(const QPolygonF &quad, QTransform &trans)
 
    Creates a transformation matrix, \a trans, that maps a four-sided polygon,
    \a quad, to a unit square. Returns \c true if the transformation is constructed
    or false if such a transformation does not exist.
 
    \sa squareToQuad(), quadToQuad()
*/
bool QTransform::quadToSquare(const QPolygonF &quad, QTransform &trans)
{
    if (!squareToQuad(quad, trans))
        return false;
 
    bool invertible = false;
    trans = trans.inverted(&invertible);
 
    return invertible;
}
 
/*!
    Creates a transformation matrix, \a trans, that maps a four-sided
    polygon, \a one, to another four-sided polygon, \a two.
    Returns \c true if the transformation is possible; otherwise returns
    false.
 
    This is a convenience method combining quadToSquare() and
    squareToQuad() methods. It allows the input quad to be
    transformed into any other quad.
 
    \sa squareToQuad(), quadToSquare()
*/
bool QTransform::quadToQuad(const QPolygonF &one,
                            const QPolygonF &two,
                            QTransform &trans)
{
    QTransform stq;
    if (!quadToSquare(one, trans))
        return false;
    if (!squareToQuad(two, stq))
        return false;
    trans *= stq;
    //qDebug()<<"Final = "<<trans;
    return true;
}
 
/*!
    Sets the matrix elements to the specified values, \a m11,
    \a m12, \a m13 \a m21, \a m22, \a m23 \a m31, \a m32 and
    \a m33. Note that this function replaces the previous values.
    QTransform provides the translate(), rotate(), scale() and shear()
    convenience functions to manipulate the various matrix elements
    based on the currently defined coordinate system.
 
    \sa QTransform()
*/
 
void QTransform::setMatrix(qreal m11, qreal m12, qreal m13,
                           qreal m21, qreal m22, qreal m23,
                           qreal m31, qreal m32, qreal m33)
{
    m_matrix[0][0] = m11; m_matrix[0][1] = m12; m_matrix[0][2] = m13;
    m_matrix[1][0] = m21; m_matrix[1][1] = m22; m_matrix[1][2] = m23;
    m_matrix[2][0] = m31; m_matrix[2][1] = m32; m_matrix[2][2] = m33;
    m_type = TxNone;
    m_dirty = TxProject;
}
 
QRect QTransform::mapRect(const QRect &rect) const
{
    TransformationType t = inline_type();
    if (t <= TxTranslate)
        return rect.translated(qRound(m_matrix[2][0]), qRound(m_matrix[2][1]));
 
    if (t <= TxScale) {
        int x = qRound(m_matrix[0][0] * rect.x() + m_matrix[2][0]);
        int y = qRound(m_matrix[1][1] * rect.y() + m_matrix[2][1]);
        int w = qRound(m_matrix[0][0] * rect.width());
        int h = qRound(m_matrix[1][1] * rect.height());
        if (w < 0) {
            w = -w;
            x -= w;
        }
        if (h < 0) {
            h = -h;
            y -= h;
        }
        return QRect(x, y, w, h);
    } else {
        qreal x = 0, y = 0;
        do_map(rect.left(), rect.top(), x, y);
        qreal xmin = x;
        qreal ymin = y;
        qreal xmax = x;
        qreal ymax = y;
        do_map(rect.right() + 1, rect.top(), x, y);
        xmin = qMin(xmin, x);
        ymin = qMin(ymin, y);
        xmax = qMax(xmax, x);
        ymax = qMax(ymax, y);
        do_map(rect.right() + 1, rect.bottom() + 1, x, y);
        xmin = qMin(xmin, x);
        ymin = qMin(ymin, y);
        xmax = qMax(xmax, x);
        ymax = qMax(ymax, y);
        do_map(rect.left(), rect.bottom() + 1, x, y);
        xmin = qMin(xmin, x);
        ymin = qMin(ymin, y);
        xmax = qMax(xmax, x);
        ymax = qMax(ymax, y);
        return QRectF(xmin, ymin, xmax-xmin, ymax-ymin).toRect();
    }
}
 
/*!
    \fn QRectF QTransform::mapRect(const QRectF &rectangle) const
 
    Creates and returns a QRectF object that is a copy of the given \a
    rectangle, mapped into the coordinate system defined by this
    matrix.
 
    The rectangle's coordinates are transformed using the following
    formulas:
 
    \snippet code/src_gui_painting_qtransform.cpp 2
 
    If rotation or shearing has been specified, this function returns
    the \e bounding rectangle. To retrieve the exact region the given
    \a rectangle maps to, use the mapToPolygon() function instead.
 
    \sa mapToPolygon(), {QTransform#Basic Matrix Operations}{Basic Matrix
    Operations}
*/
QRectF QTransform::mapRect(const QRectF &rect) const
{
    TransformationType t = inline_type();
    if (t <= TxTranslate)
        return rect.translated(m_matrix[2][0], m_matrix[2][1]);
 
    if (t <= TxScale) {
        qreal x = m_matrix[0][0] * rect.x() + m_matrix[2][0];
        qreal y = m_matrix[1][1] * rect.y() + m_matrix[2][1];
        qreal w = m_matrix[0][0] * rect.width();
        qreal h = m_matrix[1][1] * rect.height();
        if (w < 0) {
            w = -w;
            x -= w;
        }
        if (h < 0) {
            h = -h;
            y -= h;
        }
        return QRectF(x, y, w, h);
    } else {
        qreal x = 0, y = 0;
        do_map(rect.x(), rect.y(), x, y);
        qreal xmin = x;
        qreal ymin = y;
        qreal xmax = x;
        qreal ymax = y;
        do_map(rect.x() + rect.width(), rect.y(), x, y);
        xmin = qMin(xmin, x);
        ymin = qMin(ymin, y);
        xmax = qMax(xmax, x);
        ymax = qMax(ymax, y);
        do_map(rect.x() + rect.width(), rect.y() + rect.height(), x, y);
        xmin = qMin(xmin, x);
        ymin = qMin(ymin, y);
        xmax = qMax(xmax, x);
        ymax = qMax(ymax, y);
        do_map(rect.x(), rect.y() + rect.height(), x, y);
        xmin = qMin(xmin, x);
        ymin = qMin(ymin, y);
        xmax = qMax(xmax, x);
        ymax = qMax(ymax, y);
        return QRectF(xmin, ymin, xmax-xmin, ymax - ymin);
    }
}
 
/*!
    \fn QRect QTransform::mapRect(const QRect &rectangle) const
    \overload
 
    Creates and returns a QRect object that is a copy of the given \a
    rectangle, mapped into the coordinate system defined by this
    matrix. Note that the transformed coordinates are rounded to the
    nearest integer.
*/
 
/*!
    Maps the given coordinates \a x and \a y into the coordinate
    system defined by this matrix. The resulting values are put in *\a
    tx and *\a ty, respectively.
 
    The coordinates are transformed using the following formulas:
 
    \snippet code/src_gui_painting_qtransform.cpp 3
 
    The point (x, y) is the original point, and (x', y') is the
    transformed point.
 
    \sa {QTransform#Basic Matrix Operations}{Basic Matrix Operations}
*/
void QTransform::map(qreal x, qreal y, qreal *tx, qreal *ty) const
{
    do_map(x, y, *tx, *ty);
}
 
/*!
    \overload
 
    Maps the given coordinates \a x and \a y into the coordinate
    system defined by this matrix. The resulting values are put in *\a
    tx and *\a ty, respectively. Note that the transformed coordinates
    are rounded to the nearest integer.
*/
void QTransform::map(int x, int y, int *tx, int *ty) const
{
    qreal fx = 0, fy = 0;
    do_map(x, y, fx, fy);
    *tx = qRound(fx);
    *ty = qRound(fy);
}
 
/*!
  Returns the transformation type of this matrix.
 
  The transformation type is the highest enumeration value
  capturing all of the matrix's transformations. For example,
  if the matrix both scales and shears, the type would be \c TxShear,
  because \c TxShear has a higher enumeration value than \c TxScale.
 
  Knowing the transformation type of a matrix is useful for optimization:
  you can often handle specific types more optimally than handling
  the generic case.
  */
QTransform::TransformationType QTransform::type() const
{
    if (m_dirty == TxNone || m_dirty < m_type)
        return static_cast<TransformationType>(m_type);
 
    switch (static_cast<TransformationType>(m_dirty)) {
    case TxProject:
        if (!qFuzzyIsNull(m_matrix[0][2]) || !qFuzzyIsNull(m_matrix[1][2]) || !qFuzzyIsNull(m_matrix[2][2] - 1)) {
             m_type = TxProject;
             break;
        }
        Q_FALLTHROUGH();
    case TxShear:
    case TxRotate:
        if (!qFuzzyIsNull(m_matrix[0][1]) || !qFuzzyIsNull(m_matrix[1][0])) {
            const qreal dot = m_matrix[0][0] * m_matrix[1][0] + m_matrix[0][1] * m_matrix[1][1];
            if (qFuzzyIsNull(dot))
                m_type = TxRotate;
            else
                m_type = TxShear;
            break;
        }
        Q_FALLTHROUGH();
    case TxScale:
        if (!qFuzzyIsNull(m_matrix[0][0] - 1) || !qFuzzyIsNull(m_matrix[1][1] - 1)) {
            m_type = TxScale;
            break;
        }
        Q_FALLTHROUGH();
    case TxTranslate:
        if (!qFuzzyIsNull(m_matrix[2][0]) || !qFuzzyIsNull(m_matrix[2][1])) {
            m_type = TxTranslate;
            break;
        }
        Q_FALLTHROUGH();
    case TxNone:
        m_type = TxNone;
        break;
    }
 
    m_dirty = TxNone;
    return static_cast<TransformationType>(m_type);
}
 
/*!
 
    Returns the transform as a QVariant.
*/
QTransform::operator QVariant() const
{
    return QVariant::fromValue(*this);
}
 
 
/*!
    \fn bool QTransform::isInvertible() const
 
    Returns \c true if the matrix is invertible, otherwise returns \c false.
 
    \sa inverted()
*/
 
/*!
    \fn qreal QTransform::m11() const
 
    Returns the horizontal scaling factor.
 
    \sa scale(), {QTransform#Basic Matrix Operations}{Basic Matrix
    Operations}
*/
 
/*!
    \fn qreal QTransform::m12() const
 
    Returns the vertical shearing factor.
 
    \sa shear(), {QTransform#Basic Matrix Operations}{Basic Matrix
    Operations}
*/
 
/*!
    \fn qreal QTransform::m21() const
 
    Returns the horizontal shearing factor.
 
    \sa shear(), {QTransform#Basic Matrix Operations}{Basic Matrix
    Operations}
*/
 
/*!
    \fn qreal QTransform::m22() const
 
    Returns the vertical scaling factor.
 
    \sa scale(), {QTransform#Basic Matrix Operations}{Basic Matrix
    Operations}
*/
 
/*!
    \fn qreal QTransform::dx() const
 
    Returns the horizontal translation factor.
 
    \sa m31(), translate(), {QTransform#Basic Matrix Operations}{Basic Matrix
    Operations}
*/
 
/*!
    \fn qreal QTransform::dy() const
 
    Returns the vertical translation factor.
 
    \sa translate(), {QTransform#Basic Matrix Operations}{Basic Matrix
    Operations}
*/
 
 
/*!
    \fn qreal QTransform::m13() const
 
    Returns the horizontal projection factor.
 
    \sa translate(), {QTransform#Basic Matrix Operations}{Basic Matrix
    Operations}
*/
 
 
/*!
    \fn qreal QTransform::m23() const
 
    Returns the vertical projection factor.
 
    \sa translate(), {QTransform#Basic Matrix Operations}{Basic Matrix
    Operations}
*/
 
/*!
    \fn qreal QTransform::m31() const
 
    Returns the horizontal translation factor.
 
    \sa dx(), translate(), {QTransform#Basic Matrix Operations}{Basic Matrix
    Operations}
*/
 
/*!
    \fn qreal QTransform::m32() const
 
    Returns the vertical translation factor.
 
    \sa dy(), translate(), {QTransform#Basic Matrix Operations}{Basic Matrix
    Operations}
*/
 
/*!
    \fn qreal QTransform::m33() const
 
    Returns the division factor.
 
    \sa translate(), {QTransform#Basic Matrix Operations}{Basic Matrix
    Operations}
*/
 
/*!
    \fn qreal QTransform::determinant() const
 
    Returns the matrix's determinant.
*/
 
/*!
    \fn bool QTransform::isIdentity() const
 
    Returns \c true if the matrix is the identity matrix, otherwise
    returns \c false.
 
    \sa reset()
*/
 
/*!
    \fn bool QTransform::isAffine() const
 
    Returns \c true if the matrix represent an affine transformation,
    otherwise returns \c false.
*/
 
/*!
    \fn bool QTransform::isScaling() const
 
    Returns \c true if the matrix represents a scaling
    transformation, otherwise returns \c false.
 
    \sa reset()
*/
 
/*!
    \fn bool QTransform::isRotating() const
 
    Returns \c true if the matrix represents some kind of a
    rotating transformation, otherwise returns \c false.
 
    \note A rotation transformation of 180 degrees and/or 360 degrees is treated as a scaling transformation.
 
    \sa reset()
*/
 
/*!
    \fn bool QTransform::isTranslating() const
 
    Returns \c true if the matrix represents a translating
    transformation, otherwise returns \c false.
 
    \sa reset()
*/
 
/*!
    \fn bool qFuzzyCompare(const QTransform& t1, const QTransform& t2)
 
    \relates QTransform
    \since 4.6
 
    Returns \c true if \a t1 and \a t2 are equal, allowing for a small
    fuzziness factor for floating-point comparisons; false otherwise.
*/
 
 
// returns true if the transform is uniformly scaling
// (same scale in x and y direction)
// scale is set to the max of x and y scaling factors
Q_GUI_EXPORT
bool qt_scaleForTransform(const QTransform &transform, qreal *scale)
{
    const QTransform::TransformationType type = transform.type();
    if (type <= QTransform::TxTranslate) {
        if (scale)
            *scale = 1;
        return true;
    } else if (type == QTransform::TxScale) {
        const qreal xScale = qAbs(transform.m11());
        const qreal yScale = qAbs(transform.m22());
        if (scale)
            *scale = qMax(xScale, yScale);
        return qFuzzyCompare(xScale, yScale);
    }
 
    // rotate then scale: compare columns
    const qreal xScale1 = transform.m11() * transform.m11()
                         + transform.m21() * transform.m21();
    const qreal yScale1 = transform.m12() * transform.m12()
                         + transform.m22() * transform.m22();
 
    // scale then rotate: compare rows
    const qreal xScale2 = transform.m11() * transform.m11()
                         + transform.m12() * transform.m12();
    const qreal yScale2 = transform.m21() * transform.m21()
                         + transform.m22() * transform.m22();
 
    // decide the order of rotate and scale operations
    if (qAbs(xScale1 - yScale1) > qAbs(xScale2 - yScale2)) {
        if (scale)
            *scale = qSqrt(qMax(xScale1, yScale1));
 
        return type == QTransform::TxRotate && qFuzzyCompare(xScale1, yScale1);
    } else {
        if (scale)
            *scale = qSqrt(qMax(xScale2, yScale2));
 
        return type == QTransform::TxRotate && qFuzzyCompare(xScale2, yScale2);
    }
}
 
QDataStream & operator>>(QDataStream &s, QTransform::Affine &m)
{
    if (s.version() == 1) {
        float m11, m12, m21, m22, dx, dy;
        s >> m11; s >> m12; s >> m21; s >> m22; s >> dx; s >> dy;
 
        m.m_matrix[0][0] = m11;
        m.m_matrix[0][1] = m12;
        m.m_matrix[1][0] = m21;
        m.m_matrix[1][1] = m22;
        m.m_matrix[2][0] = dx;
        m.m_matrix[2][1] = dy;
    } else {
        s >> m.m_matrix[0][0];
        s >> m.m_matrix[0][1];
        s >> m.m_matrix[1][0];
        s >> m.m_matrix[1][1];
        s >> m.m_matrix[2][0];
        s >> m.m_matrix[2][1];
    }
    m.m_matrix[0][2] = 0;
    m.m_matrix[1][2] = 0;
    m.m_matrix[2][2] = 1;
    return s;
}
 
QDataStream &operator<<(QDataStream &s, const QTransform::Affine &m)
{
    if (s.version() == 1) {
        s << (float)m.m_matrix[0][0]
          << (float)m.m_matrix[0][1]
          << (float)m.m_matrix[1][0]
          << (float)m.m_matrix[1][1]
          << (float)m.m_matrix[2][0]
          << (float)m.m_matrix[2][1];
    } else {
        s << m.m_matrix[0][0]
          << m.m_matrix[0][1]
          << m.m_matrix[1][0]
          << m.m_matrix[1][1]
          << m.m_matrix[2][0]
          << m.m_matrix[2][1];
    }
    return s;
}
 
QT_END_NAMESPACE