qmatrix4x4.cpp

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// Copyright (C) 2016 The Qt Company Ltd.
// SPDX-License-Identifier: LicenseRef-Qt-Commercial OR LGPL-3.0-only OR GPL-2.0-only OR GPL-3.0-only
 
#include "qmatrix4x4.h"
#include <QtCore/qmath.h>
#include <QtCore/qvariant.h>
 
#include <QtGui/qquaternion.h>
#include <QtGui/qtransform.h>
 
#include <cmath>
 
QT_BEGIN_NAMESPACE
 
#ifndef QT_NO_MATRIX4X4
 
/*!
    \class QMatrix4x4
    \brief The QMatrix4x4 class represents a 4x4 transformation matrix in 3D space.
    \since 4.6
    \ingroup painting-3D
    \inmodule QtGui
 
    The QMatrix4x4 class in general is treated as a row-major matrix, in that the
    constructors and operator() functions take data in row-major format, as is
    familiar in C-style usage.
 
    Internally the data is stored as column-major format, so as to be optimal for
    passing to OpenGL functions, which expect column-major data.
 
    When using these functions be aware that they return data in \b{column-major}
    format:
    \list
    \li data()
    \li constData()
    \endlist
 
    \sa QVector3D, QGenericMatrix
*/
 
static const float inv_dist_to_plane = 1.0f / 1024.0f;
 
/*!
    \fn QMatrix4x4::QMatrix4x4()
 
    Constructs an identity matrix.
*/
 
/*!
    \fn QMatrix4x4::QMatrix4x4(Qt::Initialization)
    \since 5.5
    \internal
 
    Constructs a matrix without initializing the contents.
*/
 
/*!
    Constructs a matrix from the given 16 floating-point \a values.
    The contents of the array \a values is assumed to be in
    row-major order.
 
    If the matrix has a special type (identity, translate, scale, etc),
    the programmer should follow this constructor with a call to
    optimize() if they wish QMatrix4x4 to optimize further
    calls to translate(), scale(), etc.
 
    \sa copyDataTo(), optimize()
*/
QMatrix4x4::QMatrix4x4(const float *values)
{
    for (int row = 0; row < 4; ++row)
        for (int col = 0; col < 4; ++col)
            m[col][row] = values[row * 4 + col];
    flagBits = General;
}
 
/*!
    \fn QMatrix4x4::QMatrix4x4(float m11, float m12, float m13, float m14, float m21, float m22, float m23, float m24, float m31, float m32, float m33, float m34, float m41, float m42, float m43, float m44)
 
    Constructs a matrix from the 16 elements \a m11, \a m12, \a m13, \a m14,
    \a m21, \a m22, \a m23, \a m24, \a m31, \a m32, \a m33, \a m34,
    \a m41, \a m42, \a m43, and \a m44.  The elements are specified in
    row-major order.
 
    If the matrix has a special type (identity, translate, scale, etc),
    the programmer should follow this constructor with a call to
    optimize() if they wish QMatrix4x4 to optimize further
    calls to translate(), scale(), etc.
 
    \sa optimize()
*/
 
/*!
    \fn template <int N, int M> QMatrix4x4::QMatrix4x4(const QGenericMatrix<N, M, float>& matrix)
 
    Constructs a 4x4 matrix from the left-most 4 columns and top-most
    4 rows of \a matrix.  If \a matrix has less than 4 columns or rows,
    the remaining elements are filled with elements from the identity
    matrix.
 
    \sa toGenericMatrix()
*/
 
/*!
    \fn template <int N, int M> QGenericMatrix<N, M, float> QMatrix4x4::toGenericMatrix() const
 
    Constructs a NxM generic matrix from the left-most N columns and
    top-most M rows of this 4x4 matrix.  If N or M is greater than 4,
    then the remaining elements are filled with elements from the
    identity matrix.
*/
 
/*!
    \internal
*/
QMatrix4x4::QMatrix4x4(const float *values, int cols, int rows)
{
    for (int col = 0; col < 4; ++col) {
        for (int row = 0; row < 4; ++row) {
            if (col < cols && row < rows)
                m[col][row] = values[col * rows + row];
            else if (col == row)
                m[col][row] = 1.0f;
            else
                m[col][row] = 0.0f;
        }
    }
    flagBits = General;
}
 
/*!
    Constructs a 4x4 matrix from the conventional Qt 2D
    transformation matrix \a transform.
 
    If \a transform has a special type (identity, translate, scale, etc),
    the programmer should follow this constructor with a call to
    optimize() if they wish QMatrix4x4 to optimize further
    calls to translate(), scale(), etc.
 
    \sa toTransform(), optimize()
*/
QMatrix4x4::QMatrix4x4(const QTransform& transform)
{
    m[0][0] = transform.m11();
    m[0][1] = transform.m12();
    m[0][2] = 0.0f;
    m[0][3] = transform.m13();
    m[1][0] = transform.m21();
    m[1][1] = transform.m22();
    m[1][2] = 0.0f;
    m[1][3] = transform.m23();
    m[2][0] = 0.0f;
    m[2][1] = 0.0f;
    m[2][2] = 1.0f;
    m[2][3] = 0.0f;
    m[3][0] = transform.dx();
    m[3][1] = transform.dy();
    m[3][2] = 0.0f;
    m[3][3] = transform.m33();
    flagBits = General;
}
 
/*!
    \fn const float& QMatrix4x4::operator()(int row, int column) const
 
    Returns a constant reference to the element at position
    (\a row, \a column) in this matrix.
 
    \sa column(), row()
*/
 
/*!
    \fn float& QMatrix4x4::operator()(int row, int column)
 
    Returns a reference to the element at position (\a row, \a column)
    in this matrix so that the element can be assigned to.
 
    \sa optimize(), setColumn(), setRow()
*/
 
/*!
    \fn QVector4D QMatrix4x4::column(int index) const
 
    Returns the elements of column \a index as a 4D vector.
 
    \sa setColumn(), row()
*/
 
/*!
    \fn void QMatrix4x4::setColumn(int index, const QVector4D& value)
 
    Sets the elements of column \a index to the components of \a value.
 
    \sa column(), setRow()
*/
 
/*!
    \fn QVector4D QMatrix4x4::row(int index) const
 
    Returns the elements of row \a index as a 4D vector.
 
    \sa setRow(), column()
*/
 
/*!
    \fn void QMatrix4x4::setRow(int index, const QVector4D& value)
 
    Sets the elements of row \a index to the components of \a value.
 
    \sa row(), setColumn()
*/
 
/*!
    \fn bool QMatrix4x4::isAffine() const
    \since 5.5
 
    Returns \c true if this matrix is affine matrix; false otherwise.
 
    An affine matrix is a 4x4 matrix with row 3 equal to (0, 0, 0, 1),
    e.g. no projective coefficients.
 
    \sa isIdentity()
*/
 
/*!
    \fn bool QMatrix4x4::isIdentity() const
 
    Returns \c true if this matrix is the identity; false otherwise.
 
    \sa setToIdentity()
*/
 
/*!
    \fn void QMatrix4x4::setToIdentity()
 
    Sets this matrix to the identity.
 
    \sa isIdentity()
*/
 
/*!
    \fn void QMatrix4x4::fill(float value)
 
    Fills all elements of this matrx with \a value.
*/
 
static inline double matrixDet2(const double m[4][4], int col0, int col1, int row0, int row1)
{
    return m[col0][row0] * m[col1][row1] - m[col0][row1] * m[col1][row0];
}
 
 
// The 4x4 matrix inverse algorithm is based on that described at:
// http://www.j3d.org/matrix_faq/matrfaq_latest.html#Q24
// Some optimization has been done to avoid making copies of 3x3
// sub-matrices and to unroll the loops.
 
// Calculate the determinant of a 3x3 sub-matrix.
//     | A B C |
// M = | D E F |   det(M) = A * (EI - HF) - B * (DI - GF) + C * (DH - GE)
//     | G H I |
static inline double matrixDet3
    (const double m[4][4], int col0, int col1, int col2,
     int row0, int row1, int row2)
{
    return m[col0][row0] * matrixDet2(m, col1, col2, row1, row2)
            - m[col1][row0] * matrixDet2(m, col0, col2, row1, row2)
            + m[col2][row0] * matrixDet2(m, col0, col1, row1, row2);
}
 
// Calculate the determinant of a 4x4 matrix.
static inline double matrixDet4(const double m[4][4])
{
    double det;
    det  = m[0][0] * matrixDet3(m, 1, 2, 3, 1, 2, 3);
    det -= m[1][0] * matrixDet3(m, 0, 2, 3, 1, 2, 3);
    det += m[2][0] * matrixDet3(m, 0, 1, 3, 1, 2, 3);
    det -= m[3][0] * matrixDet3(m, 0, 1, 2, 1, 2, 3);
    return det;
}
 
static inline void copyToDoubles(const float m[4][4], double mm[4][4])
{
    for (int i = 0; i < 4; ++i)
        for (int j = 0; j < 4; ++j)
            mm[i][j] = double(m[i][j]);
}
 
/*!
    Returns the determinant of this matrix.
*/
double QMatrix4x4::determinant() const
{
    if ((flagBits & ~(Translation | Rotation2D | Rotation)) == Identity)
        return 1.0;
 
    double mm[4][4];
    copyToDoubles(m, mm);
    if (flagBits < Rotation2D)
        return mm[0][0] * mm[1][1] * mm[2][2]; // Translation | Scale
    if (flagBits < Perspective)
        return matrixDet3(mm, 0, 1, 2, 0, 1, 2);
    return matrixDet4(mm);
}
 
/*!
    Returns the inverse of this matrix.  Returns the identity if
    this matrix cannot be inverted; i.e. determinant() is zero.
    If \a invertible is not null, then true will be written to
    that location if the matrix can be inverted; false otherwise.
 
    If the matrix is recognized as the identity or an orthonormal
    matrix, then this function will quickly invert the matrix
    using optimized routines.
 
    \sa determinant(), normalMatrix()
*/
QMatrix4x4 QMatrix4x4::inverted(bool *invertible) const
{
    // Handle some of the easy cases first.
    if (flagBits == Identity) {
        if (invertible)
            *invertible = true;
        return QMatrix4x4();
    } else if (flagBits == Translation) {
        QMatrix4x4 inv;
        inv.m[3][0] = -m[3][0];
        inv.m[3][1] = -m[3][1];
        inv.m[3][2] = -m[3][2];
        inv.flagBits = Translation;
        if (invertible)
            *invertible = true;
        return inv;
    } else if (flagBits < Rotation2D) {
        // Translation | Scale
        if (m[0][0] == 0 || m[1][1] == 0 || m[2][2] == 0) {
            if (invertible)
                *invertible = false;
            return QMatrix4x4();
        }
        QMatrix4x4 inv;
        inv.m[0][0] = 1.0f / m[0][0];
        inv.m[1][1] = 1.0f / m[1][1];
        inv.m[2][2] = 1.0f / m[2][2];
        inv.m[3][0] = -m[3][0] * inv.m[0][0];
        inv.m[3][1] = -m[3][1] * inv.m[1][1];
        inv.m[3][2] = -m[3][2] * inv.m[2][2];
        inv.flagBits = flagBits;
 
        if (invertible)
            *invertible = true;
        return inv;
    } else if ((flagBits & ~(Translation | Rotation2D | Rotation)) == Identity) {
        if (invertible)
            *invertible = true;
        return orthonormalInverse();
    } else if (flagBits < Perspective) {
        QMatrix4x4 inv(Qt::Uninitialized);
 
        double mm[4][4];
        copyToDoubles(m, mm);
 
        double det = matrixDet3(mm, 0, 1, 2, 0, 1, 2);
        if (det == 0.0f) {
            if (invertible)
                *invertible = false;
            return QMatrix4x4();
        }
        det = 1.0f / det;
 
        inv.m[0][0] =  matrixDet2(mm, 1, 2, 1, 2) * det;
        inv.m[0][1] = -matrixDet2(mm, 0, 2, 1, 2) * det;
        inv.m[0][2] =  matrixDet2(mm, 0, 1, 1, 2) * det;
        inv.m[0][3] = 0;
        inv.m[1][0] = -matrixDet2(mm, 1, 2, 0, 2) * det;
        inv.m[1][1] =  matrixDet2(mm, 0, 2, 0, 2) * det;
        inv.m[1][2] = -matrixDet2(mm, 0, 1, 0, 2) * det;
        inv.m[1][3] = 0;
        inv.m[2][0] =  matrixDet2(mm, 1, 2, 0, 1) * det;
        inv.m[2][1] = -matrixDet2(mm, 0, 2, 0, 1) * det;
        inv.m[2][2] =  matrixDet2(mm, 0, 1, 0, 1) * det;
        inv.m[2][3] = 0;
        inv.m[3][0] = -inv.m[0][0] * m[3][0] - inv.m[1][0] * m[3][1] - inv.m[2][0] * m[3][2];
        inv.m[3][1] = -inv.m[0][1] * m[3][0] - inv.m[1][1] * m[3][1] - inv.m[2][1] * m[3][2];
        inv.m[3][2] = -inv.m[0][2] * m[3][0] - inv.m[1][2] * m[3][1] - inv.m[2][2] * m[3][2];
        inv.m[3][3] = 1;
        inv.flagBits = flagBits;
 
        if (invertible)
            *invertible = true;
        return inv;
    }
 
    QMatrix4x4 inv(Qt::Uninitialized);
 
    double mm[4][4];
    copyToDoubles(m, mm);
 
    double det = matrixDet4(mm);
    if (det == 0.0f) {
        if (invertible)
            *invertible = false;
        return QMatrix4x4();
    }
    det = 1.0f / det;
 
    inv.m[0][0] =  matrixDet3(mm, 1, 2, 3, 1, 2, 3) * det;
    inv.m[0][1] = -matrixDet3(mm, 0, 2, 3, 1, 2, 3) * det;
    inv.m[0][2] =  matrixDet3(mm, 0, 1, 3, 1, 2, 3) * det;
    inv.m[0][3] = -matrixDet3(mm, 0, 1, 2, 1, 2, 3) * det;
    inv.m[1][0] = -matrixDet3(mm, 1, 2, 3, 0, 2, 3) * det;
    inv.m[1][1] =  matrixDet3(mm, 0, 2, 3, 0, 2, 3) * det;
    inv.m[1][2] = -matrixDet3(mm, 0, 1, 3, 0, 2, 3) * det;
    inv.m[1][3] =  matrixDet3(mm, 0, 1, 2, 0, 2, 3) * det;
    inv.m[2][0] =  matrixDet3(mm, 1, 2, 3, 0, 1, 3) * det;
    inv.m[2][1] = -matrixDet3(mm, 0, 2, 3, 0, 1, 3) * det;
    inv.m[2][2] =  matrixDet3(mm, 0, 1, 3, 0, 1, 3) * det;
    inv.m[2][3] = -matrixDet3(mm, 0, 1, 2, 0, 1, 3) * det;
    inv.m[3][0] = -matrixDet3(mm, 1, 2, 3, 0, 1, 2) * det;
    inv.m[3][1] =  matrixDet3(mm, 0, 2, 3, 0, 1, 2) * det;
    inv.m[3][2] = -matrixDet3(mm, 0, 1, 3, 0, 1, 2) * det;
    inv.m[3][3] =  matrixDet3(mm, 0, 1, 2, 0, 1, 2) * det;
    inv.flagBits = flagBits;
 
    if (invertible)
        *invertible = true;
    return inv;
}
 
/*!
    Returns the normal matrix corresponding to this 4x4 transformation.
    The normal matrix is the transpose of the inverse of the top-left
    3x3 part of this 4x4 matrix.  If the 3x3 sub-matrix is not invertible,
    this function returns the identity.
 
    \sa inverted()
*/
QMatrix3x3 QMatrix4x4::normalMatrix() const
{
    QMatrix3x3 inv;
 
    // Handle the simple cases first.
    if (flagBits < Scale) {
        // Translation
        return inv;
    } else if (flagBits < Rotation2D) {
        // Translation | Scale
        if (m[0][0] == 0.0f || m[1][1] == 0.0f || m[2][2] == 0.0f)
            return inv;
        inv.data()[0] = 1.0f / m[0][0];
        inv.data()[4] = 1.0f / m[1][1];
        inv.data()[8] = 1.0f / m[2][2];
        return inv;
    } else if ((flagBits & ~(Translation | Rotation2D | Rotation)) == Identity) {
        float *invm = inv.data();
        invm[0 + 0 * 3] = m[0][0];
        invm[1 + 0 * 3] = m[0][1];
        invm[2 + 0 * 3] = m[0][2];
        invm[0 + 1 * 3] = m[1][0];
        invm[1 + 1 * 3] = m[1][1];
        invm[2 + 1 * 3] = m[1][2];
        invm[0 + 2 * 3] = m[2][0];
        invm[1 + 2 * 3] = m[2][1];
        invm[2 + 2 * 3] = m[2][2];
        return inv;
    }
 
    double mm[4][4];
    copyToDoubles(m, mm);
    double det = matrixDet3(mm, 0, 1, 2, 0, 1, 2);
    if (det == 0.0f)
        return inv;
    det = 1.0f / det;
 
    float *invm = inv.data();
 
    // Invert and transpose in a single step.
    invm[0 + 0 * 3] =  (mm[1][1] * mm[2][2] - mm[2][1] * mm[1][2]) * det;
    invm[1 + 0 * 3] = -(mm[1][0] * mm[2][2] - mm[1][2] * mm[2][0]) * det;
    invm[2 + 0 * 3] =  (mm[1][0] * mm[2][1] - mm[1][1] * mm[2][0]) * det;
    invm[0 + 1 * 3] = -(mm[0][1] * mm[2][2] - mm[2][1] * mm[0][2]) * det;
    invm[1 + 1 * 3] =  (mm[0][0] * mm[2][2] - mm[0][2] * mm[2][0]) * det;
    invm[2 + 1 * 3] = -(mm[0][0] * mm[2][1] - mm[0][1] * mm[2][0]) * det;
    invm[0 + 2 * 3] =  (mm[0][1] * mm[1][2] - mm[0][2] * mm[1][1]) * det;
    invm[1 + 2 * 3] = -(mm[0][0] * mm[1][2] - mm[0][2] * mm[1][0]) * det;
    invm[2 + 2 * 3] =  (mm[0][0] * mm[1][1] - mm[1][0] * mm[0][1]) * det;
 
    return inv;
}
 
/*!
    Returns this matrix, transposed about its diagonal.
*/
QMatrix4x4 QMatrix4x4::transposed() const
{
    QMatrix4x4 result(Qt::Uninitialized);
    for (int row = 0; row < 4; ++row) {
        for (int col = 0; col < 4; ++col) {
            result.m[col][row] = m[row][col];
        }
    }
    // When a translation is transposed, it becomes a perspective transformation.
    result.flagBits = (flagBits & Translation ? General : flagBits);
    return result;
}
 
/*!
    \fn QMatrix4x4& QMatrix4x4::operator+=(const QMatrix4x4& other)
 
    Adds the contents of \a other to this matrix.
*/
 
/*!
    \fn QMatrix4x4& QMatrix4x4::operator-=(const QMatrix4x4& other)
 
    Subtracts the contents of \a other from this matrix.
*/
 
/*!
    \fn QMatrix4x4& QMatrix4x4::operator*=(const QMatrix4x4& other)
 
    Multiplies the contents of \a other by this matrix.
*/
 
/*!
    \fn QMatrix4x4& QMatrix4x4::operator*=(float factor)
    \overload
 
    Multiplies all elements of this matrix by \a factor.
*/
 
/*!
    \overload
 
    Divides all elements of this matrix by \a divisor.
*/
QMatrix4x4& QMatrix4x4::operator/=(float divisor)
{
    m[0][0] /= divisor;
    m[0][1] /= divisor;
    m[0][2] /= divisor;
    m[0][3] /= divisor;
    m[1][0] /= divisor;
    m[1][1] /= divisor;
    m[1][2] /= divisor;
    m[1][3] /= divisor;
    m[2][0] /= divisor;
    m[2][1] /= divisor;
    m[2][2] /= divisor;
    m[2][3] /= divisor;
    m[3][0] /= divisor;
    m[3][1] /= divisor;
    m[3][2] /= divisor;
    m[3][3] /= divisor;
    flagBits = General;
    return *this;
}
 
/*!
    \fn bool QMatrix4x4::operator==(const QMatrix4x4& other) const
 
    Returns \c true if this matrix is identical to \a other; false otherwise.
    This operator uses an exact floating-point comparison.
*/
 
/*!
    \fn bool QMatrix4x4::operator!=(const QMatrix4x4& other) const
 
    Returns \c true if this matrix is not identical to \a other; false otherwise.
    This operator uses an exact floating-point comparison.
*/
 
/*!
    \fn QMatrix4x4 operator+(const QMatrix4x4& m1, const QMatrix4x4& m2)
    \relates QMatrix4x4
 
    Returns the sum of \a m1 and \a m2.
*/
 
/*!
    \fn QMatrix4x4 operator-(const QMatrix4x4& m1, const QMatrix4x4& m2)
    \relates QMatrix4x4
 
    Returns the difference of \a m1 and \a m2.
*/
 
/*!
    \fn QMatrix4x4 operator*(const QMatrix4x4& m1, const QMatrix4x4& m2)
    \relates QMatrix4x4
 
    Returns the product of \a m1 and \a m2.
*/
 
#ifndef QT_NO_VECTOR3D
 
#if QT_DEPRECATED_SINCE(6, 1)
 
/*!
    \fn QVector3D operator*(const QVector3D& vector, const QMatrix4x4& matrix)
    \relates QMatrix4x4
 
    \deprecated [6.1] Convert the QVector3D to a QVector4D with 1.0 as the w coordinate, then multiply.
 
    Returns the result of transforming \a vector according to \a matrix,
    with the matrix applied post-vector. The vector is transformed as a point.
*/
 
/*!
    \fn QVector3D operator*(const QMatrix4x4& matrix, const QVector3D& vector)
    \relates QMatrix4x4
 
    \deprecated [6.1] Use QMatrix4x4::map() instead.
 
    Returns the result of transforming \a vector according to \a matrix,
    with the matrix applied pre-vector. The vector is transformed as a
    projective point.
*/
 
#endif
 
#endif
 
#ifndef QT_NO_VECTOR4D
 
/*!
    \fn QVector4D operator*(const QVector4D& vector, const QMatrix4x4& matrix)
    \relates QMatrix4x4
 
    Returns the result of transforming \a vector according to \a matrix,
    with the matrix applied post-vector.
*/
 
/*!
    \fn QVector4D operator*(const QMatrix4x4& matrix, const QVector4D& vector)
    \relates QMatrix4x4
 
    Returns the result of transforming \a vector according to \a matrix,
    with the matrix applied pre-vector.
*/
 
#endif
 
/*!
    \fn QPoint operator*(const QPoint& point, const QMatrix4x4& matrix)
    \relates QMatrix4x4
 
    Returns the result of transforming \a point according to \a matrix,
    with the matrix applied post-point.
*/
 
/*!
    \fn QPointF operator*(const QPointF& point, const QMatrix4x4& matrix)
    \relates QMatrix4x4
 
    Returns the result of transforming \a point according to \a matrix,
    with the matrix applied post-point.
*/
 
#if QT_DEPRECATED_SINCE(6, 1)
 
/*!
    \fn QPoint operator*(const QMatrix4x4& matrix, const QPoint& point)
    \relates QMatrix4x4
 
    \deprecated [6.1] Use QMatrix4x4::map() instead.
 
    Returns the result of transforming \a point according to \a matrix,
    with the matrix applied pre-point.
*/
 
/*!
    \fn QPointF operator*(const QMatrix4x4& matrix, const QPointF& point)
    \relates QMatrix4x4
 
    \deprecated [6.1] Use QMatrix4x4::map() instead.
 
    Returns the result of transforming \a point according to \a matrix,
    with the matrix applied pre-point.
*/
 
#endif
 
/*!
    \fn QMatrix4x4 operator-(const QMatrix4x4& matrix)
    \overload
    \relates QMatrix4x4
 
    Returns the negation of \a matrix.
*/
 
/*!
    \fn QMatrix4x4 operator*(float factor, const QMatrix4x4& matrix)
    \relates QMatrix4x4
 
    Returns the result of multiplying all elements of \a matrix by \a factor.
*/
 
/*!
    \fn QMatrix4x4 operator*(const QMatrix4x4& matrix, float factor)
    \relates QMatrix4x4
 
    Returns the result of multiplying all elements of \a matrix by \a factor.
*/
 
/*!
    \relates QMatrix4x4
 
    Returns the result of dividing all elements of \a matrix by \a divisor.
*/
QMatrix4x4 operator/(const QMatrix4x4& matrix, float divisor)
{
    QMatrix4x4 m(Qt::Uninitialized);
    m.m[0][0] = matrix.m[0][0] / divisor;
    m.m[0][1] = matrix.m[0][1] / divisor;
    m.m[0][2] = matrix.m[0][2] / divisor;
    m.m[0][3] = matrix.m[0][3] / divisor;
    m.m[1][0] = matrix.m[1][0] / divisor;
    m.m[1][1] = matrix.m[1][1] / divisor;
    m.m[1][2] = matrix.m[1][2] / divisor;
    m.m[1][3] = matrix.m[1][3] / divisor;
    m.m[2][0] = matrix.m[2][0] / divisor;
    m.m[2][1] = matrix.m[2][1] / divisor;
    m.m[2][2] = matrix.m[2][2] / divisor;
    m.m[2][3] = matrix.m[2][3] / divisor;
    m.m[3][0] = matrix.m[3][0] / divisor;
    m.m[3][1] = matrix.m[3][1] / divisor;
    m.m[3][2] = matrix.m[3][2] / divisor;
    m.m[3][3] = matrix.m[3][3] / divisor;
    m.flagBits = QMatrix4x4::General;
    return m;
}
 
/*!
    \fn bool QMatrix4x4::qFuzzyCompare(const QMatrix4x4& m1, const QMatrix4x4& m2)
 
    Returns \c true if \a m1 and \a m2 are equal, allowing for a small
    fuzziness factor for floating-point comparisons; false otherwise.
*/
bool qFuzzyCompare(const QMatrix4x4& m1, const QMatrix4x4& m2) noexcept
{
    return qFuzzyCompare(m1.m[0][0], m2.m[0][0]) &&
            qFuzzyCompare(m1.m[0][1], m2.m[0][1]) &&
            qFuzzyCompare(m1.m[0][2], m2.m[0][2]) &&
            qFuzzyCompare(m1.m[0][3], m2.m[0][3]) &&
            qFuzzyCompare(m1.m[1][0], m2.m[1][0]) &&
            qFuzzyCompare(m1.m[1][1], m2.m[1][1]) &&
            qFuzzyCompare(m1.m[1][2], m2.m[1][2]) &&
            qFuzzyCompare(m1.m[1][3], m2.m[1][3]) &&
            qFuzzyCompare(m1.m[2][0], m2.m[2][0]) &&
            qFuzzyCompare(m1.m[2][1], m2.m[2][1]) &&
            qFuzzyCompare(m1.m[2][2], m2.m[2][2]) &&
            qFuzzyCompare(m1.m[2][3], m2.m[2][3]) &&
            qFuzzyCompare(m1.m[3][0], m2.m[3][0]) &&
            qFuzzyCompare(m1.m[3][1], m2.m[3][1]) &&
            qFuzzyCompare(m1.m[3][2], m2.m[3][2]) &&
            qFuzzyCompare(m1.m[3][3], m2.m[3][3]);
}
 
 
#ifndef QT_NO_VECTOR3D
 
/*!
    Multiplies this matrix by another that scales coordinates by
    the components of \a vector.
 
    \sa translate(), rotate()
*/
void QMatrix4x4::scale(const QVector3D& vector)
{
    float vx = vector.x();
    float vy = vector.y();
    float vz = vector.z();
    if (flagBits < Scale) {
        m[0][0] = vx;
        m[1][1] = vy;
        m[2][2] = vz;
    } else if (flagBits < Rotation2D) {
        m[0][0] *= vx;
        m[1][1] *= vy;
        m[2][2] *= vz;
    } else if (flagBits < Rotation) {
        m[0][0] *= vx;
        m[0][1] *= vx;
        m[1][0] *= vy;
        m[1][1] *= vy;
        m[2][2] *= vz;
    } else {
        m[0][0] *= vx;
        m[0][1] *= vx;
        m[0][2] *= vx;
        m[0][3] *= vx;
        m[1][0] *= vy;
        m[1][1] *= vy;
        m[1][2] *= vy;
        m[1][3] *= vy;
        m[2][0] *= vz;
        m[2][1] *= vz;
        m[2][2] *= vz;
        m[2][3] *= vz;
    }
    flagBits |= Scale;
}
 
#endif
 
/*!
    \overload
 
    Multiplies this matrix by another that scales coordinates by the
    components \a x, and \a y.
 
    \sa translate(), rotate()
*/
void QMatrix4x4::scale(float x, float y)
{
    if (flagBits < Scale) {
        m[0][0] = x;
        m[1][1] = y;
    } else if (flagBits < Rotation2D) {
        m[0][0] *= x;
        m[1][1] *= y;
    } else if (flagBits < Rotation) {
        m[0][0] *= x;
        m[0][1] *= x;
        m[1][0] *= y;
        m[1][1] *= y;
    } else {
        m[0][0] *= x;
        m[0][1] *= x;
        m[0][2] *= x;
        m[0][3] *= x;
        m[1][0] *= y;
        m[1][1] *= y;
        m[1][2] *= y;
        m[1][3] *= y;
    }
    flagBits |= Scale;
}
 
/*!
    \overload
 
    Multiplies this matrix by another that scales coordinates by the
    components \a x, \a y, and \a z.
 
    \sa translate(), rotate()
*/
void QMatrix4x4::scale(float x, float y, float z)
{
    if (flagBits < Scale) {
        m[0][0] = x;
        m[1][1] = y;
        m[2][2] = z;
    } else if (flagBits < Rotation2D) {
        m[0][0] *= x;
        m[1][1] *= y;
        m[2][2] *= z;
    } else if (flagBits < Rotation) {
        m[0][0] *= x;
        m[0][1] *= x;
        m[1][0] *= y;
        m[1][1] *= y;
        m[2][2] *= z;
    } else {
        m[0][0] *= x;
        m[0][1] *= x;
        m[0][2] *= x;
        m[0][3] *= x;
        m[1][0] *= y;
        m[1][1] *= y;
        m[1][2] *= y;
        m[1][3] *= y;
        m[2][0] *= z;
        m[2][1] *= z;
        m[2][2] *= z;
        m[2][3] *= z;
    }
    flagBits |= Scale;
}
 
/*!
    \overload
 
    Multiplies this matrix by another that scales coordinates by the
    given \a factor.
 
    \sa translate(), rotate()
*/
void QMatrix4x4::scale(float factor)
{
    if (flagBits < Scale) {
        m[0][0] = factor;
        m[1][1] = factor;
        m[2][2] = factor;
    } else if (flagBits < Rotation2D) {
        m[0][0] *= factor;
        m[1][1] *= factor;
        m[2][2] *= factor;
    } else if (flagBits < Rotation) {
        m[0][0] *= factor;
        m[0][1] *= factor;
        m[1][0] *= factor;
        m[1][1] *= factor;
        m[2][2] *= factor;
    } else {
        m[0][0] *= factor;
        m[0][1] *= factor;
        m[0][2] *= factor;
        m[0][3] *= factor;
        m[1][0] *= factor;
        m[1][1] *= factor;
        m[1][2] *= factor;
        m[1][3] *= factor;
        m[2][0] *= factor;
        m[2][1] *= factor;
        m[2][2] *= factor;
        m[2][3] *= factor;
    }
    flagBits |= Scale;
}
 
#ifndef QT_NO_VECTOR3D
/*!
    Multiplies this matrix by another that translates coordinates by
    the components of \a vector.
 
    \sa scale(), rotate()
*/
 
void QMatrix4x4::translate(const QVector3D& vector)
{
    float vx = vector.x();
    float vy = vector.y();
    float vz = vector.z();
    if (flagBits == Identity) {
        m[3][0] = vx;
        m[3][1] = vy;
        m[3][2] = vz;
    } else if (flagBits == Translation) {
        m[3][0] += vx;
        m[3][1] += vy;
        m[3][2] += vz;
    } else if (flagBits == Scale) {
        m[3][0] = m[0][0] * vx;
        m[3][1] = m[1][1] * vy;
        m[3][2] = m[2][2] * vz;
    } else if (flagBits == (Translation | Scale)) {
        m[3][0] += m[0][0] * vx;
        m[3][1] += m[1][1] * vy;
        m[3][2] += m[2][2] * vz;
    } else if (flagBits < Rotation) {
        m[3][0] += m[0][0] * vx + m[1][0] * vy;
        m[3][1] += m[0][1] * vx + m[1][1] * vy;
        m[3][2] += m[2][2] * vz;
    } else {
        m[3][0] += m[0][0] * vx + m[1][0] * vy + m[2][0] * vz;
        m[3][1] += m[0][1] * vx + m[1][1] * vy + m[2][1] * vz;
        m[3][2] += m[0][2] * vx + m[1][2] * vy + m[2][2] * vz;
        m[3][3] += m[0][3] * vx + m[1][3] * vy + m[2][3] * vz;
    }
    flagBits |= Translation;
}
#endif
 
/*!
    \overload
 
    Multiplies this matrix by another that translates coordinates
    by the components \a x, and \a y.
 
    \sa scale(), rotate()
*/
void QMatrix4x4::translate(float x, float y)
{
    if (flagBits == Identity) {
        m[3][0] = x;
        m[3][1] = y;
    } else if (flagBits == Translation) {
        m[3][0] += x;
        m[3][1] += y;
    } else if (flagBits == Scale) {
        m[3][0] = m[0][0] * x;
        m[3][1] = m[1][1] * y;
    } else if (flagBits == (Translation | Scale)) {
        m[3][0] += m[0][0] * x;
        m[3][1] += m[1][1] * y;
    } else if (flagBits < Rotation) {
        m[3][0] += m[0][0] * x + m[1][0] * y;
        m[3][1] += m[0][1] * x + m[1][1] * y;
    } else {
        m[3][0] += m[0][0] * x + m[1][0] * y;
        m[3][1] += m[0][1] * x + m[1][1] * y;
        m[3][2] += m[0][2] * x + m[1][2] * y;
        m[3][3] += m[0][3] * x + m[1][3] * y;
    }
    flagBits |= Translation;
}
 
/*!
    \overload
 
    Multiplies this matrix by another that translates coordinates
    by the components \a x, \a y, and \a z.
 
    \sa scale(), rotate()
*/
void QMatrix4x4::translate(float x, float y, float z)
{
    if (flagBits == Identity) {
        m[3][0] = x;
        m[3][1] = y;
        m[3][2] = z;
    } else if (flagBits == Translation) {
        m[3][0] += x;
        m[3][1] += y;
        m[3][2] += z;
    } else if (flagBits == Scale) {
        m[3][0] = m[0][0] * x;
        m[3][1] = m[1][1] * y;
        m[3][2] = m[2][2] * z;
    } else if (flagBits == (Translation | Scale)) {
        m[3][0] += m[0][0] * x;
        m[3][1] += m[1][1] * y;
        m[3][2] += m[2][2] * z;
    } else if (flagBits < Rotation) {
        m[3][0] += m[0][0] * x + m[1][0] * y;
        m[3][1] += m[0][1] * x + m[1][1] * y;
        m[3][2] += m[2][2] * z;
    } else {
        m[3][0] += m[0][0] * x + m[1][0] * y + m[2][0] * z;
        m[3][1] += m[0][1] * x + m[1][1] * y + m[2][1] * z;
        m[3][2] += m[0][2] * x + m[1][2] * y + m[2][2] * z;
        m[3][3] += m[0][3] * x + m[1][3] * y + m[2][3] * z;
    }
    flagBits |= Translation;
}
 
#ifndef QT_NO_VECTOR3D
/*!
    Multiples this matrix by another that rotates coordinates through
    \a angle degrees about \a vector.
 
    \sa scale(), translate()
*/
 
void QMatrix4x4::rotate(float angle, const QVector3D& vector)
{
    rotate(angle, vector.x(), vector.y(), vector.z());
}
 
#endif
 
/*!
    \overload
 
    Multiplies this matrix by another that rotates coordinates through
    \a angle degrees about the vector (\a x, \a y, \a z).
 
    \sa scale(), translate()
*/
void QMatrix4x4::rotate(float angle, float x, float y, float z)
{
    if (angle == 0.0f)
        return;
    float c, s;
    if (angle == 90.0f || angle == -270.0f) {
        s = 1.0f;
        c = 0.0f;
    } else if (angle == -90.0f || angle == 270.0f) {
        s = -1.0f;
        c = 0.0f;
    } else if (angle == 180.0f || angle == -180.0f) {
        s = 0.0f;
        c = -1.0f;
    } else {
        float a = qDegreesToRadians(angle);
        c = std::cos(a);
        s = std::sin(a);
    }
    if (x == 0.0f) {
        if (y == 0.0f) {
            if (z != 0.0f) {
                // Rotate around the Z axis.
                if (z < 0)
                    s = -s;
                float tmp;
                m[0][0] = (tmp = m[0][0]) * c + m[1][0] * s;
                m[1][0] = m[1][0] * c - tmp * s;
                m[0][1] = (tmp = m[0][1]) * c + m[1][1] * s;
                m[1][1] = m[1][1] * c - tmp * s;
                m[0][2] = (tmp = m[0][2]) * c + m[1][2] * s;
                m[1][2] = m[1][2] * c - tmp * s;
                m[0][3] = (tmp = m[0][3]) * c + m[1][3] * s;
                m[1][3] = m[1][3] * c - tmp * s;
 
                flagBits |= Rotation2D;
                return;
            }
        } else if (z == 0.0f) {
            // Rotate around the Y axis.
            if (y < 0)
                s = -s;
            float tmp;
            m[2][0] = (tmp = m[2][0]) * c + m[0][0] * s;
            m[0][0] = m[0][0] * c - tmp * s;
            m[2][1] = (tmp = m[2][1]) * c + m[0][1] * s;
            m[0][1] = m[0][1] * c - tmp * s;
            m[2][2] = (tmp = m[2][2]) * c + m[0][2] * s;
            m[0][2] = m[0][2] * c - tmp * s;
            m[2][3] = (tmp = m[2][3]) * c + m[0][3] * s;
            m[0][3] = m[0][3] * c - tmp * s;
 
            flagBits |= Rotation;
            return;
        }
    } else if (y == 0.0f && z == 0.0f) {
        // Rotate around the X axis.
        if (x < 0)
            s = -s;
        float tmp;
        m[1][0] = (tmp = m[1][0]) * c + m[2][0] * s;
        m[2][0] = m[2][0] * c - tmp * s;
        m[1][1] = (tmp = m[1][1]) * c + m[2][1] * s;
        m[2][1] = m[2][1] * c - tmp * s;
        m[1][2] = (tmp = m[1][2]) * c + m[2][2] * s;
        m[2][2] = m[2][2] * c - tmp * s;
        m[1][3] = (tmp = m[1][3]) * c + m[2][3] * s;
        m[2][3] = m[2][3] * c - tmp * s;
 
        flagBits |= Rotation;
        return;
    }
 
    double len = double(x) * double(x) +
                 double(y) * double(y) +
                 double(z) * double(z);
    if (!qFuzzyCompare(len, 1.0) && !qFuzzyIsNull(len)) {
        len = std::sqrt(len);
        x = float(double(x) / len);
        y = float(double(y) / len);
        z = float(double(z) / len);
    }
    float ic = 1.0f - c;
    QMatrix4x4 rot(Qt::Uninitialized);
    rot.m[0][0] = x * x * ic + c;
    rot.m[1][0] = x * y * ic - z * s;
    rot.m[2][0] = x * z * ic + y * s;
    rot.m[3][0] = 0.0f;
    rot.m[0][1] = y * x * ic + z * s;
    rot.m[1][1] = y * y * ic + c;
    rot.m[2][1] = y * z * ic - x * s;
    rot.m[3][1] = 0.0f;
    rot.m[0][2] = x * z * ic - y * s;
    rot.m[1][2] = y * z * ic + x * s;
    rot.m[2][2] = z * z * ic + c;
    rot.m[3][2] = 0.0f;
    rot.m[0][3] = 0.0f;
    rot.m[1][3] = 0.0f;
    rot.m[2][3] = 0.0f;
    rot.m[3][3] = 1.0f;
    rot.flagBits = Rotation;
    *this *= rot;
}
 
/*!
    \internal
*/
void QMatrix4x4::projectedRotate(float angle, float x, float y, float z, float distanceToPlane)
{
    // Used by QGraphicsRotation::applyTo() to perform a rotation
    // and projection back to 2D in a single step.
    if (qIsNull(distanceToPlane))
        return rotate(angle, x, y, z);
    if (angle == 0.0f)
        return;
 
    float c, s;
    if (angle == 90.0f || angle == -270.0f) {
        s = 1.0f;
        c = 0.0f;
    } else if (angle == -90.0f || angle == 270.0f) {
        s = -1.0f;
        c = 0.0f;
    } else if (angle == 180.0f || angle == -180.0f) {
        s = 0.0f;
        c = -1.0f;
    } else {
        float a = qDegreesToRadians(angle);
        c = std::cos(a);
        s = std::sin(a);
    }
 
    const qreal d = 1.0 / distanceToPlane;
    if (x == 0.0f) {
        if (y == 0.0f) {
            if (z != 0.0f) {
                // Rotate around the Z axis.
                if (z < 0)
                    s = -s;
                float tmp;
                m[0][0] = (tmp = m[0][0]) * c + m[1][0] * s;
                m[1][0] = m[1][0] * c - tmp * s;
                m[0][1] = (tmp = m[0][1]) * c + m[1][1] * s;
                m[1][1] = m[1][1] * c - tmp * s;
                m[0][2] = (tmp = m[0][2]) * c + m[1][2] * s;
                m[1][2] = m[1][2] * c - tmp * s;
                m[0][3] = (tmp = m[0][3]) * c + m[1][3] * s;
                m[1][3] = m[1][3] * c - tmp * s;
 
                flagBits |= Rotation2D;
                return;
            }
        } else if (z == 0.0f) {
            // Rotate around the Y axis.
            if (y < 0)
                s = -s;
            s *= d;
            m[0][0] = m[0][0] * c + m[3][0] * s;
            m[0][1] = m[0][1] * c + m[3][1] * s;
            m[0][2] = m[0][2] * c + m[3][2] * s;
            m[0][3] = m[0][3] * c + m[3][3] * s;
            flagBits = General;
            return;
        }
    } else if (y == 0.0f && z == 0.0f) {
        // Rotate around the X axis.
        if (x < 0)
            s = -s;
        s *= d;
        m[1][0] = m[1][0] * c - m[3][0] * s;
        m[1][1] = m[1][1] * c - m[3][1] * s;
        m[1][2] = m[1][2] * c - m[3][2] * s;
        m[1][3] = m[1][3] * c - m[3][3] * s;
        flagBits = General;
        return;
    }
    double len = double(x) * double(x) +
                 double(y) * double(y) +
                 double(z) * double(z);
    if (!qFuzzyCompare(len, 1.0) && !qFuzzyIsNull(len)) {
        len = std::sqrt(len);
        x = float(double(x) / len);
        y = float(double(y) / len);
        z = float(double(z) / len);
    }
    const float ic = 1.0f - c;
    QMatrix4x4 rot(Qt::Uninitialized);
    rot.m[0][0] = x * x * ic + c;
    rot.m[1][0] = x * y * ic - z * s;
    rot.m[2][0] = 0.0f;
    rot.m[3][0] = 0.0f;
    rot.m[0][1] = y * x * ic + z * s;
    rot.m[1][1] = y * y * ic + c;
    rot.m[2][1] = 0.0f;
    rot.m[3][1] = 0.0f;
    rot.m[0][2] = 0.0f;
    rot.m[1][2] = 0.0f;
    rot.m[2][2] = 1.0f;
    rot.m[3][2] = 0.0f;
    rot.m[0][3] = (x * z * ic - y * s) * -d;
    rot.m[1][3] = (y * z * ic + x * s) * -d;
    rot.m[2][3] = 0.0f;
    rot.m[3][3] = 1.0f;
    rot.flagBits = General;
    *this *= rot;
}
 
#if QT_VERSION < QT_VERSION_CHECK(7, 0, 0)
/*!
    \internal
*/
void QMatrix4x4::projectedRotate(float angle, float x, float y, float z)
{
    projectedRotate(angle, x, y, z, 1024.0);
}
#endif
 
/*!
    \fn int QMatrix4x4::flags() const
    \internal
*/
 
/*!
    \enum QMatrix4x4::Flag
    \internal
    \omitvalue Identity
    \omitvalue Translation
    \omitvalue Scale
    \omitvalue Rotation2D
    \omitvalue Rotation
    \omitvalue Perspective
    \omitvalue General
*/
 
#ifndef QT_NO_QUATERNION
 
/*!
    Multiples this matrix by another that rotates coordinates according
    to a specified \a quaternion.  The \a quaternion is assumed to have
    been normalized.
 
    \sa scale(), translate(), QQuaternion
*/
void QMatrix4x4::rotate(const QQuaternion& quaternion)
{
    // Algorithm from:
    // http://www.j3d.org/matrix_faq/matrfaq_latest.html#Q54
 
    QMatrix4x4 m(Qt::Uninitialized);
 
    const float f2x = quaternion.x() + quaternion.x();
    const float f2y = quaternion.y() + quaternion.y();
    const float f2z = quaternion.z() + quaternion.z();
    const float f2xw = f2x * quaternion.scalar();
    const float f2yw = f2y * quaternion.scalar();
    const float f2zw = f2z * quaternion.scalar();
    const float f2xx = f2x * quaternion.x();
    const float f2xy = f2x * quaternion.y();
    const float f2xz = f2x * quaternion.z();
    const float f2yy = f2y * quaternion.y();
    const float f2yz = f2y * quaternion.z();
    const float f2zz = f2z * quaternion.z();
 
    m.m[0][0] = 1.0f - (f2yy + f2zz);
    m.m[1][0] =         f2xy - f2zw;
    m.m[2][0] =         f2xz + f2yw;
    m.m[3][0] = 0.0f;
    m.m[0][1] =         f2xy + f2zw;
    m.m[1][1] = 1.0f - (f2xx + f2zz);
    m.m[2][1] =         f2yz - f2xw;
    m.m[3][1] = 0.0f;
    m.m[0][2] =         f2xz - f2yw;
    m.m[1][2] =         f2yz + f2xw;
    m.m[2][2] = 1.0f - (f2xx + f2yy);
    m.m[3][2] = 0.0f;
    m.m[0][3] = 0.0f;
    m.m[1][3] = 0.0f;
    m.m[2][3] = 0.0f;
    m.m[3][3] = 1.0f;
    m.flagBits = Rotation;
    *this *= m;
}
 
#endif
 
/*!
    \overload
 
    Multiplies this matrix by another that applies an orthographic
    projection for a window with boundaries specified by \a rect.
    The near and far clipping planes will be -1 and 1 respectively.
 
    \sa frustum(), perspective()
*/
void QMatrix4x4::ortho(const QRect& rect)
{
    // Note: rect.right() and rect.bottom() subtract 1 in QRect,
    // which gives the location of a pixel within the rectangle,
    // instead of the extent of the rectangle.  We want the extent.
    // QRectF expresses the extent properly.
    ortho(rect.x(), rect.x() + rect.width(), rect.y() + rect.height(), rect.y(), -1.0f, 1.0f);
}
 
/*!
    \overload
 
    Multiplies this matrix by another that applies an orthographic
    projection for a window with boundaries specified by \a rect.
    The near and far clipping planes will be -1 and 1 respectively.
 
    \sa frustum(), perspective()
*/
void QMatrix4x4::ortho(const QRectF& rect)
{
    ortho(rect.left(), rect.right(), rect.bottom(), rect.top(), -1.0f, 1.0f);
}
 
/*!
    Multiplies this matrix by another that applies an orthographic
    projection for a window with lower-left corner (\a left, \a bottom),
    upper-right corner (\a right, \a top), and the specified \a nearPlane
    and \a farPlane clipping planes.
 
    \sa frustum(), perspective()
*/
void QMatrix4x4::ortho(float left, float right, float bottom, float top, float nearPlane, float farPlane)
{
    // Bail out if the projection volume is zero-sized.
    if (left == right || bottom == top || nearPlane == farPlane)
        return;
 
    // Construct the projection.
    const float width = right - left;
    const float invheight = top - bottom;
    const float clip = farPlane - nearPlane;
    QMatrix4x4 m(Qt::Uninitialized);
    m.m[0][0] = 2.0f / width;
    m.m[1][0] = 0.0f;
    m.m[2][0] = 0.0f;
    m.m[3][0] = -(left + right) / width;
    m.m[0][1] = 0.0f;
    m.m[1][1] = 2.0f / invheight;
    m.m[2][1] = 0.0f;
    m.m[3][1] = -(top + bottom) / invheight;
    m.m[0][2] = 0.0f;
    m.m[1][2] = 0.0f;
    m.m[2][2] = -2.0f / clip;
    m.m[3][2] = -(nearPlane + farPlane) / clip;
    m.m[0][3] = 0.0f;
    m.m[1][3] = 0.0f;
    m.m[2][3] = 0.0f;
    m.m[3][3] = 1.0f;
    m.flagBits = Translation | Scale;
 
    // Apply the projection.
    *this *= m;
}
 
/*!
    Multiplies this matrix by another that applies a perspective
    frustum projection for a window with lower-left corner (\a left, \a bottom),
    upper-right corner (\a right, \a top), and the specified \a nearPlane
    and \a farPlane clipping planes.
 
    \sa ortho(), perspective()
*/
void QMatrix4x4::frustum(float left, float right, float bottom, float top, float nearPlane, float farPlane)
{
    // Bail out if the projection volume is zero-sized.
    if (left == right || bottom == top || nearPlane == farPlane)
        return;
 
    // Construct the projection.
    QMatrix4x4 m(Qt::Uninitialized);
    const float width = right - left;
    const float invheight = top - bottom;
    const float clip = farPlane - nearPlane;
    m.m[0][0] = 2.0f * nearPlane / width;
    m.m[1][0] = 0.0f;
    m.m[2][0] = (left + right) / width;
    m.m[3][0] = 0.0f;
    m.m[0][1] = 0.0f;
    m.m[1][1] = 2.0f * nearPlane / invheight;
    m.m[2][1] = (top + bottom) / invheight;
    m.m[3][1] = 0.0f;
    m.m[0][2] = 0.0f;
    m.m[1][2] = 0.0f;
    m.m[2][2] = -(nearPlane + farPlane) / clip;
    m.m[3][2] = -2.0f * nearPlane * farPlane / clip;
    m.m[0][3] = 0.0f;
    m.m[1][3] = 0.0f;
    m.m[2][3] = -1.0f;
    m.m[3][3] = 0.0f;
    m.flagBits = General;
 
    // Apply the projection.
    *this *= m;
}
 
/*!
    Multiplies this matrix by another that applies a perspective
    projection. The vertical field of view will be \a verticalAngle degrees
    within a window with a given \a aspectRatio that determines the horizontal
    field of view.
    The projection will have the specified \a nearPlane and \a farPlane clipping
    planes which are the distances from the viewer to the corresponding planes.
 
    \sa ortho(), frustum()
*/
void QMatrix4x4::perspective(float verticalAngle, float aspectRatio, float nearPlane, float farPlane)
{
    // Bail out if the projection volume is zero-sized.
    if (nearPlane == farPlane || aspectRatio == 0.0f)
        return;
 
    // Construct the projection.
    QMatrix4x4 m(Qt::Uninitialized);
    const float radians = qDegreesToRadians(verticalAngle / 2.0f);
    const float sine = std::sin(radians);
    if (sine == 0.0f)
        return;
    float cotan = std::cos(radians) / sine;
    float clip = farPlane - nearPlane;
    m.m[0][0] = cotan / aspectRatio;
    m.m[1][0] = 0.0f;
    m.m[2][0] = 0.0f;
    m.m[3][0] = 0.0f;
    m.m[0][1] = 0.0f;
    m.m[1][1] = cotan;
    m.m[2][1] = 0.0f;
    m.m[3][1] = 0.0f;
    m.m[0][2] = 0.0f;
    m.m[1][2] = 0.0f;
    m.m[2][2] = -(nearPlane + farPlane) / clip;
    m.m[3][2] = -(2.0f * nearPlane * farPlane) / clip;
    m.m[0][3] = 0.0f;
    m.m[1][3] = 0.0f;
    m.m[2][3] = -1.0f;
    m.m[3][3] = 0.0f;
    m.flagBits = General;
 
    // Apply the projection.
    *this *= m;
}
 
#ifndef QT_NO_VECTOR3D
 
/*!
    Multiplies this matrix by a viewing matrix derived from an eye
    point. The \a center value indicates the center of the view that
    the \a eye is looking at.  The \a up value indicates which direction
    should be considered up with respect to the \a eye.
 
    \note The \a up vector must not be parallel to the line of sight
    from \a eye to \a center.
*/
void QMatrix4x4::lookAt(const QVector3D& eye, const QVector3D& center, const QVector3D& up)
{
    QVector3D forward = center - eye;
    if (qFuzzyIsNull(forward.x()) && qFuzzyIsNull(forward.y()) && qFuzzyIsNull(forward.z()))
        return;
 
    forward.normalize();
    QVector3D side = QVector3D::crossProduct(forward, up).normalized();
    QVector3D upVector = QVector3D::crossProduct(side, forward);
 
    QMatrix4x4 m(Qt::Uninitialized);
    m.m[0][0] = side.x();
    m.m[1][0] = side.y();
    m.m[2][0] = side.z();
    m.m[3][0] = 0.0f;
    m.m[0][1] = upVector.x();
    m.m[1][1] = upVector.y();
    m.m[2][1] = upVector.z();
    m.m[3][1] = 0.0f;
    m.m[0][2] = -forward.x();
    m.m[1][2] = -forward.y();
    m.m[2][2] = -forward.z();
    m.m[3][2] = 0.0f;
    m.m[0][3] = 0.0f;
    m.m[1][3] = 0.0f;
    m.m[2][3] = 0.0f;
    m.m[3][3] = 1.0f;
    m.flagBits = Rotation;
 
    *this *= m;
    translate(-eye);
}
 
#endif
 
/*!
    \fn void QMatrix4x4::viewport(const QRectF &rect)
    \overload
 
    Sets up viewport transform for viewport bounded by \a rect and with near and far set
    to 0 and 1 respectively.
*/
 
/*!
    Multiplies this matrix by another that performs the scale and bias
    transformation used by OpenGL to transform from normalized device
    coordinates (NDC) to viewport (window) coordinates. That is it maps
    points from the cube ranging over [-1, 1] in each dimension to the
    viewport with it's near-lower-left corner at (\a left, \a bottom, \a nearPlane)
    and with size (\a width, \a height, \a farPlane - \a nearPlane).
 
    This matches the transform used by the fixed function OpenGL viewport
    transform controlled by the functions glViewport() and glDepthRange().
 */
void QMatrix4x4::viewport(float left, float bottom, float width, float height, float nearPlane, float farPlane)
{
    const float w2 = width / 2.0f;
    const float h2 = height / 2.0f;
 
    QMatrix4x4 m(Qt::Uninitialized);
    m.m[0][0] = w2;
    m.m[1][0] = 0.0f;
    m.m[2][0] = 0.0f;
    m.m[3][0] = left + w2;
    m.m[0][1] = 0.0f;
    m.m[1][1] = h2;
    m.m[2][1] = 0.0f;
    m.m[3][1] = bottom + h2;
    m.m[0][2] = 0.0f;
    m.m[1][2] = 0.0f;
    m.m[2][2] = (farPlane - nearPlane) / 2.0f;
    m.m[3][2] = (nearPlane + farPlane) / 2.0f;
    m.m[0][3] = 0.0f;
    m.m[1][3] = 0.0f;
    m.m[2][3] = 0.0f;
    m.m[3][3] = 1.0f;
    m.flagBits = General;
 
    *this *= m;
}
 
/*!
    \deprecated
 
    Flips between right-handed and left-handed coordinate systems
    by multiplying the y and z coordinates by -1.  This is normally
    used to create a left-handed orthographic view without scaling
    the viewport as ortho() does.
 
    \sa ortho()
*/
void QMatrix4x4::flipCoordinates()
{
    // Multiplying the y and z coordinates with -1 does NOT flip between right-handed and
    // left-handed coordinate systems, it just rotates 180 degrees around the x axis, so
    // I'm deprecating this function.
    if (flagBits < Rotation2D) {
        // Translation | Scale
        m[1][1] = -m[1][1];
        m[2][2] = -m[2][2];
    } else {
        m[1][0] = -m[1][0];
        m[1][1] = -m[1][1];
        m[1][2] = -m[1][2];
        m[1][3] = -m[1][3];
        m[2][0] = -m[2][0];
        m[2][1] = -m[2][1];
        m[2][2] = -m[2][2];
        m[2][3] = -m[2][3];
    }
    flagBits |= Scale;
}
 
/*!
    Retrieves the 16 items in this matrix and copies them to \a values
    in row-major order.
*/
void QMatrix4x4::copyDataTo(float *values) const
{
    for (int row = 0; row < 4; ++row)
        for (int col = 0; col < 4; ++col)
            values[row * 4 + col] = float(m[col][row]);
}
 
/*!
    Returns the conventional Qt 2D transformation matrix that
    corresponds to this matrix.
 
    The returned QTransform is formed by simply dropping the
    third row and third column of the QMatrix4x4.  This is suitable
    for implementing orthographic projections where the z coordinate
    should be dropped rather than projected.
*/
QTransform QMatrix4x4::toTransform() const
{
    return QTransform(m[0][0], m[0][1], m[0][3],
                      m[1][0], m[1][1], m[1][3],
                      m[3][0], m[3][1], m[3][3]);
}
 
/*!
    Returns the conventional Qt 2D transformation matrix that
    corresponds to this matrix.
 
    If \a distanceToPlane is non-zero, it indicates a projection
    factor to use to adjust for the z coordinate.  The value of
    1024 corresponds to the projection factor used
    by QTransform::rotate() for the x and y axes.
 
    If \a distanceToPlane is zero, then the returned QTransform
    is formed by simply dropping the third row and third column
    of the QMatrix4x4.  This is suitable for implementing
    orthographic projections where the z coordinate should
    be dropped rather than projected.
*/
QTransform QMatrix4x4::toTransform(float distanceToPlane) const
{
    if (distanceToPlane == 1024.0f) {
        // Optimize the common case with constants.
        return QTransform(m[0][0], m[0][1], m[0][3] - m[0][2] * inv_dist_to_plane,
                          m[1][0], m[1][1], m[1][3] - m[1][2] * inv_dist_to_plane,
                          m[3][0], m[3][1], m[3][3] - m[3][2] * inv_dist_to_plane);
    } else if (distanceToPlane != 0.0f) {
        // The following projection matrix is pre-multiplied with "matrix":
        //      | 1 0 0 0 |
        //      | 0 1 0 0 |
        //      | 0 0 1 0 |
        //      | 0 0 d 1 |
        // where d = -1 / distanceToPlane.  After projection, row 3 and
        // column 3 are dropped to form the final QTransform.
        float d = 1.0f / distanceToPlane;
        return QTransform(m[0][0], m[0][1], m[0][3] - m[0][2] * d,
                          m[1][0], m[1][1], m[1][3] - m[1][2] * d,
                          m[3][0], m[3][1], m[3][3] - m[3][2] * d);
    } else {
        // Orthographic projection: drop row 3 and column 3.
        return QTransform(m[0][0], m[0][1], m[0][3],
                          m[1][0], m[1][1], m[1][3],
                          m[3][0], m[3][1], m[3][3]);
    }
}
 
/*!
    \fn QPoint QMatrix4x4::map(const QPoint& point) const
 
    Maps \a point by multiplying this matrix by \a point.
    The matrix is applied pre-point.
 
    \sa mapRect()
*/
 
/*!
    \fn QPointF QMatrix4x4::map(const QPointF& point) const
 
    Maps \a point by post-multiplying this matrix by \a point.
    The matrix is applied pre-point.
 
    \sa mapRect()
*/
 
#ifndef QT_NO_VECTOR3D
 
/*!
    \fn QVector3D QMatrix4x4::map(const QVector3D& point) const
 
    Maps \a point by multiplying this matrix by \a point extended to a 4D
    vector by assuming 1.0 for the w coordinate. The matrix is applied
    pre-point.
 
    \note This function is not the same as mapVector(). For points, always use
    map(). mapVector() is suitable for vectors (directions) only.
 
    \sa mapRect(), mapVector()
*/
 
/*!
    \fn QVector3D QMatrix4x4::mapVector(const QVector3D& vector) const
 
    Maps \a vector by multiplying the top 3x3 portion of this matrix
    by \a vector.  The translation and projection components of
    this matrix are ignored. The matrix is applied pre-vector.
 
    \sa map()
*/
 
#endif
 
#ifndef QT_NO_VECTOR4D
 
/*!
    \fn QVector4D QMatrix4x4::map(const QVector4D& point) const;
 
    Maps \a point by multiplying this matrix by \a point.
    The matrix is applied pre-point.
 
    \sa mapRect()
*/
 
#endif
 
/*!
    Maps \a rect by multiplying this matrix by the corners
    of \a rect and then forming a new rectangle from the results.
    The returned rectangle will be an ordinary 2D rectangle
    with sides parallel to the horizontal and vertical axes.
 
    \sa map()
*/
QRect QMatrix4x4::mapRect(const QRect& rect) const
{
    if (flagBits < Scale) {
        // Translation
        return QRect(qRound(rect.x() + m[3][0]),
                     qRound(rect.y() + m[3][1]),
                     rect.width(), rect.height());
    } else if (flagBits < Rotation2D) {
        // Translation | Scale
        float x = rect.x() * m[0][0] + m[3][0];
        float y = rect.y() * m[1][1] + m[3][1];
        float w = rect.width() * m[0][0];
        float h = rect.height() * m[1][1];
        if (w < 0) {
            w = -w;
            x -= w;
        }
        if (h < 0) {
            h = -h;
            y -= h;
        }
        return QRect(qRound(x), qRound(y), qRound(w), qRound(h));
    }
 
    QPoint tl = map(rect.topLeft());
    QPoint tr = map(QPoint(rect.x() + rect.width(), rect.y()));
    QPoint bl = map(QPoint(rect.x(), rect.y() + rect.height()));
    QPoint br = map(QPoint(rect.x() + rect.width(),
                           rect.y() + rect.height()));
 
    int xmin = qMin(qMin(tl.x(), tr.x()), qMin(bl.x(), br.x()));
    int xmax = qMax(qMax(tl.x(), tr.x()), qMax(bl.x(), br.x()));
    int ymin = qMin(qMin(tl.y(), tr.y()), qMin(bl.y(), br.y()));
    int ymax = qMax(qMax(tl.y(), tr.y()), qMax(bl.y(), br.y()));
 
    return QRect(xmin, ymin, xmax - xmin, ymax - ymin);
}
 
/*!
    Maps \a rect by multiplying this matrix by the corners
    of \a rect and then forming a new rectangle from the results.
    The returned rectangle will be an ordinary 2D rectangle
    with sides parallel to the horizontal and vertical axes.
 
    \sa map()
*/
QRectF QMatrix4x4::mapRect(const QRectF& rect) const
{
    if (flagBits < Scale) {
        // Translation
        return rect.translated(m[3][0], m[3][1]);
    } else if (flagBits < Rotation2D) {
        // Translation | Scale
        float x = rect.x() * m[0][0] + m[3][0];
        float y = rect.y() * m[1][1] + m[3][1];
        float w = rect.width() * m[0][0];
        float h = rect.height() * m[1][1];
        if (w < 0) {
            w = -w;
            x -= w;
        }
        if (h < 0) {
            h = -h;
            y -= h;
        }
        return QRectF(x, y, w, h);
    }
 
    QPointF tl = map(rect.topLeft()); QPointF tr = map(rect.topRight());
    QPointF bl = map(rect.bottomLeft()); QPointF br = map(rect.bottomRight());
 
    float xmin = qMin(qMin(tl.x(), tr.x()), qMin(bl.x(), br.x()));
    float xmax = qMax(qMax(tl.x(), tr.x()), qMax(bl.x(), br.x()));
    float ymin = qMin(qMin(tl.y(), tr.y()), qMin(bl.y(), br.y()));
    float ymax = qMax(qMax(tl.y(), tr.y()), qMax(bl.y(), br.y()));
 
    return QRectF(QPointF(xmin, ymin), QPointF(xmax, ymax));
}
 
/*!
    \fn float *QMatrix4x4::data()
 
    Returns a pointer to the raw data of this matrix.
 
    \sa constData(), optimize()
*/
 
/*!
    \fn const float *QMatrix4x4::data() const
 
    Returns a constant pointer to the raw data of this matrix.
    This raw data is stored in column-major format.
 
    \sa constData()
*/
 
/*!
    \fn const float *QMatrix4x4::constData() const
 
    Returns a constant pointer to the raw data of this matrix.
    This raw data is stored in column-major format.
 
    \sa data()
*/
 
// Helper routine for inverting orthonormal matrices that consist
// of just rotations and translations.
QMatrix4x4 QMatrix4x4::orthonormalInverse() const
{
    QMatrix4x4 result(Qt::Uninitialized);
 
    result.m[0][0] = m[0][0];
    result.m[1][0] = m[0][1];
    result.m[2][0] = m[0][2];
 
    result.m[0][1] = m[1][0];
    result.m[1][1] = m[1][1];
    result.m[2][1] = m[1][2];
 
    result.m[0][2] = m[2][0];
    result.m[1][2] = m[2][1];
    result.m[2][2] = m[2][2];
 
    result.m[0][3] = 0.0f;
    result.m[1][3] = 0.0f;
    result.m[2][3] = 0.0f;
 
    result.m[3][0] = -(result.m[0][0] * m[3][0] + result.m[1][0] * m[3][1] + result.m[2][0] * m[3][2]);
    result.m[3][1] = -(result.m[0][1] * m[3][0] + result.m[1][1] * m[3][1] + result.m[2][1] * m[3][2]);
    result.m[3][2] = -(result.m[0][2] * m[3][0] + result.m[1][2] * m[3][1] + result.m[2][2] * m[3][2]);
    result.m[3][3] = 1.0f;
 
    result.flagBits = flagBits;
 
    return result;
}
 
/*!
    Optimize the usage of this matrix from its current elements.
 
    Some operations such as translate(), scale(), and rotate() can be
    performed more efficiently if the matrix being modified is already
    known to be the identity, a previous translate(), a previous
    scale(), etc.
 
    Normally the QMatrix4x4 class keeps track of this special type internally
    as operations are performed.  However, if the matrix is modified
    directly with operator()(int, int) or data(), then QMatrix4x4 will
    lose track of the special type and will revert to the safest but least
    efficient operations thereafter.
 
    By calling optimize() after directly modifying the matrix,
    the programmer can force QMatrix4x4 to recover the special type if
    the elements appear to conform to one of the known optimized types.
 
    \sa operator()(int, int), data(), translate()
*/
void QMatrix4x4::optimize()
{
    // If the last row is not (0, 0, 0, 1), the matrix is not a special type.
    flagBits = General;
    if (m[0][3] != 0 || m[1][3] != 0 || m[2][3] != 0 || m[3][3] != 1)
        return;
 
    flagBits &= ~Perspective;
 
    // If the last column is (0, 0, 0, 1), then there is no translation.
    if (m[3][0] == 0 && m[3][1] == 0 && m[3][2] == 0)
        flagBits &= ~Translation;
 
    // If the two first elements of row 3 and column 3 are 0, then any rotation must be about Z.
    if (!m[0][2] && !m[1][2] && !m[2][0] && !m[2][1]) {
        flagBits &= ~Rotation;
        // If the six non-diagonal elements in the top left 3x3 matrix are 0, there is no rotation.
        if (!m[0][1] && !m[1][0]) {
            flagBits &= ~Rotation2D;
            // Check for identity.
            if (m[0][0] == 1 && m[1][1] == 1 && m[2][2] == 1)
                flagBits &= ~Scale;
        } else {
            // If the columns are orthonormal and form a right-handed system, then there is no scale.
            double mm[4][4];
            copyToDoubles(m, mm);
            double det = matrixDet2(mm, 0, 1, 0, 1);
            double lenX = mm[0][0] * mm[0][0] + mm[0][1] * mm[0][1];
            double lenY = mm[1][0] * mm[1][0] + mm[1][1] * mm[1][1];
            double lenZ = mm[2][2];
            if (qFuzzyCompare(det, 1.0) && qFuzzyCompare(lenX, 1.0)
                    && qFuzzyCompare(lenY, 1.0) && qFuzzyCompare(lenZ, 1.0))
            {
                flagBits &= ~Scale;
            }
        }
    } else {
        // If the columns are orthonormal and form a right-handed system, then there is no scale.
        double mm[4][4];
        copyToDoubles(m, mm);
        double det = matrixDet3(mm, 0, 1, 2, 0, 1, 2);
        double lenX = mm[0][0] * mm[0][0] + mm[0][1] * mm[0][1] + mm[0][2] * mm[0][2];
        double lenY = mm[1][0] * mm[1][0] + mm[1][1] * mm[1][1] + mm[1][2] * mm[1][2];
        double lenZ = mm[2][0] * mm[2][0] + mm[2][1] * mm[2][1] + mm[2][2] * mm[2][2];
        if (qFuzzyCompare(det, 1.0) && qFuzzyCompare(lenX, 1.0)
                && qFuzzyCompare(lenY, 1.0) && qFuzzyCompare(lenZ, 1.0))
        {
            flagBits &= ~Scale;
        }
    }
}
 
/*!
    Returns the matrix as a QVariant.
*/
QMatrix4x4::operator QVariant() const
{
    return QVariant::fromValue(*this);
}
 
#ifndef QT_NO_DEBUG_STREAM
 
QDebug operator<<(QDebug dbg, const QMatrix4x4 &m)
{
    QDebugStateSaver saver(dbg);
    // Create a string that represents the matrix type.
    QByteArray bits;
    if (m.flagBits == QMatrix4x4::Identity) {
        bits = "Identity";
    } else if (m.flagBits == QMatrix4x4::General) {
        bits = "General";
    } else {
        if ((m.flagBits & QMatrix4x4::Translation) != 0)
            bits += "Translation,";
        if ((m.flagBits & QMatrix4x4::Scale) != 0)
            bits += "Scale,";
        if ((m.flagBits & QMatrix4x4::Rotation2D) != 0)
            bits += "Rotation2D,";
        if ((m.flagBits & QMatrix4x4::Rotation) != 0)
            bits += "Rotation,";
        if ((m.flagBits & QMatrix4x4::Perspective) != 0)
            bits += "Perspective,";
        bits.chop(1);
    }
 
    // Output in row-major order because it is more human-readable.
    dbg.nospace() << "QMatrix4x4(type:" << bits.constData() << Qt::endl
        << qSetFieldWidth(10)
        << m(0, 0) << m(0, 1) << m(0, 2) << m(0, 3) << Qt::endl
        << m(1, 0) << m(1, 1) << m(1, 2) << m(1, 3) << Qt::endl
        << m(2, 0) << m(2, 1) << m(2, 2) << m(2, 3) << Qt::endl
        << m(3, 0) << m(3, 1) << m(3, 2) << m(3, 3) << Qt::endl
        << qSetFieldWidth(0) << ')';
    return dbg;
}
 
#endif
 
#ifndef QT_NO_DATASTREAM
 
/*!
    \fn QDataStream &operator<<(QDataStream &stream, const QMatrix4x4 &matrix)
    \relates QMatrix4x4
 
    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 &stream, const QMatrix4x4 &matrix)
{
    for (int row = 0; row < 4; ++row)
        for (int col = 0; col < 4; ++col)
            stream << matrix(row, col);
    return stream;
}
 
/*!
    \fn QDataStream &operator>>(QDataStream &stream, QMatrix4x4 &matrix)
    \relates QMatrix4x4
 
    Reads a 4x4 matrix from the given \a stream into the given \a matrix
    and returns a reference to the stream.
 
    \sa {Serializing Qt Data Types}
*/
 
QDataStream &operator>>(QDataStream &stream, QMatrix4x4 &matrix)
{
    float x;
    for (int row = 0; row < 4; ++row) {
        for (int col = 0; col < 4; ++col) {
            stream >> x;
            matrix(row, col) = x;
        }
    }
    matrix.optimize();
    return stream;
}
 
#endif // QT_NO_DATASTREAM
 
#endif // QT_NO_MATRIX4X4
 
QT_END_NAMESPACE