path: root/src/kernel/qwmatrix.cpp

/****************************************************************************
**
** Implementation of QWMatrix class
**
** Created : 941020
**
** Copyright (C) 1992-2008 Trolltech ASA.  All rights reserved.
**
** This file is part of the kernel module of the Qt GUI Toolkit.
**
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** Software Foundation and appearing in the files LICENSE.GPL2
** and LICENSE.GPL3 included in the packaging of this file.
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** and the KDE Free Qt Foundation.
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**********************************************************************/

#include "qwmatrix.h"
#include "qdatastream.h"
#include "qregion.h"
#if defined(Q_WS_X11)
double qsincos( double, bool calcCos );		// defined in qpainter_x11.cpp
#else
#include <math.h>
#endif

#include <limits.h>

#ifndef QT_NO_WMATRIX

/*!
    \class QWMatrix qwmatrix.h
    \brief The QWMatrix class specifies 2D transformations of a
    coordinate system.

    \ingroup graphics
    \ingroup images

    The standard coordinate system of a \link QPaintDevice paint
    device\endlink has the origin located at the top-left position. X
    values increase to the right; Y values increase downward.

    This coordinate system is the default for the QPainter, which
    renders graphics in a paint device. A user-defined coordinate
    system can be specified by setting a QWMatrix for the painter.

    Example:
    \code
	MyWidget::paintEvent( QPaintEvent * )
	{
            QPainter p;                      // our painter
            QWMatrix m;                      // our transformation matrix
            m.rotate( 22.5 );                // rotated coordinate system
            p.begin( this );                 // start painting
            p.setWorldMatrix( m );           // use rotated coordinate system
            p.drawText( 30,20, "detator" );  // draw rotated text at 30,20
            p.end();                         // painting done
	}
    \endcode

    A matrix specifies how to translate, scale, shear or rotate the
    graphics; the actual transformation is performed by the drawing
    routines in QPainter and by QPixmap::xForm().

    The QWMatrix class contains a 3x3 matrix of the form:
    <table align=center border=1 cellpadding=1 cellspacing=0>
    <tr align=center><td>m11</td><td>m12</td><td>&nbsp;0 </td></tr>
    <tr align=center><td>m21</td><td>m22</td><td>&nbsp;0 </td></tr>
    <tr align=center><td>dx</td> <td>dy</td> <td>&nbsp;1 </td></tr>
    </table>

    A matrix transforms a point in the plane to another point:
    \code
	x' = m11*x + m21*y + dx
	y' = m22*y + m12*x + dy
    \endcode

    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 \link
    QWMatrix::invert() inverted matrix\endlink.

    The elements \e dx and \e dy specify horizontal and vertical
    translation. The elements \e m11 and \e m22 specify horizontal and
    vertical scaling. The elements \e m12 and \e m21 specify
    horizontal and vertical shearing.

    The identity matrix has \e m11 and \e m22 set to 1; all others are
    set to 0. This matrix maps a point to itself.

    Translation is the simplest transformation. Setting \e dx and \e
    dy will move the coordinate system \e dx units along the X axis
    and \e dy units along the Y axis.

    Scaling can be done by setting \e m11 and \e m22. For example,
    setting \e m11 to 2 and \e m22 to 1.5 will double the height and
    increase the width by 50%.

    Shearing is controlled by \e m12 and \e m21. Setting these
    elements to values different from zero will twist the coordinate
    system.

    Rotation is achieved by carefully setting both the shearing
    factors and the scaling factors. The QWMatrix also has a function
    that sets \link rotate() rotation \endlink directly.

    QWMatrix lets you combine transformations like this:
    \code
	QWMatrix m;           // identity matrix
	m.translate(10, -20); // first translate (10,-20)
	m.rotate(25);         // then rotate 25 degrees
	m.scale(1.2, 0.7);    // finally scale it
    \endcode

    Here's the same example using basic matrix operations:
    \code
	double a    = pi/180 * 25;         // convert 25 to radians
	double sina = sin(a);
	double cosa = cos(a);
	QWMatrix m1(1, 0, 0, 1, 10, -20);  // translation matrix
	QWMatrix m2( cosa, sina,           // rotation matrix
		    -sina, cosa, 0, 0 );
	QWMatrix m3(1.2, 0, 0, 0.7, 0, 0); // scaling matrix
	QWMatrix m;
	m = m3 * m2 * m1;                  // combine all transformations
    \endcode

    \l QPainter has functions to translate, scale, shear and rotate the
    coordinate system without using a QWMatrix. Although these
    functions are very convenient, it can be more efficient to build a
    QWMatrix and call QPainter::setWorldMatrix() if you want to perform
    more than a single transform operation.

    \sa QPainter::setWorldMatrix(), QPixmap::xForm()
*/

bool qt_old_transformations = TRUE;

/*!
    \enum QWMatrix::TransformationMode

    \keyword transformation matrix

    QWMatrix offers two transformation modes. Calculations can either
    be done in terms of points (Points mode, the default), or in
    terms of area (Area mode).

    In Points mode the transformation is applied to the points that
    mark out the shape's bounding line. In Areas mode the
    transformation is applied in such a way that the area of the
    contained region is correctly transformed under the matrix.

    \value Points transformations are applied to the shape's points.
    \value Areas transformations are applied (e.g. to the width and
    height) so that the area is transformed.

    Example:

    Suppose we have a rectangle,
    \c{QRect( 10, 20, 30, 40 )} and a transformation matrix
    \c{QWMatrix( 2, 0, 0, 2, 0, 0 )} to double the rectangle's size.

    In Points mode, the matrix will transform the top-left (10,20) and
    the bottom-right (39,59) points producing a rectangle with its
    top-left point at (20,40) and its bottom-right point at (78,118),
    i.e. with a width of 59 and a height of 79.

    In Areas mode, the matrix will transform the top-left point in
    the same way as in Points mode to (20/40), and double the width
    and height, so the bottom-right will become (69,99), i.e. a width
    of 60 and a height of 80.

    Because integer arithmetic is used (for speed), rounding
    differences mean that the modes will produce slightly different
    results given the same shape and the same transformation,
    especially when scaling up. This also means that some operations
    are not commutative.

    Under Points mode, \c{matrix * ( region1 | region2 )} is not equal to
    \c{matrix * region1 | matrix * region2}. Under Area mode, \c{matrix *
    (pointarray[i])} is not neccesarily equal to
    \c{(matrix * pointarry)[i]}.

    \img xform.png Comparison of Points and Areas TransformationModes
*/

/*!
    Sets the transformation mode that QWMatrix and painter
    transformations use to \a m.

    \sa QWMatrix::TransformationMode
*/
void QWMatrix::setTransformationMode( QWMatrix::TransformationMode m )
{
    if ( m == QWMatrix::Points )
	qt_old_transformations = TRUE;
    else
	qt_old_transformations = FALSE;
}


/*!
    Returns the current transformation mode.

    \sa QWMatrix::TransformationMode
*/
QWMatrix::TransformationMode QWMatrix::transformationMode()
{
    return (qt_old_transformations ? QWMatrix::Points : QWMatrix::Areas );
}


// some defines to inline some code
#define MAPDOUBLE( x, y, nx, ny ) \
{ \
    double fx = x; \
    double fy = y; \
    nx = _m11*fx + _m21*fy + _dx; \
    ny = _m12*fx + _m22*fy + _dy; \
}

#define MAPINT( x, y, nx, ny ) \
{ \
    double fx = x; \
    double fy = y; \
    nx = qRound(_m11*fx + _m21*fy + _dx); \
    ny = qRound(_m12*fx + _m22*fy + _dy); \
}

/*****************************************************************************
  QWMatrix member functions
 *****************************************************************************/

/*!
    Constructs an identity matrix. All elements are set to zero except
    \e m11 and \e m22 (scaling), which are set to 1.
*/

QWMatrix::QWMatrix()
{
    _m11 = _m22 = 1.0;
    _m12 = _m21 = _dx = _dy = 0.0;
}

/*!
    Constructs a matrix with the elements, \a m11, \a m12, \a m21, \a
    m22, \a dx and \a dy.
*/

QWMatrix::QWMatrix( double m11, double m12, double m21, double m22,
		    double dx, double dy )
{
    _m11 = m11;	 _m12 = m12;
    _m21 = m21;	 _m22 = m22;
    _dx	 = dx;	 _dy  = dy;
}


/*!
    Sets the matrix elements to the specified values, \a m11, \a m12,
    \a m21, \a m22, \a dx and \a dy.
*/

void QWMatrix::setMatrix( double m11, double m12, double m21, double m22,
			  double dx, double dy )
{
    _m11 = m11;	 _m12 = m12;
    _m21 = m21;	 _m22 = m22;
    _dx	 = dx;	 _dy  = dy;
}


/*!
    \fn double QWMatrix::m11() const

    Returns the X scaling factor.
*/

/*!
    \fn double QWMatrix::m12() const

    Returns the vertical shearing factor.
*/

/*!
    \fn double QWMatrix::m21() const

    Returns the horizontal shearing factor.
*/

/*!
    \fn double QWMatrix::m22() const

    Returns the Y scaling factor.
*/

/*!
    \fn double QWMatrix::dx() const

    Returns the horizontal translation.
*/

/*!
    \fn double QWMatrix::dy() const

    Returns the vertical translation.
*/


/*!
    \overload

    Transforms ( \a x, \a y ) to ( \a *tx, \a *ty ) using the
    following formulae:

    \code
	*tx = m11*x + m21*y + dx
	*ty = m22*y + m12*x + dy
    \endcode
*/

void QWMatrix::map( double x, double y, double *tx, double *ty ) const
{
    MAPDOUBLE( x, y, *tx, *ty );
}

/*!
    Transforms ( \a x, \a y ) to ( \a *tx, \a *ty ) using the formulae:

    \code
	*tx = m11*x + m21*y + dx  (rounded to the nearest integer)
	*ty = m22*y + m12*x + dy  (rounded to the nearest integer)
    \endcode
*/

void QWMatrix::map( int x, int y, int *tx, int *ty ) const
{
    MAPINT( x, y, *tx, *ty );
}

/*!
    \fn QPoint QWMatrix::map( const QPoint &p ) const

    \overload

    Transforms \a p to using the formulae:

    \code
	retx = m11*px + m21*py + dx  (rounded to the nearest integer)
	rety = m22*py + m12*px + dy  (rounded to the nearest integer)
    \endcode
*/

/*!
  \fn QRect QWMatrix::map( const QRect &r ) const

  \obsolete

  Please use \l QWMatrix::mapRect() instead.

  Note that this method does return the bounding rectangle of the \a r, when
  shearing or rotations are used.
*/

/*!
    \fn QPointArray QWMatrix::map( const QPointArray &a ) const

    \overload

    Returns the point array \a a transformed by calling map for each point.
*/


/*!
    \fn QRegion QWMatrix::map( const QRegion &r ) const

    \overload

    Transforms the region \a r.

    Calling this method can be rather expensive, if rotations or
    shearing are used.
*/

/*!
    \fn QRegion QWMatrix::mapToRegion( const QRect &rect ) const

    Returns the transformed rectangle \a rect.

    A rectangle which has been rotated or sheared may result in a
    non-rectangular region being returned.

    Calling this method can be expensive, if rotations or shearing are
    used. If you just need to know the bounding rectangle of the
    returned region, use mapRect() which is a lot faster than this
    function.

    \sa QWMatrix::mapRect()
*/


/*!
    Returns the transformed rectangle \a rect.

    The bounding rectangle is returned if rotation or shearing has
    been specified.

    If you need to know the exact region \a rect maps to use \l
    operator*().

    \sa operator*()
*/

QRect QWMatrix::mapRect( const QRect &rect ) const
{
    QRect result;
    if( qt_old_transformations ) {
	if ( _m12 == 0.0F && _m21 == 0.0F ) {
	    result = QRect( map(rect.topLeft()), map(rect.bottomRight()) ).normalize();
	} else {
	    QPointArray a( rect );
	    a = map( a );
	    result = a.boundingRect();
	}
    } else {
	if ( _m12 == 0.0F && _m21 == 0.0F ) {
	    int x = qRound( _m11*rect.x() + _dx );
	    int y = qRound( _m22*rect.y() + _dy );
	    int w = qRound( _m11*rect.width() );
	    int h = qRound( _m22*rect.height() );
	    if ( w < 0 ) {
		w = -w;
		x -= w-1;
	    }
	    if ( h < 0 ) {
		h = -h;
		y -= h-1;
	    }
	    result = QRect( x, y, w, h );
	} else {

	    // see mapToPolygon for explanations of the algorithm.
	    double x0, y0;
	    double x, y;
	    MAPDOUBLE( rect.left(), rect.top(), x0, y0 );
	    double xmin = x0;
	    double ymin = y0;
	    double xmax = x0;
	    double ymax = y0;
	    MAPDOUBLE( rect.right() + 1, rect.top(), x, y );
	    xmin = QMIN( xmin, x );
	    ymin = QMIN( ymin, y );
	    xmax = QMAX( xmax, x );
	    ymax = QMAX( ymax, y );
	    MAPDOUBLE( rect.right() + 1, rect.bottom() + 1, x, y );
	    xmin = QMIN( xmin, x );
	    ymin = QMIN( ymin, y );
	    xmax = QMAX( xmax, x );
	    ymax = QMAX( ymax, y );
	    MAPDOUBLE( rect.left(), rect.bottom() + 1, x, y );
	    xmin = QMIN( xmin, x );
	    ymin = QMIN( ymin, y );
	    xmax = QMAX( xmax, x );
	    ymax = QMAX( ymax, y );
	    double w = xmax - xmin;
	    double h = ymax - ymin;
	    xmin -= ( xmin - x0 ) / w;
	    ymin -= ( ymin - y0 ) / h;
	    xmax -= ( xmax - x0 ) / w;
	    ymax -= ( ymax - y0 ) / h;
	    result = QRect( qRound(xmin), qRound(ymin), qRound(xmax)-qRound(xmin)+1, qRound(ymax)-qRound(ymin)+1 );
	}
    }
    return result;
}


/*!
  \internal
*/
QPoint QWMatrix::operator *( const QPoint &p ) const
{
    double fx = p.x();
    double fy = p.y();
    return QPoint( qRound(_m11*fx + _m21*fy + _dx),
		   qRound(_m12*fx + _m22*fy + _dy) );
}


struct QWMDoublePoint {
    double x;
    double y;
};

/*!
  \internal
*/
QPointArray QWMatrix::operator *( const QPointArray &a ) const
{
    if( qt_old_transformations ) {
	QPointArray result = a.copy();
	int x, y;
	for ( int i=0; i<(int)result.size(); i++ ) {
	    result.point( i, &x, &y );
	    MAPINT( x, y, x, y );
	    result.setPoint( i, x, y );
	}
	return result;
    } else {
	int size = a.size();
	int i;
	QMemArray<QWMDoublePoint> p( size );
	QPoint *da = a.data();
	QWMDoublePoint *dp = p.data();
	double xmin = INT_MAX;
	double ymin = xmin;
	double xmax = INT_MIN;
	double ymax = xmax;
	int xminp = 0;
	int yminp = 0;
	for( i = 0; i < size; i++ ) {
	    dp[i].x = da[i].x();
	    dp[i].y = da[i].y();
	    if ( dp[i].x < xmin ) {
		xmin = dp[i].x;
		xminp = i;
	    }
	    if ( dp[i].y < ymin ) {
		ymin = dp[i].y;
		yminp = i;
	    }
	    xmax = QMAX( xmax, dp[i].x );
	    ymax = QMAX( ymax, dp[i].y );
	}
	double w = QMAX( xmax - xmin, 1 );
	double h = QMAX( ymax - ymin, 1 );
	for( i = 0; i < size; i++ ) {
	    dp[i].x += (dp[i].x - xmin)/w;
	    dp[i].y += (dp[i].y - ymin)/h;
	    MAPDOUBLE( dp[i].x, dp[i].y, dp[i].x, dp[i].y );
	}

	// now apply correction back for transformed values...
	xmin = INT_MAX;
	ymin = xmin;
	xmax = INT_MIN;
	ymax = xmax;
	for( i = 0; i < size; i++ ) {
	    xmin = QMIN( xmin, dp[i].x );
	    ymin = QMIN( ymin, dp[i].y );
	    xmax = QMAX( xmax, dp[i].x );
	    ymax = QMAX( ymax, dp[i].y );
	}
	w = QMAX( xmax - xmin, 1 );
	h = QMAX( ymax - ymin, 1 );

	QPointArray result( size );
	QPoint *dr = result.data();
	for( i = 0; i < size; i++ ) {
	    dr[i].setX( qRound( dp[i].x - (dp[i].x - dp[xminp].x)/w ) );
	    dr[i].setY( qRound( dp[i].y - (dp[i].y - dp[yminp].y)/h ) );
	}
	return result;
    }
}

/*!
\internal
*/
QRegion QWMatrix::operator * (const QRect &rect ) const
{
    QRegion result;
    if ( isIdentity() ) {
	result = rect;
    } else if ( _m12 == 0.0F && _m21 == 0.0F ) {
	if( qt_old_transformations ) {
	    result = QRect( map(rect.topLeft()), map(rect.bottomRight()) ).normalize();
	} else {
	    int x = qRound( _m11*rect.x() + _dx );
	    int y = qRound( _m22*rect.y() + _dy );
	    int w = qRound( _m11*rect.width() );
	    int h = qRound( _m22*rect.height() );
	    if ( w < 0 ) {
		w = -w;
		x -= w - 1;
	    }
	    if ( h < 0 ) {
		h = -h;
		y -= h - 1;
	    }
	    result = QRect( x, y, w, h );
	}
    } else {
	result = QRegion( mapToPolygon( rect ) );
    }
    return result;

}

/*!
    Returns the transformed rectangle \a rect as a polygon.

    Polygons and rectangles behave slightly differently
    when transformed (due to integer rounding), so
    \c{matrix.map( QPointArray( rect ) )} is not always the same as
    \c{matrix.mapToPolygon( rect )}.
*/
QPointArray QWMatrix::mapToPolygon( const QRect &rect ) const
{
    QPointArray a( 4 );
    if ( qt_old_transformations ) {
	a = QPointArray( rect );
	return operator *( a );
    }
    double x[4], y[4];
    if ( _m12 == 0.0F && _m21 == 0.0F ) {
	x[0] = qRound( _m11*rect.x() + _dx );
        y[0] = qRound( _m22*rect.y() + _dy );
	double w = qRound( _m11*rect.width() );
	double h = qRound( _m22*rect.height() );
	if ( w < 0 ) {
	    w = -w;
	    x[0] -= w - 1.;
	}
	if ( h < 0 ) {
	    h = -h;
	    y[0] -= h - 1.;
	}
	x[1] = x[0]+w-1;
	x[2] = x[1];
	x[3] = x[0];
	y[1] = y[0];
	y[2] = y[0]+h-1;
	y[3] = y[2];
    } else {
	MAPINT( rect.left(), rect.top(), x[0], y[0] );
	MAPINT( rect.right() + 1, rect.top(), x[1], y[1] );
	MAPINT( rect.right() + 1, rect.bottom() + 1, x[2], y[2] );
	MAPINT( rect.left(), rect.bottom() + 1, x[3], y[3] );

	/*
	Including rectangles as we have are evil.

        We now have a rectangle that is one pixel to wide and one to
        high. the tranformed position of the top-left corner is
        correct. All other points need some adjustments.

	Doing this mathematically exact would force us to calculate some square roots,
	something we don't want for the sake of speed.

        Instead we use an approximation, that converts to the correct
        answer when m12 -> 0 and m21 -> 0, and accept smaller
        errors in the general transformation case.

        The solution is to calculate the width and height of the
        bounding rect, and scale the points 1/2/3 by (xp-x0)/xw pixel direction
        to point 0.
        */

	double xmin = x[0];
	double ymin = y[0];
	double xmax = x[0];
	double ymax = y[0];
	int i;
	for( i = 1; i< 4; i++ ) {
	    xmin = QMIN( xmin, x[i] );
	    ymin = QMIN( ymin, y[i] );
	    xmax = QMAX( xmax, x[i] );
	    ymax = QMAX( ymax, y[i] );
	}
	double w = xmax - xmin;
	double h = ymax - ymin;

	for( i = 1; i < 4; i++ ) {
	    x[i] -= (x[i] - x[0])/w;
	    y[i] -= (y[i] - y[0])/h;
	}
    }
#if 0
    int i;
    for( i = 0; i< 4; i++ )
	qDebug("coords(%d) = (%f/%f) (%d/%d)", i, x[i], y[i], qRound(x[i]), qRound(y[i]) );
    qDebug( "width=%f, height=%f", sqrt( (x[1]-x[0])*(x[1]-x[0]) + (y[1]-y[0])*(y[1]-y[0]) ),
	    sqrt( (x[0]-x[3])*(x[0]-x[3]) + (y[0]-y[3])*(y[0]-y[3]) ) );
#endif
    // all coordinates are correctly, tranform 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;
}

/*!
\internal
*/
QRegion QWMatrix::operator * (const QRegion &r ) const
{
    if ( isIdentity() )
	return r;
    QMemArray<QRect> rects = r.rects();
    QRegion result;
    QRect *rect = rects.data();
    int i = rects.size();
    if ( _m12 == 0.0F && _m21 == 0.0F && _m11 > 1.0F && _m22 > 1.0F ) {
	// simple case, no rotation
	while ( i ) {
	    int x = qRound( _m11*rect->x() + _dx );
	    int y = qRound( _m22*rect->y() + _dy );
	    int w = qRound( _m11*rect->width() );
	    int h = qRound( _m22*rect->height() );
	    if ( w < 0 ) {
		w = -w;
		x -= w-1;
	    }
	    if ( h < 0 ) {
		h = -h;
		y -= h-1;
	    }
	    *rect = QRect( x, y, w, h );
	    rect++;
	    i--;
	}
	result.setRects( rects.data(), rects.size() );
    } else {
	while ( i ) {
	    result |= operator *( *rect );
	    rect++;
	    i--;
	}
    }
    return result;
}

/*!
    Resets the matrix to an identity matrix.

    All elements are set to zero, except \e m11 and \e m22 (scaling)
    which are set to 1.

    \sa isIdentity()
*/

void QWMatrix::reset()
{
    _m11 = _m22 = 1.0;
    _m12 = _m21 = _dx = _dy = 0.0;
}

/*!
    Returns TRUE if the matrix is the identity matrix; otherwise returns FALSE.

    \sa reset()
*/
bool QWMatrix::isIdentity() const
{
    return _m11 == 1.0 && _m22 == 1.0 && _m12 == 0.0 && _m21 == 0.0
	&& _dx == 0.0 && _dy == 0.0;
}

/*!
    Moves the coordinate system \a dx along the X-axis and \a dy along
    the Y-axis.

    Returns a reference to the matrix.

    \sa scale(), shear(), rotate()
*/

QWMatrix &QWMatrix::translate( double dx, double dy )
{
    _dx += dx*_m11 + dy*_m21;
    _dy += dy*_m22 + dx*_m12;
    return *this;
}

/*!
    Scales the coordinate system unit by \a sx horizontally and \a sy
    vertically.

    Returns a reference to the matrix.

    \sa translate(), shear(), rotate()
*/

QWMatrix &QWMatrix::scale( double sx, double sy )
{
    _m11 *= sx;
    _m12 *= sx;
    _m21 *= sy;
    _m22 *= sy;
    return *this;
}

/*!
    Shears the coordinate system by \a sh horizontally and \a sv
    vertically.

    Returns a reference to the matrix.

    \sa translate(), scale(), rotate()
*/

QWMatrix &QWMatrix::shear( double sh, double sv )
{
    double tm11 = sv*_m21;
    double tm12 = sv*_m22;
    double tm21 = sh*_m11;
    double tm22 = sh*_m12;
    _m11 += tm11;
    _m12 += tm12;
    _m21 += tm21;
    _m22 += tm22;
    return *this;
}

const double deg2rad = 0.017453292519943295769;	// pi/180

/*!
    Rotates the coordinate system \a a degrees counterclockwise.

    Returns a reference to the matrix.

    \sa translate(), scale(), shear()
*/

QWMatrix &QWMatrix::rotate( double a )
{
    double b = deg2rad*a;			// convert to radians
#if defined(Q_WS_X11)
    double sina = qsincos(b,FALSE);		// fast and convenient
    double cosa = qsincos(b,TRUE);
#else
    double sina = sin(b);
    double cosa = cos(b);
#endif
    double tm11 = cosa*_m11 + sina*_m21;
    double tm12 = cosa*_m12 + sina*_m22;
    double tm21 = -sina*_m11 + cosa*_m21;
    double tm22 = -sina*_m12 + cosa*_m22;
    _m11 = tm11; _m12 = tm12;
    _m21 = tm21; _m22 = tm22;
    return *this;
}

/*!
    \fn bool QWMatrix::isInvertible() const

    Returns TRUE if the matrix is invertible; otherwise returns FALSE.

    \sa invert()
*/

/*!
    \fn double QWMatrix::det() const

    Returns the matrix's determinant.
*/


/*!
    Returns the inverted matrix.

    If the matrix is singular (not invertible), the identity matrix is
    returned.

    If \a invertible is not 0: the value of \a *invertible is set
    to TRUE if the matrix is invertible; otherwise \a *invertible is
    set to FALSE.

    \sa isInvertible()
*/

QWMatrix QWMatrix::invert( bool *invertible ) const
{
    double determinant = det();
    if ( determinant == 0.0 ) {
	if ( invertible )
	    *invertible = FALSE;		// singular matrix
	QWMatrix defaultMatrix;
	return defaultMatrix;
    }
    else {					// invertible matrix
	if ( invertible )
	    *invertible = TRUE;
	double dinv = 1.0/determinant;
	QWMatrix imatrix( (_m22*dinv),	(-_m12*dinv),
			  (-_m21*dinv), ( _m11*dinv),
			  ((_m21*_dy - _m22*_dx)*dinv),
			  ((_m12*_dx - _m11*_dy)*dinv) );
	return imatrix;
    }
}


/*!
    Returns TRUE if this matrix is equal to \a m; otherwise returns FALSE.
*/

bool QWMatrix::operator==( const QWMatrix &m ) const
{
    return _m11 == m._m11 &&
	   _m12 == m._m12 &&
	   _m21 == m._m21 &&
	   _m22 == m._m22 &&
	   _dx == m._dx &&
	   _dy == m._dy;
}

/*!
    Returns TRUE if this matrix is not equal to \a m; otherwise returns FALSE.
*/

bool QWMatrix::operator!=( const QWMatrix &m ) const
{
    return _m11 != m._m11 ||
	   _m12 != m._m12 ||
	   _m21 != m._m21 ||
	   _m22 != m._m22 ||
	   _dx != m._dx ||
	   _dy != m._dy;
}

/*!
    Returns the result of multiplying this matrix by matrix \a m.
*/

QWMatrix &QWMatrix::operator*=( const QWMatrix &m )
{
    double tm11 = _m11*m._m11 + _m12*m._m21;
    double tm12 = _m11*m._m12 + _m12*m._m22;
    double tm21 = _m21*m._m11 + _m22*m._m21;
    double tm22 = _m21*m._m12 + _m22*m._m22;

    double tdx  = _dx*m._m11  + _dy*m._m21 + m._dx;
    double tdy =  _dx*m._m12  + _dy*m._m22 + m._dy;

    _m11 = tm11; _m12 = tm12;
    _m21 = tm21; _m22 = tm22;
    _dx = tdx; _dy = tdy;
    return *this;
}

/*!
    \overload
    \relates QWMatrix
    Returns the product of \a m1 * \a m2.

    Note that matrix multiplication is not commutative, i.e. a*b !=
    b*a.
*/

QWMatrix operator*( const QWMatrix &m1, const QWMatrix &m2 )
{
    QWMatrix result = m1;
    result *= m2;
    return result;
}

/*****************************************************************************
  QWMatrix stream functions
 *****************************************************************************/
#ifndef QT_NO_DATASTREAM
/*!
    \relates QWMatrix

    Writes the matrix \a m to the stream \a s and returns a reference
    to the stream.

    \sa \link datastreamformat.html Format of the QDataStream operators \endlink
*/

QDataStream &operator<<( QDataStream &s, const QWMatrix &m )
{
    if ( s.version() == 1 )
	s << (float)m.m11() << (float)m.m12() << (float)m.m21()
	  << (float)m.m22() << (float)m.dx()  << (float)m.dy();
    else
	s << m.m11() << m.m12() << m.m21() << m.m22()
	  << m.dx() << m.dy();
    return s;
}

/*!
    \relates QWMatrix

    Reads the matrix \a m from the stream \a s and returns a reference
    to the stream.

    \sa \link datastreamformat.html Format of the QDataStream operators \endlink
*/

QDataStream &operator>>( QDataStream &s, QWMatrix &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.setMatrix( m11, m12, m21, m22, dx, dy );
    }
    else {
	double m11, m12, m21, m22, dx, dy;
	s >> m11;  s >> m12;  s >> m21;  s >> m22;
	s >> dx;   s >> dy;
	m.setMatrix( m11, m12, m21, m22, dx, dy );
    }
    return s;
}
#endif // QT_NO_DATASTREAM

#endif // QT_NO_WMATRIX