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# QMatrix Class Reference
[QtGui module]

The QMatrix class specifies 2D transformations of a coordinate system. More...

    #include <QMatrix>

- List of all members, including inherited members
- Qt 3 support members

### Public Functions

- QMatrix ()
- QMatrix ( qreal m11, qreal m12, qreal m21, qreal m22, qreal dx, qreal dy )
- QMatrix ( const QMatrix & matrix )
- qreal m11 () const
- qreal m12 () const
- qreal m21 () const
- qreal m22 () const
- qreal det () const
- qreal dx () const
- qreal dy () const
- QMatrix inverted ( bool * invertible = 0 ) const
- bool isIdentity () const
- bool isInvertible () const
- void map ( qreal x, qreal y, qreal * tx, qreal * ty ) const
- void map ( int x, int y, int * tx, int * ty ) const
- QPointF map ( const QPointF & point ) const
- QPoint map ( const QPoint & point ) const
- QLineF map ( const QLineF & line ) const
- QLine map ( const QLine & line ) const
- QPolygonF map ( const QPolygonF & polygon ) const
- QPolygon map ( const QPolygon & polygon ) const
- QRegion map ( const QRegion & region ) const
- QPainterPath map ( const QPainterPath & path ) const
- QRectF mapRect ( const QRectF & rectangle ) const
- QRect mapRect ( const QRect & rectangle ) const
- QPolygon mapToPolygon ( const QRect & rectangle ) const
- void reset ()
- QMatrix & rotate ( qreal degrees )
- QMatrix & scale ( qreal sx, qreal sy )
- void setMatrix ( qreal m11, qreal m12, qreal m21, qreal m22, qreal dx, qreal dy )
- QMatrix & shear ( qreal sh, qreal sv )
- QMatrix & translate ( qreal dx, qreal dy )
- bool operator!= ( const QMatrix & matrix ) const
- QMatrix operator* ( const QMatrix & matrix ) const
- QMatrix & operator*= ( const QMatrix & matrix )
- QMatrix & operator= ( const QMatrix & matrix )
- bool operator== ( const QMatrix & matrix ) const

### Related Non-Members

- QPoint operator* ( const QPoint & point, const QMatrix & matrix )
- QPointF operator* ( const QPointF & point, const QMatrix & matrix )
- QLineF operator* ( const QLineF & line, const QMatrix & matrix )
- QLine operator* ( const QLine & line, const QMatrix & matrix )
- QPolygon operator* ( const QPolygon & polygon, const QMatrix & matrix )
- QPolygonF operator* ( const QPolygonF & polygon, const QMatrix & matrix )
- QRegion operator* ( const QRegion & region, const QMatrix & matrix )
- QPainterPath operator* ( const QPainterPath & path, const QMatrix & matrix )
- QDataStream & operator<< ( QDataStream & stream, const QMatrix & matrix )
- QDataStream & operator>> ( QDataStream & stream, QMatrix & matrix )

## Detailed Description

The QMatrix class specifies 2D transformations of a coordinate system.

A matrix specifies how to translate, scale, shear or rotate the coordinate system, and is typically used when rendering graphics.

A QMatrix object can be built using the setMatrix(), scale(), rotate(), translate() and shear() functions. Alternatively, it can be built by applying 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 QMatrix class supports mapping of graphic primitives: A given point, line, polygon, region, or painter path can be mapped to the coordinate system defined by 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 polygon (mapped to the coordinate system defined by this matrix), using the mapToPolygon() function.

QMatrix provides the isIdentity() function which returns true if the matrix is the identity matrix, and the isInvertible() function which returns true if the matrix is non-singular (i.e. AB = BA = I). The inverted() function returns an inverted copy of this matrix if it is invertible (otherwise it returns the identity matrix). In addition, QMatrix provides the det() function returning the matrix's determinant.

Finally, the QMatrix class supports matrix multiplication, and objects of the class can be streamed as well as compared.

- Rendering Graphics
- Basic Matrix Operations

### 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 x values increase to the right; y values increase downward. For a complete description, see the coordinate system documentation.

QPainter has functions to translate, scale, shear and rotate the coordinate system without using a QMatrix. For example:

        void SimpleTransformation::paintEvent(QPaintEvent *)
        {
            QPainter painter(this);
            painter.setPen(QPen(Qt::blue, 1, Qt::DashLine));
            painter.drawRect(0, 0, 100, 100);

            painter.rotate(45);

            painter.setFont(QFont("Helvetica", 24));
            painter.setPen(QPen(Qt::black, 1));
            painter.drawText(20, 10, "QMatrix");
        }

Although these functions are very convenient, it can be more efficient to build a QMatrix and call QPainter::setMatrix() if you want to perform more than a single transform operation. For example:

        void CombinedTransformation::paintEvent(QPaintEvent *)
        {
            QPainter painter(this);
            painter.setPen(QPen(Qt::blue, 1, Qt::DashLine));
            painter.drawRect(0, 0, 100, 100);

            QMatrix matrix;
            matrix.translate(50, 50);
            matrix.rotate(45);
            matrix.scale(0.5, 1.0);
            painter.setMatrix(matrix);

            painter.setFont(QFont("Helvetica", 24));
            painter.setPen(QPen(Qt::black, 1));
            painter.drawText(20, 10, "QMatrix");
        }

### Basic Matrix Operations

A QMatrix object contains a 3 x 3 matrix. The `dx` and `dy` elements specify horizontal and vertical translation. The `m11` and `m22` elements specify horizontal and vertical scaling. And finally, the `m21` and `m12` elements specify horizontal and vertical shearing.

QMatrix transforms a point in the plane to another point using the following formulas:

        x' = m11*x + m21*y + dx
        y' = m22*y + m12*x + dy

The point (x, y) is the original point, and (x', y') is the transformed point. (x', y') can be transformed back to (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 also be manipulated using the translate(), rotate(), scale() and shear() convenience functions, The currently set values can be retrieved using the m11(), m12(), m21(), m22(), dx() and dy() functions.

Translation is the simplest transformation. Setting `dx` and `dy` will move the coordinate system `dx` units along the X axis and `dy` units along the Y axis. Scaling can be done by setting `m11` and `m22`. For example, setting `m11` to 2 and `m22` to 1.5 will double the height and increase the width by 50%. The identity matrix has `m11` and `m22` set to 1 (all others are set to 0) mapping a point to itself. Shearing is controlled by `m12` and `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.

Here's the combined transformations example using basic matrix operations:

        void BasicOperations::paintEvent(QPaintEvent *)
        {
            double pi = 3.14;

            double a    = pi/180 * 45.0;
            double sina = sin(a);
            double cosa = cos(a);

            QMatrix translationMatrix(1, 0, 0, 1, 50.0, 50.0);
            QMatrix rotationMatrix(cosa, sina, -sina, cosa, 0, 0);
            QMatrix scalingMatrix(0.5, 0, 0, 1.0, 0, 0);

            QMatrix matrix;
            matrix =  scalingMatrix * rotationMatrix * translationMatrix;

            QPainter painter(this);
            painter.setPen(QPen(Qt::blue, 1, Qt::DashLine));
            painter.drawRect(0, 0, 100, 100);

            painter.setMatrix(matrix);

            painter.setFont(QFont("Helvetica", 24));
            painter.setPen(QPen(Qt::black, 1));
            painter.drawText(20, 10, "QMatrix");
        }

See also QPainter, The Coordinate System, and Affine Transformations Demo.

## Member Function Documentation

### QMatrix::QMatrix ()

Constructs an identity matrix.

All elements are set to zero except `m11` and `m22` (specifying the scale), which are set to 1.

See also reset().

### QMatrix::QMatrix ( qreal m11, qreal m12, qreal m21, qreal m22, qreal dx, qreal dy )

Constructs a matrix with the elements, m11, m12, m21, m22, dx and dy.

See also setMatrix().

### QMatrix::QMatrix ( const QMatrix & matrix )

Constructs a matrix that is a copy of the given matrix.

### qreal QMatrix::m11 () const

Returns the horizontal scaling factor.

See also scale() and Basic Matrix Operations.

### qreal QMatrix::m12 () const

Returns the vertical shearing factor.

See also shear() and Basic Matrix Operations.

### qreal QMatrix::m21 () const

Returns the horizontal shearing factor.

See also shear() and Basic Matrix Operations.

### qreal QMatrix::m22 () const

Returns the vertical scaling factor.

See also scale() and Basic Matrix Operations.

### qreal QMatrix::det () const

Returns the matrix's determinant.

### qreal QMatrix::dx () const

Returns the horizontal translation factor.

See also translate() and Basic Matrix Operations.

### qreal QMatrix::dy () const

Returns the vertical translation factor.

See also translate() and Basic Matrix Operations.

### QMatrix QMatrix::inverted ( bool * invertible = 0 ) const

Returns an inverted copy of this matrix.

If the matrix is singular (not invertible), the returned matrix is the identity matrix. If invertible is valid (i.e. not 0), its value is set to true if the matrix is invertible, otherwise it is set to false.

See also isInvertible().

### bool QMatrix::isIdentity () const

Returns true if the matrix is the identity matrix, otherwise returns false.

See also reset().

### bool QMatrix::isInvertible () const

Returns true if the matrix is invertible, otherwise returns false.

See also inverted().

### void QMatrix::map ( qreal x, qreal y, qreal * tx, qreal * ty ) const

Maps the given coordinates x and y into the coordinate system defined by this matrix. The resulting values are put in *tx and *ty, respectively.

The coordinates are transformed using the following formulas:

        x' = m11*x + m21*y + dx
        y' = m22*y + m12*x + dy

The point (x, y) is the original point, and (x', y') is the transformed point.

See also Basic Matrix Operations.

### void QMatrix::map ( int x, int y, int * tx, int * ty ) const

This is an overloaded member function, provided for convenience.

Maps the given coordinates x and y into the coordinate system defined by this matrix. The resulting values are put in *tx and *ty, respectively. Note that the transformed coordinates are rounded to the nearest integer.

### QPointF QMatrix::map ( const QPointF & point ) const

This is an overloaded member function, provided for convenience.

Creates and returns a QPointF object that is a copy of the given point, mapped into the coordinate system defined by this matrix.

### QPoint QMatrix::map ( const QPoint & point ) const

This is an overloaded member function, provided for convenience.

Creates and returns a QPoint object that is a copy of the given point, mapped into the coordinate system defined by this matrix. Note that the transformed coordinates are rounded to the nearest integer.

### QLineF QMatrix::map ( const QLineF & line ) const

This is an overloaded member function, provided for convenience.

Creates and returns a QLineF object that is a copy of the given line, mapped into the coordinate system defined by this matrix.

### QLine QMatrix::map ( const QLine & line ) const

This is an overloaded member function, provided for convenience.

Creates and returns a QLine object that is a copy of the given line, mapped into the coordinate system defined by this matrix. Note that the transformed coordinates are rounded to the nearest integer.

### QPolygonF QMatrix::map ( const QPolygonF & polygon ) const

This is an overloaded member function, provided for convenience.

Creates and returns a QPolygonF object that is a copy of the given polygon, mapped into the coordinate system defined by this matrix.

### QPolygon QMatrix::map ( const QPolygon & polygon ) const

This is an overloaded member function, provided for convenience.

Creates and returns a QPolygon object that is a copy of the given polygon, mapped into the coordinate system defined by this matrix. Note that the transformed coordinates are rounded to the nearest integer.

### QRegion QMatrix::map ( const QRegion & region ) const

This is an overloaded member function, provided for convenience.

Creates and returns a QRegion object that is a copy of the given region, mapped into the coordinate system defined by this matrix.

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

### QPainterPath QMatrix::map ( const QPainterPath & path ) const

This is an overloaded member function, provided for convenience.

Creates and returns a QPainterPath object that is a copy of the given path, mapped into the coordinate system defined by this matrix.

### QRectF QMatrix::mapRect ( const QRectF & rectangle ) const

Creates and returns a QRectF object that is a copy of the given rectangle, mapped into the coordinate system defined by this matrix.

The rectangle's coordinates are transformed using the following formulas:

        x' = m11*x + m21*y + dx
        y' = m22*y + m12*x + dy

If rotation or shearing has been specified, this function returns the bounding rectangle. To retrieve the exact region the given rectangle maps to, use the mapToPolygon() function instead.

See also mapToPolygon() and Basic Matrix Operations.

### QRect QMatrix::mapRect ( const QRect & rectangle ) const

This is an overloaded member function, provided for convenience.

Creates and returns a QRect object that is a copy of the given rectangle, mapped into the coordinate system defined by this matrix. Note that the transformed coordinates are rounded to the nearest integer.

### QPolygon QMatrix::mapToPolygon ( const QRect & rectangle ) const

Creates and returns a QPolygon representation of the given rectangle, mapped into the coordinate system defined by this matrix.

The rectangle's coordinates are transformed using the following formulas:

        x' = m11*x + m21*y + dx
        y' = m22*y + m12*x + dy

Polygons and rectangles behave slightly differently when transformed (due to integer rounding), so `matrix.map(QPolygon(rectangle))` is not always the same as `matrix.mapToPolygon(rectangle)`.

See also mapRect() and Basic Matrix Operations.

### void QMatrix::reset ()

Resets the matrix to an identity matrix, i.e. all elements are set to zero, except `m11` and `m22` (specifying the scale) which are set to 1.

See also QMatrix(), isIdentity(), and Basic Matrix Operations.

### QMatrix & QMatrix::rotate ( qreal degrees )

Rotates the coordinate system the given degrees counterclockwise.

Note that if you apply a QMatrix to a point defined in widget coordinates, the direction of the rotation will be clockwise because the y-axis points downwards.

Returns a reference to the matrix.

See also setMatrix().

### QMatrix & QMatrix::scale ( qreal sx, qreal sy )

Scales the coordinate system by sx horizontally and sy vertically, and returns a reference to the matrix.

See also setMatrix().

### void QMatrix::setMatrix ( qreal m11, qreal m12, qreal m21, qreal m22, qreal dx, qreal dy )

Sets the matrix elements to the specified values, m11, m12, m21, m22, dx and dy.

Note that this function replaces the previous values. QMatrix provide the translate(), rotate(), scale() and shear() convenience functions to manipulate the various matrix elements based on the currently defined coordinate system.

See also QMatrix().

### QMatrix & QMatrix::shear ( qreal sh, qreal sv )

Shears the coordinate system by sh horizontally and sv vertically, and returns a reference to the matrix.

See also setMatrix().

### QMatrix & QMatrix::translate ( qreal dx, qreal dy )

Moves the coordinate system dx along the x axis and dy along the y axis, and returns a reference to the matrix.

See also setMatrix().

### bool QMatrix::operator!= ( const QMatrix & matrix ) const

Returns true if this matrix is not equal to the given matrix, otherwise returns false.

### QMatrix QMatrix::operator* ( const QMatrix & matrix ) const

Returns the result of multiplying this matrix by the given matrix.

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

### QMatrix & QMatrix::operator*= ( const QMatrix & matrix )

Returns the result of multiplying this matrix by the given matrix.

### QMatrix & QMatrix::operator= ( const QMatrix & matrix )

Assigns the given matrix's values to this matrix.

### bool QMatrix::operator== ( const QMatrix & matrix ) const

Returns true if this matrix is equal to the given matrix, otherwise returns false.

## Related Non-Members

### QPoint operator* ( const QPoint & point, const QMatrix & matrix )

This is an overloaded member function, provided for convenience.

This is the same as matrix.map(point).

See also QMatrix::map().

### QPointF operator* ( const QPointF & point, const QMatrix & matrix )

This is an overloaded member function, provided for convenience.

Same as matrix.map(point).

See also QMatrix::map().

### QLineF operator* ( const QLineF & line, const QMatrix & matrix )

This is an overloaded member function, provided for convenience.

This is the same as matrix.map(line).

See also QMatrix::map().

### QLine operator* ( const QLine & line, const QMatrix & matrix )

This is an overloaded member function, provided for convenience.

This is the same as matrix.map(line).

See also QMatrix::map().

### QPolygon operator* ( const QPolygon & polygon, const QMatrix & matrix )

This is an overloaded member function, provided for convenience.

This is the same as matrix.map(polygon).

See also QMatrix::map().

### QPolygonF operator* ( const QPolygonF & polygon, const QMatrix & matrix )

This is an overloaded member function, provided for convenience.

This is the same as matrix.map(polygon).

See also QMatrix::map().

### QRegion operator* ( const QRegion & region, const QMatrix & matrix )

This is an overloaded member function, provided for convenience.

This is the same as matrix.map(region).

See also QMatrix::map().

### QPainterPath operator* ( const QPainterPath & path, const QMatrix & matrix )

This is an overloaded member function, provided for convenience.

This is the same as matrix.map(path).

See also QMatrix::map().

### QDataStream & operator<< ( QDataStream & stream, const QMatrix & matrix )

This is an overloaded member function, provided for convenience.

Writes the given matrix to the given stream and returns a reference to the stream.

See also Format of the QDataStream Operators.

### QDataStream & operator>> ( QDataStream & stream, QMatrix & matrix )

This is an overloaded member function, provided for convenience.

Reads the given matrix from the given stream and returns a reference to the stream.

See also Format of the QDataStream Operators.