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# QTransform Class Reference
[QtGui module]

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

     #include <QTransform>

This class was introduced in Qt 4.3.

- List of all members, including inherited members

### Public Types

- enum TransformationType { TxNone, TxTranslate, TxScale, TxRotate, TxShear, TxProject }

### Public Functions

- QTransform ()
- QTransform ( qreal h11, qreal h12, qreal h13, qreal h21, qreal h22, qreal h23, qreal h31, qreal h32, qreal h33 = 1.0 )
- QTransform ( qreal h11, qreal h12, qreal h21, qreal h22, qreal dx, qreal dy )
- QTransform ( const QMatrix & matrix )
- qreal m11 () const
- qreal m12 () const
- qreal m13 () const
- qreal m21 () const
- qreal m22 () const
- qreal m23 () const
- qreal m31 () const
- qreal m32 () const
- qreal m33 () const
- QTransform adjoint () const
- qreal det () const
- qreal determinant () const
- qreal dx () const
- qreal dy () const
- QTransform inverted ( bool * invertible = 0 ) const
- bool isAffine () const
- bool isIdentity () const
- bool isInvertible () const
- bool isRotating () const
- bool isScaling () const
- bool isTranslating () const
- void map ( qreal x, qreal y, qreal * tx, qreal * ty ) const
- QPointF map ( const QPointF & p ) const
- QPoint map ( const QPoint & point ) const
- QLine map ( const QLine & l ) const
- QLineF map ( const QLineF & 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
- void map ( int x, int y, int * tx, int * ty ) const
- QRectF mapRect ( const QRectF & rectangle ) const
- QRect mapRect ( const QRect & rectangle ) const
- QPolygon mapToPolygon ( const QRect & rectangle ) const
- void reset ()
- QTransform & rotate ( qreal angle, Qt::Axis axis = Qt::ZAxis )
- QTransform & rotateRadians ( qreal angle, Qt::Axis axis = Qt::ZAxis )
- QTransform & scale ( qreal sx, qreal sy )
- void setMatrix ( qreal m11, qreal m12, qreal m13, qreal m21, qreal m22, qreal m23, qreal m31, qreal m32, qreal m33 )
- QTransform & shear ( qreal sh, qreal sv )
- const QMatrix & toAffine () const
- QTransform & translate ( qreal dx, qreal dy )
- QTransform transposed () const
- TransformationType type () const
- operator QVariant () const
- bool operator!= ( const QTransform & matrix ) const
- QTransform operator* ( const QTransform & matrix ) const
- QTransform & operator*= ( const QTransform & matrix )
- QTransform & operator*= ( qreal scalar )
- QTransform & operator+= ( qreal scalar )
- QTransform & operator-= ( qreal scalar )
- QTransform & operator/= ( qreal scalar )
- QTransform & operator= ( const QTransform & matrix )
- bool operator== ( const QTransform & matrix ) const

### Static Public Members

- bool quadToQuad ( const QPolygonF & one, const QPolygonF & two, QTransform & trans )
- bool quadToSquare ( const QPolygonF & quad, QTransform & trans )
- bool squareToQuad ( const QPolygonF & quad, QTransform & trans )

### Related Non-Members

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

## Detailed Description

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

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

QTransform differs from QMatrix in that it is a true 3x3 matrix, allowing perpective transformations. QTransform's toAffine() method allows casting QTransform to QMatrix. If a perspective transformation has been specified on the matrix, then the conversion to an affine QMatrix will cause loss of data.

QTransform is the recommended transformation class in Qt.

A QTransform 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 QTransform class supports mapping of graphic primitives: A given point, line, polygon, region, or painter path can be mapped to the coordinate system defined by 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.

QTransform 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, QTransform provides the det() function returning the matrix's determinant.

Finally, the QTransform 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 QTransform. 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, "QTransform");
     }

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

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

         QTransform transform;
         transform.translate(50, 50);
         transform.rotate(45);
         transform.scale(0.5, 1.0);
         painter.setTransform(transform);

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

### Basic Matrix Operations

A QTransform 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.

QTransform 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);

         QTransform translationTransform(1, 0, 0, 1, 50.0, 50.0);
         QTransform rotationTransform(cosa, sina, -sina, cosa, 0, 0);
         QTransform scalingTransform(0.5, 0, 0, 1.0, 0, 0);

         QTransform transform;
         transform = scalingTransform * rotationTransform * translationTransform;

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

         painter.setTransform(transform);

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

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

## Member Type Documentation

### enum QTransform::TransformationType

Constant | Value
--- | ---
`QTransform::TxNone` | `0x00`
`QTransform::TxTranslate` | `0x01`
`QTransform::TxScale` | `0x02`
`QTransform::TxRotate` | `0x04`
`QTransform::TxShear` | `0x08`
`QTransform::TxProject` | `0x10`

## Member Function Documentation

### QTransform::QTransform ()

Constructs an identity matrix.

All elements are set to zero except `m11` and `m22` (specifying the scale) and `m13` which are set to 1.

See also reset().

### QTransform::QTransform ( qreal h11, qreal h12, qreal h13, qreal h21, qreal h22, qreal h23, qreal h31, qreal h32, qreal h33 = 1.0 )

Constructs a matrix with the elements, h11, h12, h13, h21, h22, h23, h31, h32, h33.

See also setMatrix().

### QTransform::QTransform ( qreal h11, qreal h12, qreal h21, qreal h22, qreal dx, qreal dy )

Constructs a matrix with the elements, h11, h12, h21, h22, dx and dy.

See also setMatrix().

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

Constructs a matrix that is a copy of the given matrix. Note that the `m13`, `m23`, and `m33` elements are set to 0, 0, and 1 respectively.

### qreal QTransform::m11 () const

Returns the horizontal scaling factor.

See also scale() and Basic Matrix Operations.

### qreal QTransform::m12 () const

Returns the vertical shearing factor.

See also shear() and Basic Matrix Operations.

### qreal QTransform::m13 () const

Returns the horizontal projection factor.

See also translate() and Basic Matrix Operations.

### qreal QTransform::m21 () const

Returns the horizontal shearing factor.

See also shear() and Basic Matrix Operations.

### qreal QTransform::m22 () const

Returns the vertical scaling factor.

See also scale() and Basic Matrix Operations.

### qreal QTransform::m23 () const

Returns the vertical projection factor.

See also translate() and Basic Matrix Operations.

### qreal QTransform::m31 () const

Returns the horizontal translation factor.

See also dx(), translate(), and Basic Matrix Operations.

### qreal QTransform::m32 () const

Returns the vertical translation factor.

See also dy(), translate(), and Basic Matrix Operations.

### qreal QTransform::m33 () const

Returns the division factor.

See also translate() and Basic Matrix Operations.

### QTransform QTransform::adjoint () const

Returns the adjoint of this matrix.

### qreal QTransform::det () const

Returns the matrix's determinant.

### qreal QTransform::determinant () const

Returns the matrix's determinant.

### qreal QTransform::dx () const

Returns the horizontal translation factor.

See also m31(), translate(), and Basic Matrix Operations.

### qreal QTransform::dy () const

Returns the vertical translation factor.

See also translate() and Basic Matrix Operations.

### QTransform QTransform::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 QTransform::isAffine () const

Returns true if the matrix represent an affine transformation, otherwise returns false.

### bool QTransform::isIdentity () const

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

See also reset().

### bool QTransform::isInvertible () const

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

See also inverted().

### bool QTransform::isRotating () const

Returns true if the matrix represents some kind of a rotating transformation, otherwise returns false.

See also reset().

### bool QTransform::isScaling () const

Returns true if the matrix represents a scaling transformation, otherwise returns false.

See also reset().

### bool QTransform::isTranslating () const

Returns true if the matrix represents a translating transformation, otherwise returns false.

See also reset().

### void QTransform::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.

### QPointF QTransform::map ( const QPointF & p ) const

This is an overloaded member function, provided for convenience.

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

### QPoint QTransform::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.

### QLine QTransform::map ( const QLine & l ) const

This is an overloaded member function, provided for convenience.

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

### QLineF QTransform::map ( const QLineF & 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 QTransform::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 QTransform::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 QTransform::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 QTransform::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.

### void QTransform::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.

### QRectF QTransform::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 (is not affine) {
                 w' = m13*x + m23*y + m33
                 x' /= w'
                 y' /= w'
             }

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 QTransform::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 QTransform::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
             if (is not affine) {
                 w' = m13*x + m23*y + m33
                 x' /= w'
                 y' /= w'
             }

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.

### bool QTransform::quadToQuad ( const QPolygonF & one, const QPolygonF & two, QTransform & trans ) `[static]`

Creates a transformation matrix, trans, that maps a four-sided polygon, one, to another four-sided polygon, two. Returns true if the transformation is possible; otherwise returns false.

This is a convenience method combining quadToSquare() and squareToQuad() methods. It allows the input quad to be transformed into any other quad.

See also squareToQuad() and quadToSquare().

### bool QTransform::quadToSquare ( const QPolygonF & quad, QTransform & trans ) `[static]`

Creates a transformation matrix, trans, that maps a four-sided polygon, quad, to a unit square. Returns true if the transformation is constructed or false if such a transformation does not exist.

See also squareToQuad() and quadToQuad().

### void QTransform::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 QTransform(), isIdentity(), and Basic Matrix Operations.

### QTransform & QTransform::rotate ( qreal angle, Qt::Axis axis = Qt::ZAxis )

Rotates the coordinate system counterclockwise by the given angle about the specified axis and returns a reference to the matrix.

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

The angle is specified in degrees.

See also setMatrix().

### QTransform & QTransform::rotateRadians ( qreal angle, Qt::Axis axis = Qt::ZAxis )

Rotates the coordinate system counterclockwise by the given angle about the specified axis and returns a reference to the matrix.

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

The angle is specified in radians.

See also setMatrix().

### QTransform & QTransform::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 QTransform::setMatrix ( qreal m11, qreal m12, qreal m13, qreal m21, qreal m22, qreal m23, qreal m31, qreal m32, qreal m33 )

Sets the matrix elements to the specified values, m11, m12, m13 m21, m22, m23 m31, m32 and m33. Note that this function replaces the previous values. QMatrix provides the translate(), rotate(), scale() and shear() convenience functions to manipulate the various matrix elements based on the currently defined coordinate system.

See also QTransform().

### QTransform & QTransform::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().

### bool QTransform::squareToQuad ( const QPolygonF & quad, QTransform & trans ) `[static]`

Creates a transformation matrix, trans, that maps a unit square to a four-sided polygon, quad. Returns true if the transformation is constructed or false if such a transformation does not exist.

See also quadToSquare() and quadToQuad().

### const QMatrix & QTransform::toAffine () const

Returns the QTransform cast to a QMatrix.

### QTransform & QTransform::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().

### QTransform QTransform::transposed () const

Returns the transpose of this matrix.

### TransformationType QTransform::type () const

Returns the transformation type of this matrix.

### QTransform::operator QVariant () const

Returns the transform as a QVariant.

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

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

### QTransform QTransform::operator* ( const QTransform & 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.

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

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

### QTransform & QTransform::operator*= ( qreal scalar )

This is an overloaded member function, provided for convenience.

Returns the result of performing an element-wise multiplication of this matrix with the given scalar.

### QTransform & QTransform::operator+= ( qreal scalar )

Returns the matrix obtained by adding the given scalar to each element of this matrix.

### QTransform & QTransform::operator-= ( qreal scalar )

Returns the matrix obtained by subtracting the given scalar from each element of this matrix.

### QTransform & QTransform::operator/= ( qreal scalar )

Returns the result of performing an element-wise division of this matrix by the given scalar.

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

Assigns the given matrix's values to this matrix.

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

This is an overloaded member function, provided for convenience.

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

See also QTransform::map().

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

This is an overloaded member function, provided for convenience.

Same as matrix.map(point).

See also QTransform::map().

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

This is an overloaded member function, provided for convenience.

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

See also QTransform::map().

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

This is an overloaded member function, provided for convenience.

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

See also QTransform::map().

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

This is an overloaded member function, provided for convenience.

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

See also QTransform::map().

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

This is an overloaded member function, provided for convenience.

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

This function was introduced in Qt 4.3.

See also QTransform::map().

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

This is an overloaded member function, provided for convenience.

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

See also QTransform::map().

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

This is an overloaded member function, provided for convenience.

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

This function was introduced in Qt 4.3.

See also QTransform::map().

### QDataStream & operator<< ( QDataStream & stream, const QTransform & 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.

This function was introduced in Qt 4.3.

See also Format of the QDataStream Operators.

### QDataStream & operator>> ( QDataStream & stream, QTransform & 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.

This function was introduced in Qt 4.3.

See also Format of the QDataStream Operators.