QTransform Class▲

  • Header: QTransform

  • CMake:

    find_package(Qt6 REQUIRED COMPONENTS Gui)

    target_link_libraries(mytarget PRIVATE Qt6::Gui)

  • qmake: QT += gui

  • Group: QTransform is part of Painting Classes

Detailed Description▲

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

A QTransform object can be built using the setMatrix(), scale(), rotate(), translate() and shear() functions. Alternatively, it can be built by applying 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), and adjoint() returns the matrix's classical adjoint. In addition, QTransform provides the determinant() function which returns the matrix's determinant.

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

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:

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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:

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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▲

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A QTransform object contains a 3 x 3 matrix. The m31 (dx) and m32 (dy) elements specify horizontal and vertical translation. The m11 and m22 elements specify horizontal and vertical scaling. The m21 and m12 elements specify horizontal and vertical shearing. And finally, the m13 and m23 elements specify horizontal and vertical projection, with m33 as an additional projection factor.

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

 
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x' = m11*x + m21*y + dx
y' = m22*y + m12*x + dy
if (!isAffine()) {
    w' = m13*x + m23*y + m33
    x' /= w'
    y' /= w'
}

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 can also be manipulated using the translate(), rotate(), scale() and shear() convenience functions. The currently set values can be retrieved using the m11(), m12(), m13(), m21(), m22(), m23(), m31(), m32(), m33(), dx() and dy() functions.

Translation is the simplest transformation. Setting 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, m22, and m33 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 setting both the shearing factors and the scaling factors. Perspective transformation is achieved by setting both the projection factors and the scaling factors.

Combining Transforms▲

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

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void BasicOperations::paintEvent(QPaintEvent *)
{
    const double a = qDegreesToRadians(45.0);
    double sina = sin(a);
    double cosa = cos(a);

    QTransform scale(0.5, 0, 0, 1.0, 0, 0);
    QTransform rotate(cosa, sina, -sina, cosa, 0, 0);
    QTransform translate(1, 0, 0, 1, 50.0, 50.0);

    QTransform transform = scale * rotate * translate;

    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");
}

The combined transform first scales each operand, then rotates it, and finally translates it, just as in the order in which the product of its factors is written. This means the point to which the transforms are applied is implicitly multiplied on the left with the transform to its right.

Relation to Matrix Notation▲

The matrix notation in QTransform is the transpose of a commonly-taught convention which represents transforms and points as matrices and vectors. That convention multiplies its matrix on the left and column vector to the right. In other words, when several transforms are applied to a point, the right-most matrix acts directly on the vector first. Then the next matrix to the left acts on the result of the first operation - and so on. As a result, that convention multiplies the matrices that make up a composite transform in the reverse of the order in QTransform, as you can see in Combining Transforms. Transposing the matrices, and combining them to the right of a row vector that represents the point, lets the matrices of transforms appear, in their product, in the order in which we think of the transforms being applied to the point.

See