The standard coordinate system of a paint device 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 TQPainter, which renders graphics in a paint device. A user-defined coordinate system can be specified by setting a TQWMatrix for the painter.
A matrix specifies how to translate, scale, shear or rotate the graphics; the actual transformation is performed by the drawing routines in TQPainter and by TQPixmap::xForm().
A matrix transforms a point in the plane to another point:
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x' = m11*x + m21*y + dx
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y' = m22*y + m12*x + dy
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The point \fI(x, y)\fR is the original point, and \fI(x', y')\fR is the transformed point. \fI(x', y')\fR can be transformed back to \fI(x, y)\fR by performing the same operation on the inverted matrix.
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The elements \fIdx\fR and \fIdy\fR specify horizontal and vertical translation. The elements \fIm11\fR and \fIm22\fR specify horizontal and vertical scaling. The elements \fIm12\fR and \fIm21\fR specify horizontal and vertical shearing.
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The identity matrix has \fIm11\fR and \fIm22\fR set to 1; all others are set to 0. This matrix maps a point to itself.
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Translation is the simplest transformation. Setting \fIdx\fR and \fIdy\fR will move the coordinate system \fIdx\fR units along the X axis and \fIdy\fR units along the Y axis.
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Scaling can be done by setting \fIm11\fR and \fIm22\fR. For example, setting \fIm11\fR to 2 and \fIm22\fR to 1.5 will double the height and increase the width by 50%.
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Shearing is controlled by \fIm12\fR and \fIm21\fR. 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 TQWMatrix also has a function that sets rotation directly.
TQPainter has functions to translate, scale, shear and rotate the coordinate system without using a TQWMatrix. Although these functions are very convenient, it can be more efficient to build a TQWMatrix and call TQPainter::setWorldMatrix() if you want to perform more than a single transform operation.
TQWMatrix 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.
Suppose we have a rectangle, \fCTQRect( 10, 20, 30, 40 )\fR and a transformation matrix \fCTQWMatrix( 2, 0, 0, 2, 0, 0 )\fR 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.
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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.
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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.
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Under Points mode, \fCmatrix * ( region1 | region2 )\fR is not equal to \fCmatrix * region1 | matrix * region2\fR. Under Area mode, \fCmatrix * (pointarray[i])\fR is not neccesarily equal to \fC(matrix * pointarry)[i]\fR.
Polygons and rectangles behave slightly differently when transformed (due to integer rounding), so \fCmatrix.map( TQPointArray( rect ) )\fR is not always the same as \fCmatrix.mapToPolygon( rect )\fR.
.SH "TQRegion TQWMatrix::mapToRegion ( const TQRect & rect ) const"
A rectangle which has been rotated or sheared may result in a non-rectangular region being returned.
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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.
