hom_mat2d_slant (Operator)

Name

`hom_mat2d_slant` — Add a slant to a homogeneous 2D transformation matrix.

Signature

`hom_mat2d_slant( : : HomMat2D, Theta, Axis, Px, Py : HomMat2DSlant)`

Description

`hom_mat2d_slant` adds a slant by the angle `Theta` to the homogeneous 2D transformation matrix `HomMat2D` and returns the resulting matrix in `HomMat2DSlant`. A slant is an affine transformation in which one coordinate axis remains fixed, while the other coordinate axis is rotated counterclockwise by an angle `Theta`. The parameter `Axis` determines which coordinate axis is slanted. For `Axis` = 'x', the x-axis is slanted and the y-axis remains fixed, while for `Axis` = 'y' the y-axis is slanted and the x-axis remains fixed. The slanting is performed relative to the global (i.e., fixed) coordinate system; this corresponds to the following chains of transformation matrices:

The point (`Px`,`Py`) is the fixed point of the transformation, i.e., this point remains unchanged when transformed using `HomMat2DSlant`. To obtain this behavior, first a translation is added to the input transformation matrix that moves the fixed point onto the origin of the global coordinate system. Then, the slant is added, and finally a translation that moves the fixed point back to its original position. This corresponds to the following chain of transformations for `Axis` = 'x':

To perform the transformation in the local coordinate system, i.e., the one described by `HomMat2D`, use `hom_mat2d_slant_local`.

Attention

It should be noted that homogeneous transformation matrices refer to a general right-handed mathematical coordinate system. If a homogeneous transformation matrix is used to transform images, regions, XLD contours, or any other data that has been extracted from images, the row coordinates of the transformation must be passed in the x coordinates, while the column coordinates must be passed in the y coordinates. Consequently, the order of passing row and column coordinates follows the usual order (`Row`,`Column`). This convention is essential to obtain a right-handed coordinate system for the transformation of iconic data, and consequently to ensure in particular that rotations are performed in the correct mathematical direction.

Note that homogeneous matrices are stored row-by-row as a tuple; the last row is usually not stored because it is identical for all homogeneous matrices that describe an affine transformation. For example, the homogeneous matrix is stored as the tuple [ra, rb, tc, rd, re, tf]. However, it is also possible to process full 3×3 matrices, which represent a projective 2D transformation.

Execution Information

• Multithreading type: reentrant (runs in parallel with non-exclusive operators).
• Processed without parallelization.

Parameters

`HomMat2D` (input_control)  hom_mat2d `→` (real)

Input transformation matrix.

`Theta` (input_control)  angle.rad `→` (real / integer)

Slant angle.

Default value: 0.78

Suggested values: 0.1, 0.2, 0.3, 0.4, 0.78, 1.57, 3.14

Typical range of values: ```0 ≤ Theta ≤ 6.28318530718```

`Axis` (input_control)  string `→` (string)

Coordinate axis that is slanted.

Default value: 'x'

List of values: 'x', 'y'

`Px` (input_control)  point.x `→` (real / integer)

Fixed point of the transformation (x coordinate).

Default value: 0

Suggested values: 0, 16, 32, 64, 128, 256, 512, 1024

`Py` (input_control)  point.y `→` (real / integer)

Fixed point of the transformation (y coordinate).

Default value: 0

Suggested values: 0, 16, 32, 64, 128, 256, 512, 1024

`HomMat2DSlant` (output_control)  hom_mat2d `→` (real)

Output transformation matrix.

Result

If the parameters are valid, the operator `hom_mat2d_slant` returns TRUE. If necessary, an exception is raised.

Possible Predecessors

`hom_mat2d_identity`, `hom_mat2d_translate`, `hom_mat2d_scale`, `hom_mat2d_rotate`, `hom_mat2d_slant`, `hom_mat2d_reflect`

Possible Successors

`hom_mat2d_translate`, `hom_mat2d_scale`, `hom_mat2d_rotate`, `hom_mat2d_slant`, `hom_mat2d_reflect`

`hom_mat2d_slant_local`