Add a slant to a homogeneous 2D transformation matrix.

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:

                               / cos(Theta) 0 0 \
  Axis = 'x':  HomMat2DSlant = | sin(Theta) 1 0 | * HomMat2D
                               \    0       0 1 /

                               / 1 -sin(Theta) 0 \
  Axis = 'y':  HomMat2DSlant = | 0  cos(Theta) 0 | * HomMat2D
                               \ 0     0       1 /

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':

                  / 1 0 +Px \   / cos(Theta) 0 0 \   / 1 0 -Px \ 
  HomMat2DSlant = | 0 1 +Py | * | sin(Theta) 1 0 | * | 0 1 -Py | * HomMat2D
                  \ 0 0  1  /   \    0       0 1 /   \ 0 0  1  /   

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

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

    / ra rb tc \
    | rd re tf |
    \ 0  0  1  /

If the parameters are valid, the operator hom_mat2d_slant returns 2 (H_MSG_TRUE). If necessary, an exception is raised.

hom_mat2d_slant is reentrant and processed without parallelization.

hom_mat2d_identity, hom_mat2d_translate, hom_mat2d_scale, hom_mat2d_rotate, hom_mat2d_slant

hom_mat2d_translate, hom_mat2d_scale, hom_mat2d_rotate, hom_mat2d_slant

hom_mat2d_slant_local

Foundation