Operators

# hom_mat2d_reflect (Operator)

## Name

hom_mat2d_reflect — Add a reflection to a homogeneous 2D transformation matrix.

## Signature

hom_mat2d_reflect( : : HomMat2D, Px, Py, Qx, Qy : HomMat2DReflect)

## Description

hom_mat2d_reflect adds a reflection about the axis given by the two points (Px,Py) and (Qx,Qy) to the homogeneous 2D transformation matrix HomMat2D and returns the resulting matrix in HomMat2DReflect. The reflection is described by a 2×2 reflection matrix M. It is performed relative to the global (i.e., fixed) coordinate system; this corresponds to the following chain of transformation matrices:

```                    / Mxx Mxy 0 \
HomMat2DReflect = | Mxy Myy 0 | * HomMat2D
\ 0   0   1 /

2      T
M = | Mxx Mxy | = I - ---- v*v
| Mxy Myy |        T
v *v

T
where v = (Py-Qy,Qx-Px)  is the normal vector to the axis.
```

The axis (Px,Py)-(Qx,Qy) is fixed in the transformation, i.e., the points on the axis remain unchanged when transformed using HomMat2DReflect. To obtain this behavior, first a translation is added to the input transformation matrix that moves the axis onto the origin of the global coordinate system. Then, the reflection is added, and finally a translation that moves the axis back to its original position. This corresponds to the following chain of transformations:

```                    / 1 0 +Px \   / Mxx Mxy 0 \   / 1 0 -Px \
HomMat2DReflect = | 0 1 +Py | * | Mxy Myy 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_reflect_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

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

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.

## Parallelization

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

## Parameters

HomMat2D (input_control)  hom_mat2d (real)

Input transformation matrix.

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

First point of the axis (x coordinate).

Default value: 0

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

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

First point of the axis (y coordinate).

Default value: 0

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

Qx (input_control)  point.x (real / integer)

Second point of the axis (x coordinate).

Default value: 16

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

Qy (input_control)  point.y (real / integer)

Second point of the axis (y coordinate).

Default value: 32

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

HomMat2DReflect (output_control)  hom_mat2d (real)

Output transformation matrix.

## Result

hom_mat2d_reflect returns 2 (H_MSG_TRUE) if both points on the axis are not identical. If necessary, an exception is raised.