This is true for any two-dimensional rigid-body transformation, and this form (with zeros on the diagonal) can be used as an alternative to singular value decomposition for proving that a matrix defines a rigid-body transformation.
The matrix logarithm can be decomposed into more elementary transformations. Analogous to the situation that applies with scalar values, the decomposition is based on the notion that the component matrices should be sum to the matrix logarithm being decomposed. The decomposition strategy anticipates the consequences of subsequent matrix exponentiation to define instantaneous elementary transformations that act simultaneously to produce the original matrix.
For the example matrix, a suitable decomposition is
In the two-dimensional case, the instantaneous elementary rotational transformation will always be identical to the conventional elementary rotational transformation obtained by sequential compositions. However, the instantaneous elementary translational matrix provides information that is directly related to the movement of the origin of the coordinate system as it is rotated around the fixed point by the original transformation. The vector defined by this translation is exactly tangent to the path traveled by the origin around the fixed point derived earlier and the total length of the translation describes the total length of the circular arc that the origin would travel in rotating as part of a rigid body around the fixed point. The term instantaneous elementary transformation is used here to denote the fact that the rotations and translations must be viewed as occurring progressively and simultaneously to produce a trajectory that leads to the correct final transformation.
In the numerical example, the origin travels 9.8614 units as it rotates around the fixed point at (53.4352, — 18.3601).
The transformation is illustrated in Fig. 2.
Issues in three dimensions are similar to those in two dimensions, but include additional subtleties. Six independent parameters are required to describe a general three-dimensional rigid-body rotation, and the ones typically used are three
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