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or if S =

0 —1

Consequently, singular value decomposition of the upper left two-by-two submatrix, followed by inspection of the resulting matrix S, provides a trivial way to determine whether a two-dimensional transformation matrix describes a rigid-body rotation. As discussed later, an analogous approach is suitable for three-dimensional rigid-body rotations. It should be emphasized that the final row of the two-dimensional transformation matrix must have elements 0, 0 and 1 for this approach to be valid.

To this point, all of the methods for describing or decomposing a two-dimensional rigid-body transformation have relied upon sequences of more elementary or more fundamental transformations that combine to give the same end result as the original transformation. As mentioned previously, these sequences give rise to very artificial intermediate transformations that are more strongly tied to the underlying coordinate system than to the rigid-body movement itself. The resulting descriptions are not particularly helpful in visualizing the fundamental geometry of the transformation. From the geometric standpoint, it turns out that any two-dimensional rigid-body transformation that includes a rotational component can be fully described as a simple rotation around some fixed point in space. Knowing this, it is easy to envision the halfway point of a 10° clockwise rotation about a point as being a 5° clockwise rotation about that same point. The entire transformation can be envisioned as one continuous, smoothly evolving process that involves the simultaneous, rather than the sequential, application of elementary movements.

The point in space that remains fixed during a two-dimensional rotational rigid-body transformation can be derived by computing the real eigenvectors of the transformation matrix. Given any transformation matrix T (this is even true for transformation matrices that do not describe rigid-body movements), a real eigenvector of T must satisfy the following equation, where k is a scalar and is referred to as the eigenvalue of the eigenvector:

0 0

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