ary transformations all correspond to very mechanical-looking movements, none of which are likely to describe the real movements of a biological system (see Figs 4 and 5). The matrix formulation omits the irrelevant intervening movements and simply describes the final results of the sequence. As in the two-dimensional case, matrix inversion and matrix multiplication can be used to derive other related rigid-body transformations.

In the three-dimensional case, it is more difficult than in the two-dimensional case to determine whether a given matrix describes a rigid-body transformation. The interaction of the three rotational angles eliminates the simple symmetries of the matrix that are present in two dimensions. Singular-value decomposition of the upper left three-by-three submatrix provides a quick way to verify that a rigid-body transformation is described. The earlier discussion of singular value decomposition of the upper left two-by-two submatrix in the two-dimensional case generalizes fully to the three-dimensional case.

For the numerical example, singular-value decomposition gives

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