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Similarly to the two-dimensional case, the elementary instantaneous translational matrix describes a translational vector with a direction that corresponds to the initial instantaneous path that would be taken by the origin as it simultaneously spiraled around and translated along the axis of the transformation in a continuous movement that would end at the correct final position. The total length of this vector corresponds to the total distance that the origin would follow along this corkscrew-like path.

The matrix logarithm provides an alternative to singular value decomposition for determining whether a given matrix represents a rigid-body transformation. If the diagonal elements are all zero and the upper left three-by-three submatrix has the property of being equal to its transpose multiplied by negative one, the matrix represents a rigid-body transformation.

The matrix logarithm also provides a simple mechanism for decomposing a transformation into a smooth continuous path. If the matrix logarithm is divided by some integer, the matrix exponential of the result will describe a transformation that is an integer fraction of the original. Repeated sequential application of this transformation will eventually generate the same result as the original transformation. Furthermore, if the object being rotated is distributed symmetrically around the rotational axis of the transformation, it can be shown that this is the most parsimonious route for the transformation to follow in terms of total distance traveled by all points in the object.

Figure 6 shows the transformation used in the numerical example decomposed into three identical sequential transformations computed using the matrix logarithm and matrix exponential functions.

The relationship of the instantaneous elementary rotational transformations to the orientation and magnitude of the actual resultant rotation provides a very convenient method for specifying rotations around a specific axis that does not correspond to one of the cardinal axes of the coordinate system. The values can be placed into the proper positions of a matrix that will have a matrix exponential that is the desired transformation matrix.

For example, to produce a 7° rotation around the vector running from the origin to the point (2,3,5) requires instantaneous simultaneous elementary rotations of 7°* (2/V22 T 32 T 52) = 2.2711° = 0.0396 radians around the x-axis, 7°*(3/V22 T 32 T 52) = 3.4066° = 0.0595 radians around the y-axis, and

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