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This sequence is illustrated schematically in Fig. 4. Trigonometric computations allow rotational angles and translational distances to be computed for any of the many other possible s«qM««c«s of elementary transformations. For example, the identical transformation can be achieved by first translating by 2.2045 units along x, 2.7793 units along y, and 5.0414 units along z, then rotating around the z-axis by

FIGURE 4 Rotational component of the numerical example illustrating three-dimensional rigid-body movements. Here, the rotations are applied sequentially around the x-axis, the y-axis, and then the z-axis. Note that the initial and final positions are identical to those in Figs 5 and 6, but the intervening positions are different. Wire-frame models of a brain undergoing transformation are shown as viewed along the coordinate axes in the first three rows. The view portrayed in the fourth row is parallel to the axis of rotation of the complete transformation, here called the "fixed axis" because it is unchanged in the absence of translations. The right hemisphere is shown in red and the left hemisphere in blue. The view along the z-axis is seen looking down on the brain from above. The view along the x-axis is seen looking toward the right hemisphere from the right. The view along the y-axis is seen looking at the back of the brain from behind. The view along the fixed axis looks down on the right frontal lobe from a right supero-anterior position. The two small round dots associated with each figure lie on the fixed axis. As a result, they are both superimposed over the origin in the starting and final positions on the fourth row. Note that these two points are in identical positions before and after the complete transformation, but that they are moved to other positions by the intermediate steps. The orientation of the fixed axis can be determined by finding the real eigenvectors of the complete transformation. The repositioning required to compose a view along the fixed axis is given by the transpose of the U matrix obtained by Schur decomposition of the original transformation. See also Plate 65.

14.4198°, then rotating around the y-axis by 9.0397°, and finally rotating around the x-axis by 9.4030°. This sequence is illustrated in Fig. 5.

Unlike the two-dimensional case, where only translations change when the order of the sequence of elementary transformations is altered, in the three-dimensional case the rotational angles are altered as well. A movie of the sequence of transformations would show that these sequences of element-

FIGURE 5 Alternative decomposition of the rotational component of the numerical example illustrating three-dimensional rigid-body movements. Here, the rotations are applied sequentially around the z-axis, the y-axis, and then the x-axis. This is the reverse order of the rotations in Fig. 4, and the magnitudes of the rotations have been adjusted so that the final results are identical. See Fig. 4 legend for additional details. See also Plate 66.

FIGURE 5 Alternative decomposition of the rotational component of the numerical example illustrating three-dimensional rigid-body movements. Here, the rotations are applied sequentially around the z-axis, the y-axis, and then the x-axis. This is the reverse order of the rotations in Fig. 4, and the magnitudes of the rotations have been adjusted so that the final results are identical. See Fig. 4 legend for additional details. See also Plate 66.

0.9565 0.2459 - 0.2208 0.9619 0.1908 - 0.1196 0 0 1 0 0 2" 0 10 3

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