The technique of using three consecutive shears to rotate a 2-D image---illustrated by the following diagram---has been known for a while.
For 3-D images, there is an obvious nine-shear solution, obtained by treating the 3-D rotation as the product of three 2-D rotations. Our algorithm, which uses only three shears, is a sizable improvement over that bound, and is actually the best that can be attained---just two shears won't do. The algorithm, illustrated by the following figure, proceeds much as in the two-dimensional case. Namely, a rotation where are the rotation's Euler angles, is decomposed as the product of three shears . Each of the three shears is given by matrix of the form (or a transpose of this form) and is thus characterized by three real parameters (rather than just one as in the two-dimensional case), which are in turn functions of the Euler angles. This result may be stated more abstractly as a linear algebra identity. In dimensions, any orthogonal matrix can be written in the form , where and are upper triangular matrices and is a lower triangular matrix---all three matrices having all 1's along the main diagonals. The question of whether this identity generalizes to is still open.
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