Three-Dimensional Rotations by Three Shears

We show that a rotation in three dimensions can be achieved by a composition of three shears, the first and third along a specified axis and the second along another given axis orthogonal to the first; this process is invertible. The resulting rotation algorithm is practical for the processing of fine-grained digital images, and is well adapted to the access constraints of common storage media such as dynamicRAMor magnetic disk. For a 2-D image, rotation by composition of three shears is well known. For 3-D, an obvious nine-shear decomposition has been mentioned in the literature. Our three-shear decomposition is a sizable improvement over that, and is the best that can be attained—just two shears won't do. Also, we give a brief summary of how the present three-shear decomposition approach generalizes to any linear transformations of unit determinant in any number of dimensions.