The six independent parameters end up dispersed across 12 variable elements ofthe transformation matrix. The constraints among the transformation matrix elements all pertain to the three-by-three submatrix in the upper left, which is fully determined by the three rotational parameters. The matrix formulation avoids potential ambiguities associated with specification of elementary transformations. As with the two-dimensional case, matrix inversion and multiplication can be used to derive a new matrix that describes the results of sequential application of other rigid-body transformations or their inverses.

As a numerical example, consider a s«qM««c« of transformations in which an image is first rotated around the x-axis by 7°, then rotated around the y-axis by 11°, then rotated around the z-axis by 13°, and finally translated 2 units along x, 3 units along y, and 5 units along z:


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