Three-Dimensional Rotations by Three Shears

✍ Scribed by Tommaso Toffoli; Jason Quick

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 317 KB
• Volume: 59
• Category: Article
• ISSN: 1077-3169
• DOI: 10.1006/gmip.1997.0420

✦ Synopsis

Algorithms that achieve spatial transformations by means of shears are useful in many computer graphics

We show that a rotation in three dimensions can be achieved by a composition of three shears, the first and third along a contexts, as discussed in 9]. Briefly, when a discrete specified axis and the second along another given axis orthogopixel array is subjected to a continuous address transformanal to the first; this process is invertible. The resulting rotation tion the transformed addresses usually fall between array algorithm is practical for the processing of fine-grained digital points; to construct the new pixel value at a given array images, and is well adapted to the access constraints of common point one must take a suitably weighted average of the storage media such as dynamic RAM or magnetic disk. For a pixels that fell near that point. Though shears are continu-2-D image, rotation by composition of three shears is well ous transformations, they maintain uniform pixel spacing known. For 3-D, an obvious nine-shear decomposition has been along each coordinate axis. Thus, it becomes possible to mentioned in the literature. Our three-shear decomposition is approximate a shear by a discrete, one-to-one address a sizable improvement over that, and is the best that can be transformation whereby each new pixel falls within just attained-just two shears won't do. Also, we give a brief sumone pixel spacing of the ideal position specified by the mary of how the present three-shear decomposition approach shear itself. In this case, if the image resolution is high generalizes to any linear transformations of unit determinant in any number of dimensions. © 1997 Academic Press enough, there is no need to ''look inside the pixel'' and do value arithmetic on their contents: it is enough to ship the pixels to their destinations in a ''data-blind'' fashion,