Co-planar shape-fitting using the Rotating Squares algorithm

Rotating Squares is a method based upon the Marching Cubes algorithm of Lorensen and Cline (1987). It is used as a method for storing a piecewise linear approximation of a two dimensional object (Donafee, 2000). Approximations to two distinct shapes can then be easily compared to find common boundary segments. However, it is also possible to use the algorithm to determine how to align the two shapes in a plane, such that they have the longest possible boundary segment in common. It is this issue that is addressed in the paper. Examples are given in which, while finding alike boundary segments, the Rotating Squares algorithm does not provide an edge along which the two shapes can be aligned without intruding upon one another. We present a simple test that can be used to indicate any space-sharing violation. Since the motivation for this work arose in three dimensions we go on to give considerations necessary for lifting the algorithms to three dimensions.