On different topological classes of spherical geodesic paths and circles in Z3

A discrete spherical geodesic path (DSGP) between two voxels s and t lying on a discrete sphere is a/the shortest path from s to t, comprising voxels of the discrete sphere intersected by the discrete geodesic plane passing through s, t, and the center of the sphere. We consider two classes of discretization, namely naive and standard, for both the sphere and the geodesic plane, which gives rise to four distinct topological classes of DSGP. We show that the naive-naive class does not guarantee the existence of a DSGP, whereas the other three classes do. We derive the upper bounds of the distance of a DSGP belonging to each class, from the real sphere and the real plane, for different neighborhood conditions. We propose an efficient integer-based algorithm to compute the DSGP for any class-and-neighborhood combination. Novel number-theoretic characterization of discrete sphere has been used for searching the voxels comprising a DSGP. The algorithm is output-sensitive, having its time and space complexities both linear in the length of the DSGP. It can also be extended for constructing discrete 3D circles of arbitrary orientations, specified by a few appropriate input parameters. Experimental results and related analysis demonstrate its efficiency and versatility.