On Covering a Solid Sphere with Concentric Spheres in ${\mathbb Z}^3$

TL;DR

This paper characterizes and fills absentee-voxels in digital spheres to produce complete, hole-free spherical surfaces, and extends the analysis to covering solid spheres with smaller spheres, with results on absentee counts.

Contribution

It provides a novel characterization of absentee-voxels in digital spheres and algorithms to fill them, improving the realism of digital spherical models.

Findings

Absentee-voxels count varies quadratically with radius.

Filling absentee-voxels results in more realistic digital spheres.

Covering a solid sphere with smaller spheres increases absentee count cubically.

Abstract

We show that a digital sphere, constructed by the circular sweep of a digital semicircle (generatrix) around its diameter, consists of some holes (absentee-voxels), which appear on its spherical surface of revolution. This incompleteness calls for a proper characterization of the absentee-voxels whose restoration will yield a complete spherical surface without any holes. In this paper, we present a characterization of such absentee-voxels using certain techniques of digital geometry and show that their count varies quadratically with the radius of the semicircular generatrix. Next, we design an algorithm to fill these absentee-voxels so as to generate a spherical surface of revolution, which is more realistic from the viewpoint of visual perception. We further show that covering a solid sphere by a set of complete spheres also results in an asymptotically larger count of absentees, which is cubic in the radius of the sphere. The characterization and generation of complete solid spheres without any holes can also be accomplished in a similar fashion. We furnish test results to substantiate our theoretical findings.