Single-sized spheres on surfaces (S4)

Abstract

Surface representations play a major role in a variety of applications throughout a diverse collection of fields, such as biology, chemistry, physics, or architecture. From a simulation point of view, it is important to simulate the surface as good as possible, including the usage of a wide range of different approximating elements. However, when it comes to manufacturing, it is desirable to have as few different building blocks as possible, as these can then be produced cost-efficiently.Our paper adds a procedure to be used in the simulation of natural phenomena as well as within the designers' creative toolbox. It represents a surface via a collection of equally sized spheres. In the first part of the paper, we give a mathematically precise definition of the underlying problem: How to cover as much as possible of a surface by single-sized spheres. This leads to questions regarding the extremal intersection area of spheres with reasonably well-behaved surfaces, for which we present upper and lower bounds. From these, we deduce how many spheres of a certain, fixed radius can be used at least and at most when interpolating a surface.Following these theoretical results, we compare a depth-first, a breadth-first, and a heuristic algorithm for the generation of surface coverings by single-sized spheres. As opposed to the mathematical description, we show that our algorithms also work for surfaces with boundary elements or sharp features such as edges or corners. We prove the applicability of our algorithm by a multitude of experiments and compare our procedure to ellipsoidal and multi-sized sphere methods.