An Upper Bound on the Average Size of Silhouettes

TL;DR

This paper provides a theoretical upper bound showing that the average silhouette size of certain polyhedra grows proportionally to the square root of their complexity, explaining a common visual phenomenon in computer graphics.

Contribution

It introduces the first theoretical proof that polyhedra approximating surfaces have average silhouette sizes of order O(√n), extending understanding beyond convex shapes.

Findings

Silhouettes of certain polyhedra are on average proportional to √n.

Theoretical evidence supports the observed small silhouette phenomenon.

Applicable to non-convex, non-differentiable, and bounded surfaces.

Abstract

It is a widely observed phenomenon in computer graphics that the size of the silhouette of a polyhedron is much smaller than the size of the whole polyhedron. This paper provides, for the first time, theoretical evidence supporting this for a large class of objects, namely for polyhedra that approximate surfaces in some reasonable way; the surfaces may be non-convex and non-differentiable and they may have boundaries. We prove that such polyhedra have silhouettes of expected size $O(\sqrt{n})$ where the average is taken over all points of view and n is the complexity of the polyhedron.