Observations on Silhouette Sizes

Silhouettes have many applications in computer graphics such as non-photorealistic edge rendering, fur rendering, shadow volume creation, and anti-aliasing. The number of edges, s, in the silhouette of a model observed from a point is therefore useful in analyzing such algorithms. This paper examines, from a theoretical viewpoint, a menagerie of objects with interesting silhouettes (including those with minimal and maximal silhouettes). It shows that the relationship between and s and the number of triangles in a model, f, is bounded above by s = O(f) and below by s = Ω(1), and that the expected value of s over all observation points at infinity is proportional to the sum of the dihedral angles. In practice, the models used with silhouette-based rendering algorithms are triangle meshes that are manually constructed or result from scans of human-made objects. They consist of only surface geometry with few cracks; there is no internal detail like the engine under a car’s hood. Geometric and aesthetic constraints on these models appear to create an inherent relationship between f and s. Measurements of the actual silhouettes of real-world 3D models with polygon counts varied across six orders of magnitude show them to follow the relationship s ~ f . Furthermore, the expected value of s at infinity is a good approximation of the expected silhouette size for a viewer at a finite location.