The Power of Orthogonal Duals (Invited Talk)

Abstract

Triangle meshes have found widespread acceptance in computer graphics as a simple, convenient, and versatile representation of surfaces. In particular, computing on such simplicial meshes is a workhorse in a variety of graphics applications. In this context, mesh duals (tied to Poincaré duality and extending the well known relationship between Delaunay triangulations and Voronoi diagrams) are often useful, be it for physical simulation of fluids or parameterization. However, the precise embedding of a dual diagram with respect to its triangulation (i.e., the placement of dual vertices) has mostly remained a matter of taste or a numerical after-thought, and barycentric versus circumcentric duals are often the only options chosen in practice. In this chapter we discuss the notion of orthogonal dual diagrams, and show through a series of recent works that exploring the full space of orthogonal dual diagrams to a given simplicial complex is not only powerful and numerically beneficial, but it also reveals (using tools from algebraic topology and computational geometry) discrete analogs to continuous properties.