Triangle and tetrahedral meshes have found widespread acceptance in computer graphics as a simple, convenient, and versatile representation of surfaces and volumes. In particular, computing on such simplicial meshes is a workhorse in a variety of graphics applications. In this context, mesh duals (tied to Poincare duality and extending the well-known relationship between Delaunay triangulations and Voronoi diagrams) are often useful, from physical simulation of fluids to mesh 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 vs. circumcentric duals are often the only options chosen in practice. In this talk we discuss the notion of orthogonal dual diagrams, and show through a series of recent works that exploring the full space of orthogonal primal/dual meshes is not only powerful and numerically beneficial, but it also reveals (using tools from algebraic topology and computational geometry) discrete analogs to continuous properties. Applications varying from point sampling and fluid dynamics, to barycentric coordinates and self-supporting masonry will be covered.
09:30 - 10:30