On Generalized Barycentric Coordinates and Their Applications in Geometric Modeling

Generalized barycentric coordinate systems allow us to express the position of a point in space with respect to a given polygon or higher dimensional polytope. In such a system, a coordinate exists for each vertex of the polytope such that its vertices are represented by unit vectors $\vect{e}_i$ (where the coordinate associated with the respective vertex is 1, and all other coordinates are 0). Coordinates thus have a geometric meaning, which allows for the simplification of a number of tasks in geometry processing. Coordinate systems with respect to triangles have been around since the 19\textsuperscript{th} century, and have since been generalized; however, all of them have certain drawbacks, and are often restricted to special types of polytopes. We eliminate most of these restrictions and introduce a definition for 3D mean value coordinates that is valid for arbitrary polyhedra in $\realspace{3}$, with a straightforward generalization to higher dimensions. Furthermore, we extend the notion of barycentric coordinates in such a way as to allow Hermite interpolation and investigate the capabilities of generalized barycentric coordinates for constructing generalized B\'ezier surfaces. Finally, we show that barycentric coordinates can be used to obtain a novel formula for curvature computation on surfaces