On Generalized Barycentric Coordinates and Their Applications in Geometric Modeling

Abstract

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 ei (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 simpli cation of a number of tasks in geometry processing. Coordinate systems with respect to triangles have been around since the 19th 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 de nition for 3D mean value coordinates that is valid for arbitrary polyhedra in ?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 Bézier surfaces. Finally, we show that barycentric coordinates can be used to obtain a novel formula for curvature computation on surfaces.