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#### On Generalized Barycentric Coordinates and Their Applications in Geometric Modeling

##### MPS-Authors
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Langer,  Torsten
Computer Graphics, MPI for Informatics, Max Planck Society;
International Max Planck Research School, MPI for Informatics, Max Planck Society;

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##### Citation

Langer, T. (2008). On Generalized Barycentric Coordinates and Their Applications in Geometric Modeling. PhD Thesis, Universität des Saarlandes, Saarbrücken.

Cite as: https://hdl.handle.net/11858/00-001M-0000-000F-1A90-6
##### 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 \$\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.