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

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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.