| dc.description.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. | en_US |
