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#### Spherical Barycentric Coordinates

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##### Citation

Langer, T., Belyaev, A., & Seidel, H.-P. (2006). Spherical Barycentric Coordinates.
In D. W. Fellner, S. N. Spencer, A. Sheffer, & K. Polthier (*SGP
2006 : Fourth Eurographics Symposium on Geometry Processing* (pp. 81-88). Aire-la-Ville, Switzerland: Eurographics.

Cite as: https://hdl.handle.net/11858/00-001M-0000-000F-23FD-B

##### Abstract

We develop spherical barycentric coordinates. Analogous to classical,

planar barycentric coordinates that describe the positions of points in a plane

with respect to

the vertices of a given planar polygon, spherical barycentric coordinates

describe the positions

of points on a sphere with respect to the vertices of a given spherical

polygon.

In particular, we introduce spherical mean value coordinates that inherit many

good properties of their planar counterparts.

Furthermore, we present a construction that gives a simple and intuitive

geometric interpretation for

classical barycentric coordinates, like Wachspress coordinates, mean value

coordinates, and discrete

harmonic coordinates.

One of the most interesting consequences is the possibility to

construct mean value coordinates for arbitrary polygonal meshes.

So far, this was only possible for triangular meshes. Furthermore, spherical

barycentric coordinates

can be used for all applications where only planar barycentric coordinates were

available up to now.

They include B\'ezier surfaces, parameterization, free-form deformations, and

interpolation of rotations.

planar barycentric coordinates that describe the positions of points in a plane

with respect to

the vertices of a given planar polygon, spherical barycentric coordinates

describe the positions

of points on a sphere with respect to the vertices of a given spherical

polygon.

In particular, we introduce spherical mean value coordinates that inherit many

good properties of their planar counterparts.

Furthermore, we present a construction that gives a simple and intuitive

geometric interpretation for

classical barycentric coordinates, like Wachspress coordinates, mean value

coordinates, and discrete

harmonic coordinates.

One of the most interesting consequences is the possibility to

construct mean value coordinates for arbitrary polygonal meshes.

So far, this was only possible for triangular meshes. Furthermore, spherical

barycentric coordinates

can be used for all applications where only planar barycentric coordinates were

available up to now.

They include B\'ezier surfaces, parameterization, free-form deformations, and

interpolation of rotations.