# A coordinate system on a surface: definition, properties and applications

1 PRISME - Geometry, Algorithms and Robotics
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : Coordinate systems associated to a finite set of sample points have been extensively studied, especially in the context of interpolation of multivariate scattered data. Notably, Sibson proposed the so-called natural neighbor coordinates that are defined from the Voronoi diagram of the sample points. A drawback of those coordinate systems is that their definition domain is restricted to the convex hull of the sample points. This makes them difficult to use when the sample points belong to a surface. To overcome this difficulty, we propose a new system of coordinates. Given a closed surface $S$, i.e. a $(d-1)$-manifold of $\mathbb{R} ^d$, the coordinate system is defined everywhere on the surface, is continuous, and is local even if the sampling density is finite. Moreover, it is inherently $(d-1)$-dimensional while the previous systems are $d$-dimensional. No assumption is made about the ordering, the connectivity or topology of the sample points nor of the surface. We illustrate our results with an application to interpolation over a surface.
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Cited literature [26 references]

https://hal.inria.fr/inria-00072058
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Submitted on : Tuesday, May 23, 2006 - 7:40:03 PM
Last modification on : Saturday, January 27, 2018 - 1:31:27 AM
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### Identifiers

• HAL Id : inria-00072058, version 1

### Citation

Jean-Daniel Boissonnat, Julia Flötotto. A coordinate system on a surface: definition, properties and applications. RR-4530, INRIA. 2002. ⟨inria-00072058⟩

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