Surfaces over Dirichlet Tessellations
Computer Aided Geometric Design
Interpolation of scattered data on closed surfaces
Computer Aided Geometric Design
Guaranteed-quality mesh generation for curved surfaces
SCG '93 Proceedings of the ninth annual symposium on Computational geometry
Properties of local coordinates based on Dirichlet tessellations
Geometric modelling
Triangulating topological spaces
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
Natural neighbor interpolation on the sphere
An international conference on curves and surfaces on Wavelets, images, and surface fitting
Bernstein-Be´zier polynomials on spheres and sphere-like surfaces
Computer Aided Geometric Design
Fitting scattered data on sphere-like surfaces using spherical splines
Journal of Computational and Applied Mathematics - Special issue on scattered data
Systems of coordinates associated with points scattered in the plane
Computer Aided Geometric Design
Scattered data fitting on the sphere
Proceedings of the international conference on Mathematical methods for curves and surfaces II Lillehammer, 1997
Algorithmic geometry
Voronoi-based interpolation with higher continuity
Proceedings of the sixteenth annual symposium on Computational geometry
Delaunay triangulations and Voronoi diagrams for Riemannian manifolds
Proceedings of the sixteenth annual symposium on Computational geometry
Proceedings of the conference on Visualization '01
Natural neighbor coordinates of points on a surface
Computational Geometry: Theory and Applications
Estimating differential quantities using polynomial fitting of osculating jets
Proceedings of the 2003 Eurographics/ACM SIGGRAPH symposium on Geometry processing
Estimating differential quantities using polynomial fitting of osculating jets
Computer Aided Geometric Design
Shape Simplification Based on the Medial Axis Transform
Proceedings of the 14th IEEE Visualization 2003 (VIS'03)
3D noise mapping in urban areas
International Journal of Geographical Information Science
Estimating differential quantities using polynomial fitting of osculating jets
Computer Aided Geometric Design
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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 make 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, i.e. a manifold of, 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 1-dimensional while the previous systems are 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.