Barycentric Coordinates and Transfinite Interpolation 9:30 Kai Hormann Generalized Barycentric Coordinates 9:55 Scott Schaefer Barycentric Coordinates for Closed Curves 10:20 Michael Floater Hermite Mean Value Interpolation 10:45 Tao Ju A General, Geometric Construction of Coordinates in any Dimensions 11:10 Solveig Bruvoll Transfinite Mean Value Interpolation over Volumetric Domains 11:35 N. Sukumar Barycentric Finite Element Methods Tenth SIAM Conference on Geometric Design and Computing San Antonio, Texas, November 4–8, 2007 Minisymposium M-14B Kai Hormann and Michael Floater Organizers
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Barycentric Coordinates and Transfinite Interpolation
9:30 Kai HormannGeneralized Barycentric Coordinates
9:55 Scott SchaeferBarycentric Coordinates for Closed Curves
10:20 Michael FloaterHermite Mean Value Interpolation
10:45 Tao JuA General, Geometric Construction of Coordinates in any Dimensions
11:10 Solveig BruvollTransfinite Mean Value Interpolation over Volumetric Domains
11:35 N. SukumarBarycentric Finite Element Methods
Tenth SIAM Conference onGeometric Design and Computing
San Antonio, Texas, November 4–8, 2007 Minisymposium M-14B
Kai Hormann and Michael FloaterOrganizers
Generalized Barycentric Coordinates
Kai Hormann
Clausthal University of Technology
San Antonio, November 8, 2007
Generalized Barycentric CoordinatesKai Hormann
▶ Introduction▶ History
▶ Related Work
▶ Barycentric Coordinates for Planar Polygons▶ Convex Polygons
▶ Star-Shaped Polygons
▶ Arbitrary Polygons
▶ Conclusion▶ Applications
▶ Future Work
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
HistoryRelated Work
Generalized Barycentric CoordinatesKai Hormann
Generalized Barycentric CoordinatesKai Hormann
Generalized Barycentric CoordinatesKai Hormann
August Ferdinand Möbius [1827]
▶ is the barycentre of the points with weights if and only if
▶ are the barycentric coordinates of
▶ unique up to common factor for triangles
Barycentric coordinates
HistoryRelated Work
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
Generalized Barycentric CoordinatesKai Hormann
Generalized Barycentric CoordinatesKai Hormann
Barycentric coordinates for triangles
Normalized barycentric coordinates
Properties
▶ linearity
▶ positivity
▶ Lagrange property
Application
▶ linear interpolation of data
HistoryRelated Work
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
Generalized Barycentric CoordinatesKai Hormann
Generalization of barycentric coordinates
Finite-element-method with polygonal elements
▶ convex [Wachspress 1975]
▶ weakly convex [Malsch & Dasgupta 2004]
▶ arbitrary [Sukumar & Malsch 2006]
Interpolation of scattered data
▶ natural neighbour interpolants [Sibson 1980]
▶ – " – of higher order [Hiyoshi & Sugihara 2000]
▶ Dirichlet tessellations [Farin 1990]
HistoryRelated Work
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
Generalized Barycentric CoordinatesKai Hormann
Generalization of barycentric coordinates
Parameterization of piecewise linear surfaces
▶ shape preserving coordinates [Floater 1997]
▶ discrete harmonic (DH) coordinates [Eck et al. 1995]