Post on 22-Jul-2020
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]
▶ mean value (MV) coordinates [Floater 2003]
Other applications
▶ discrete minimal surfaces [Pinkall & Polthier 1993]
▶ computer graphics [Meyer et al. 2002]
▶ mesh deformation [Ju et al. 2005]
HistoryRelated Work
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
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
Convex PolygonsStar-Shaped PolygonsArbitrary Polygons
Generalized Barycentric CoordinatesKai Hormann
Arbitrary polygons
Homogeneous coordinates
Normalized coordinates
Properties
▶ partition of unity
▶ reproduction
for all
linear precision
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
Convex PolygonsStar-Shaped PolygonsArbitrary Polygons
Generalized Barycentric CoordinatesKai Hormann
Convex polygons
Theorem [FHK’06]: If all , then
▶ positivity
▶ Lagrange property
▶ linear along boundary
Application
▶ interpolation of data given at the vertices
▶ inside the convex hull of the
▶ direct and efficient evaluation
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
Convex PolygonsStar-Shaped PolygonsArbitrary Polygons
Generalized Barycentric CoordinatesKai Hormann
Theorem [FHK’06]: All homogeneous coordinates can be written as
with certain real functions .
Three-point coordinates
▶ with
Theorem [H’07]: Such a generating function
exists for all three-point coordinates.
Normal form of homogeneous coordinates
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
Convex PolygonsStar-Shaped PolygonsArbitrary Polygons
Generalized Barycentric CoordinatesKai Hormann
Three-point coordinates
Theorem [FHK’06]: if and only if is
▶ positive
▶ monotonic
▶ convex
▶ sub-linear
Examples
▶ WP coordinates
▶ MV coordinates
▶ DH coordinates
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
Convex PolygonsStar-Shaped PolygonsArbitrary Polygons
Generalized Barycentric CoordinatesKai Hormann
Non-convex polygons
Poles, if , because
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
Convex PolygonsStar-Shaped PolygonsArbitrary Polygons
Wachspress mean value discrete harmonic
Generalized Barycentric CoordinatesKai Hormann
Theorem [H’07]: if and only if is
▶ positive
▶ super-linear
Examples
▶ MV coordinates
▶ DH coordinates
Theorem [H’07]: for some if is
▶ strictly super-linear
Star-shaped polygons
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
Convex PolygonsStar-Shaped PolygonsArbitrary Polygons
Generalized Barycentric CoordinatesKai Hormann
Mean value coordinates
Properties
▶ well-defined everywhere in
▶ Lagrange property
▶ linear along boundary
▶ linear precision for
▶ smoothness at , otherwise
▶ similarity invariance for
Application
▶ direct interpolation of data
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
Convex PolygonsStar-Shaped PolygonsArbitrary Polygons
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
ApplicationsFuture Work
Generalized Barycentric CoordinatesKai Hormann
Colour interpolation
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
ApplicationsFuture Work
Generalized Barycentric CoordinatesKai Hormann
Vector fields
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
ApplicationsFuture Work
Generalized Barycentric CoordinatesKai Hormann
Transfinite interpolation
mean value coordinates radial basis functions
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
ApplicationsFuture Work
Generalized Barycentric CoordinatesKai Hormann
Image warping
original image warped imagemask
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
ApplicationsFuture Work
Generalized Barycentric CoordinatesKai Hormann
Smooth shading
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
ApplicationsFuture Work
Generalized Barycentric CoordinatesKai Hormann
Implementation
▶ efficient and robust evaluation of the function
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
ApplicationsFuture Work
Generalized Barycentric CoordinatesKai Hormann
Smooth distance function
Function approximates the distance function
▶ and along the boundary
▶ smooth, except at the vertices
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
ApplicationsFuture Work
Generalized Barycentric CoordinatesKai Hormann
Open questions
▶ Positive coordinates inside arbitrary polygons
▶ positive MV coordinates [Lipman et al. 2007]
▶ only C0-continuous
▶ harmonic coordinates [Joshi et al. 2007]
▶ hard to compute
▶ Relation to boundary value problems [Belyaev 2006]
▶ Bijectivity of MV mappings
▶ convex → convex ✔▶ non-convex → convex ✔▶ (non-)convex → non-convex ✘
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
ApplicationsFuture Work
✔
✘
Kai Hormann
Thank you for your attention ☺
Generalized Barycentric Coordinates