tenth siam conference on geometric design and computing ... · kai hormann generalized barycentric...
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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
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Generalized Barycentric Coordinates
Kai Hormann
Clausthal University of Technology
San Antonio, November 8, 2007
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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
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Generalized Barycentric CoordinatesKai Hormann
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Generalized Barycentric CoordinatesKai Hormann
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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
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Generalized Barycentric CoordinatesKai Hormann
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
![Page 19: Tenth SIAM Conference on Geometric Design and Computing ... · Kai Hormann Generalized Barycentric Coordinates Open questions Positive coordinates inside arbitrary polygons positive](https://reader033.vdocuments.mx/reader033/viewer/2022050219/5f649d593d23767a8b0e2379/html5/thumbnails/19.jpg)
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
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Generalized Barycentric CoordinatesKai Hormann
Colour interpolation
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
ApplicationsFuture Work
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Generalized Barycentric CoordinatesKai Hormann
Vector fields
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
ApplicationsFuture Work
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Generalized Barycentric CoordinatesKai Hormann
Transfinite interpolation
mean value coordinates radial basis functions
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
ApplicationsFuture Work
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Generalized Barycentric CoordinatesKai Hormann
Image warping
original image warped imagemask
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
ApplicationsFuture Work
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Generalized Barycentric CoordinatesKai Hormann
Smooth shading
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
ApplicationsFuture Work
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Generalized Barycentric CoordinatesKai Hormann
Implementation
▶ efficient and robust evaluation of the function
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
ApplicationsFuture Work
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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
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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
✔
✘
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Kai Hormann
Thank you for your attention ☺
Generalized Barycentric Coordinates