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Barycentric Coordinates and Transfinite Interpolation
9:30 Kai Hormann
Generalized Barycentric Coordinates
9:55 Scott Schaefer
Barycentric Coordinates for Closed Curves
10:20 Michael FloaterHermite 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 onGeometric Design and Computing
San Antonio, Texas, November 48, 2007Minisymposium 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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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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August Ferdinand Mbius [1827]
is the barycentre of the points with
weights if and only if
are the barycentric coordinates of
unique up to common factor fortriangles
Barycentric coordinates
HistoryRelated Work
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
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Barycentric coordinates for trianglesNormalized barycentric coordinates
Properties
linearity
positivity
Lagrange property
Application
linear interpolation of data
HistoryRelated Work
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
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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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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 [Meyeret al. 2002]
mesh deformation [Ju et al. 2005]
HistoryRelated Work
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
d i l
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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
I d i C P l
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Arbitrary polygonsHomogeneous coordinates
Normalized coordinates
Properties
partition of unity
reproduction
for all
linear precision
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
Convex PolygonsStar-Shaped PolygonsArbitrary Polygons
I t d ti C P l
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Convex polygonsTheorem [FHK06]: 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
I t d ti C P l
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Theorem [FHK06]: All homogeneous coordinates can be written
as
with certain real functions .
Three-point coordinates
with
Theorem [H07]: 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
I t d ti C P l
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Three-point coordinatesTheorem [FHK06]: 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
Introduction Convex Polygons
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Non-convex polygons
Poles, if , because
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
Convex PolygonsStar-Shaped PolygonsArbitrary Polygons
Wachspress mean value discrete harmonic
Introduction Convex Polygons
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Theorem [H07]: if and only if is
positive
super-linear
Examples MV coordinates
DH coordinates
Theorem [H07]: for some if is
strictly super-linear
Star-shaped polygons
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
Convex PolygonsStar-Shaped PolygonsArbitrary Polygons
Introduction Convex Polygons
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Mean value coordinatesProperties
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
Introduction
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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
Introduction
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Colour interpolation
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
ApplicationsFuture Work
Introduction
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Vector fields
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
ApplicationsFuture Work
Introduction
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Transfinite interpolation
mean value coordinates radial basis functions
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
ApplicationsFuture Work
Introductionli i
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Image warping
original image warped imagemask
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
ApplicationsFuture Work
IntroductionA li i
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Smooth shading
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
ApplicationsFuture Work
IntroductionA li ti
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Implementation
efficient and robust evaluation of the function
IntroductionBarycentric Coordinates for Planar Polygons
Conclusion
ApplicationsFuture Work
IntroductionA li ti
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Smooth distance functionFunction approximates the distance function
and along the boundary
smooth, except at the vertices
t oduct oBarycentric Coordinates for Planar Polygons
Conclusion
ApplicationsFuture Work
IntroductionA li ti
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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
Barycentric Coordinates for Planar PolygonsConclusion
ApplicationsFuture Work
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Kai Hormann
Thank you for your attention
Generalized Barycentric Coordinates