Hormann.pdf

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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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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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6/28Generalized Barycentric CoordinatesKai Hormann

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


Conclusion



Barycentric coordinates for trianglesNormalized barycentric coordinates

Properties

linearity

positivity

Lagrange property

Application

linear interpolation of data

HistoryRelated Work


Conclusion



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


Conclusion



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


Conclusion

d i l



Introduction

History

Related Work


Convex Polygons


Arbitrary Polygons


Future Work


Conclusion

Convex PolygonsStar-Shaped PolygonsArbitrary Polygons

I d i C P l



Arbitrary polygonsHomogeneous coordinates

Normalized coordinates

Properties

partition of unity

reproduction

for all

linear precision


Conclusion


I t d ti C P l



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


Conclusion


I t d ti C P l



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


Conclusion


I t d ti C P l



Three-point coordinatesTheorem [FHK06]: if and only if is

positive

monotonic

convex

sub-linear

Examples

WP coordinates

MV coordinates

DH coordinates


Conclusion


Introduction Convex Polygons



Non-convex polygons

Poles, if , because


Conclusion


Wachspress mean value discrete harmonic




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


Conclusion





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


Conclusion


Introduction



Introduction

History

Related Work


Convex Polygons


Arbitrary Polygons


Future Work


Conclusion

ApplicationsFuture Work

Introduction



Colour interpolation


Conclusion


Introduction



Vector fields


Conclusion


Introduction



Transfinite interpolation

mean value coordinates radial basis functions


Conclusion


Introductionli i



Image warping

original image warped imagemask


Conclusion


IntroductionA li i


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Smooth shading


Conclusion


IntroductionA li ti


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Implementation

efficient and robust evaluation of the function


Conclusion


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


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



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Kai Hormann

Thank you for your attention


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