Generalized Barycentric Coordinates
Generalized Barycentric Coordinates
Generalized Barycentric Coordinates
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Tenth SIAM Conference on<br />
Geometric Design and Computing<br />
San Antonio, Texas, November 4–8, 2007<br />
Minisymposium M-14B<br />
<strong>Barycentric</strong> <strong>Coordinates</strong> and Transfinite Interpolation<br />
9:30 Kai Hormann<br />
<strong>Generalized</strong> <strong>Barycentric</strong> <strong>Coordinates</strong><br />
9:55 Scott Schaefer<br />
<strong>Barycentric</strong> <strong>Coordinates</strong> for Closed Curves<br />
10:20 Michael Floater<br />
Hermite Mean Value Interpolation<br />
10:45 Tao Ju<br />
A General, Geometric Construction of <strong>Coordinates</strong> in any Dimensions<br />
11:10 Solveig Bruvoll<br />
Transfinite Mean Value Interpolation over Volumetric Domains<br />
11:35 N. Sukumar<br />
<strong>Barycentric</strong> Finite Element Methods<br />
Organizers<br />
Kai Hormann and Michael Floater
<strong>Generalized</strong> <strong>Barycentric</strong> <strong>Coordinates</strong><br />
Kai Hormann<br />
Clausthal University of Technology<br />
San Antonio, November 8, 2007
Introduction<br />
<strong>Barycentric</strong> <strong>Coordinates</strong> for Planar Polygons<br />
Conclusion<br />
History<br />
Related Work<br />
▶ Introduction<br />
▶ History<br />
▶ Related Work<br />
▶ <strong>Barycentric</strong> <strong>Coordinates</strong> for Planar Polygons<br />
▶ Convex Polygons<br />
▶ Star-Shaped Polygons<br />
▶ Arbitrary Polygons<br />
▶ Conclusion<br />
▶ Applications<br />
▶ Future Work<br />
Kai Hormann<br />
<strong>Generalized</strong> <strong>Barycentric</strong> <strong>Coordinates</strong>
Kai Hormann<br />
<strong>Generalized</strong> <strong>Barycentric</strong> <strong>Coordinates</strong>
Kai Hormann<br />
<strong>Generalized</strong> <strong>Barycentric</strong> <strong>Coordinates</strong>
Introduction<br />
<strong>Barycentric</strong> <strong>Coordinates</strong> for Planar Polygons<br />
Conclusion<br />
<strong>Barycentric</strong> coordinates<br />
History<br />
Related Work<br />
August Ferdinand Möbius [1827]<br />
▶<br />
is the barycentre of the points with<br />
weights<br />
if and only if<br />
▶<br />
are the barycentric coordinates of<br />
▶ unique up to common factor for triangles<br />
Kai Hormann<br />
<strong>Generalized</strong> <strong>Barycentric</strong> <strong>Coordinates</strong>
Kai Hormann<br />
<strong>Generalized</strong> <strong>Barycentric</strong> <strong>Coordinates</strong>
Introduction<br />
<strong>Barycentric</strong> <strong>Coordinates</strong> for Planar Polygons<br />
Conclusion<br />
<strong>Barycentric</strong> coordinates for triangles<br />
Normalized barycentric coordinates<br />
History<br />
Related Work<br />
Properties<br />
▶ linearity<br />
▶ positivity<br />
▶ Lagrange property<br />
Application<br />
▶ linear interpolation of data<br />
Kai Hormann<br />
<strong>Generalized</strong> <strong>Barycentric</strong> <strong>Coordinates</strong>
Introduction<br />
<strong>Barycentric</strong> <strong>Coordinates</strong> for Planar Polygons<br />
Conclusion<br />
History<br />
Related Work<br />
Generalization of barycentric coordinates<br />
Finite-element-method with polygonal elements<br />
▶ convex [Wachspress 1975]<br />
▶ weakly convex [Malsch & Dasgupta 2004]<br />
▶ arbitrary [Sukumar & Malsch 2006]<br />
Interpolation of scattered data<br />
▶ natural neighbour interpolants [Sibson 1980]<br />
▶ –"– of higher order [Hiyoshi & Sugihara 2000]<br />
▶ Dirichlet tessellations [Farin 1990]<br />
Kai Hormann<br />
<strong>Generalized</strong> <strong>Barycentric</strong> <strong>Coordinates</strong>
Introduction<br />
<strong>Barycentric</strong> <strong>Coordinates</strong> for Planar Polygons<br />
Conclusion<br />
History<br />
Related Work<br />
Generalization of barycentric coordinates<br />
Parameterization of piecewise linear surfaces<br />
▶ shape preserving coordinates [Floater 1997]<br />
▶ discrete harmonic (DH) coordinates [Eck et al. 1995]<br />
▶ mean value (MV) coordinates [Floater 2003]<br />
Other applications<br />
▶ discrete minimal surfaces [Pinkall & Polthier 1993]<br />
▶ computer graphics [Meyer et al. 2002]<br />
▶ mesh deformation [Ju et al. 2005]<br />
Kai Hormann<br />
<strong>Generalized</strong> <strong>Barycentric</strong> <strong>Coordinates</strong>
Introduction<br />
<strong>Barycentric</strong> <strong>Coordinates</strong> for Planar Polygons<br />
Conclusion<br />
Convex Polygons<br />
Star-Shaped Polygons<br />
Arbitrary Polygons<br />
▶ Introduction<br />
▶ History<br />
▶ Related Work<br />
▶ <strong>Barycentric</strong> <strong>Coordinates</strong> for Planar Polygons<br />
▶ Convex Polygons<br />
▶ Star-Shaped Polygons<br />
▶ Arbitrary Polygons<br />
▶ Conclusion<br />
▶ Applications<br />
▶ Future Work<br />
Kai Hormann<br />
<strong>Generalized</strong> <strong>Barycentric</strong> <strong>Coordinates</strong>
Introduction<br />
<strong>Barycentric</strong> <strong>Coordinates</strong> for Planar Polygons<br />
Conclusion<br />
Arbitrary polygons<br />
Convex Polygons<br />
Star-Shaped Polygons<br />
Arbitrary Polygons<br />
Homogeneous coordinates<br />
Normalized coordinates<br />
Properties<br />
▶ partition of unity<br />
▶ reproduction<br />
linear precision<br />
for all<br />
Kai Hormann<br />
<strong>Generalized</strong> <strong>Barycentric</strong> <strong>Coordinates</strong>
Introduction<br />
<strong>Barycentric</strong> <strong>Coordinates</strong> for Planar Polygons<br />
Conclusion<br />
Convex polygons<br />
Convex Polygons<br />
Star-Shaped Polygons<br />
Arbitrary Polygons<br />
Theorem [FHK’06]: If all , then<br />
▶ positivity<br />
▶ Lagrange property<br />
▶ linear along boundary<br />
Application<br />
▶ interpolation of data given at the vertices<br />
▶ inside the convex hull of the<br />
▶ direct and efficient evaluation<br />
Kai Hormann<br />
<strong>Generalized</strong> <strong>Barycentric</strong> <strong>Coordinates</strong>
Introduction<br />
<strong>Barycentric</strong> <strong>Coordinates</strong> for Planar Polygons<br />
Conclusion<br />
Convex Polygons<br />
Star-Shaped Polygons<br />
Arbitrary Polygons<br />
Normal form of homogeneous coordinates<br />
Theorem [FHK’06]: All homogeneous coordinates can be written<br />
as<br />
with certain real functions .<br />
Three-point coordinates<br />
▶<br />
with<br />
Theorem [H’07]: Such a generating function<br />
exists for all three-point coordinates.<br />
Kai Hormann<br />
<strong>Generalized</strong> <strong>Barycentric</strong> <strong>Coordinates</strong>
Introduction<br />
<strong>Barycentric</strong> <strong>Coordinates</strong> for Planar Polygons<br />
Conclusion<br />
Three-point coordinates<br />
Theorem [FHK’06]:<br />
Convex Polygons<br />
Star-Shaped Polygons<br />
Arbitrary Polygons<br />
if and only if is<br />
▶ positive<br />
▶ monotonic<br />
▶ convex<br />
▶ sub-linear<br />
Examples<br />
▶ WP coordinates<br />
▶ MV coordinates<br />
▶ DH coordinates<br />
Kai Hormann<br />
<strong>Generalized</strong> <strong>Barycentric</strong> <strong>Coordinates</strong>
Introduction<br />
<strong>Barycentric</strong> <strong>Coordinates</strong> for Planar Polygons<br />
Conclusion<br />
Non-convex polygons<br />
Convex Polygons<br />
Star-Shaped Polygons<br />
Arbitrary Polygons<br />
Wachspress mean value discrete harmonic<br />
Poles, if<br />
, because<br />
Kai Hormann<br />
<strong>Generalized</strong> <strong>Barycentric</strong> <strong>Coordinates</strong>
Introduction<br />
<strong>Barycentric</strong> <strong>Coordinates</strong> for Planar Polygons<br />
Conclusion<br />
Star-shaped polygons<br />
Convex Polygons<br />
Star-Shaped Polygons<br />
Arbitrary Polygons<br />
Theorem [H’07]:<br />
if and only if is<br />
▶ positive<br />
▶ super-linear<br />
Examples<br />
▶ MV coordinates<br />
▶ DH coordinates<br />
Theorem [H’07]:<br />
for some if is<br />
▶ strictly super-linear<br />
Kai Hormann<br />
<strong>Generalized</strong> <strong>Barycentric</strong> <strong>Coordinates</strong>
Introduction<br />
<strong>Barycentric</strong> <strong>Coordinates</strong> for Planar Polygons<br />
Conclusion<br />
Mean value coordinates<br />
Convex Polygons<br />
Star-Shaped Polygons<br />
Arbitrary Polygons<br />
Properties<br />
▶ well-defined everywhere in<br />
▶ Lagrange property<br />
▶ linear along boundary<br />
▶ linear precision<br />
▶ smoothness at , otherwise<br />
▶ similarity invariance for<br />
for<br />
Application<br />
▶ direct interpolation of data<br />
Kai Hormann<br />
<strong>Generalized</strong> <strong>Barycentric</strong> <strong>Coordinates</strong>
Introduction<br />
<strong>Barycentric</strong> <strong>Coordinates</strong> for Planar Polygons<br />
Conclusion<br />
Applications<br />
Future Work<br />
▶ Introduction<br />
▶ History<br />
▶ Related Work<br />
▶ <strong>Barycentric</strong> <strong>Coordinates</strong> for Planar Polygons<br />
▶ Convex Polygons<br />
▶ Star-Shaped Polygons<br />
▶ Arbitrary Polygons<br />
▶ Conclusion<br />
▶ Applications<br />
▶ Future Work<br />
Kai Hormann<br />
<strong>Generalized</strong> <strong>Barycentric</strong> <strong>Coordinates</strong>
Introduction<br />
<strong>Barycentric</strong> <strong>Coordinates</strong> for Planar Polygons<br />
Conclusion<br />
Colour interpolation<br />
Applications<br />
Future Work<br />
Kai Hormann<br />
<strong>Generalized</strong> <strong>Barycentric</strong> <strong>Coordinates</strong>
Introduction<br />
<strong>Barycentric</strong> <strong>Coordinates</strong> for Planar Polygons<br />
Conclusion<br />
Vector fields<br />
Applications<br />
Future Work<br />
Kai Hormann<br />
<strong>Generalized</strong> <strong>Barycentric</strong> <strong>Coordinates</strong>
Introduction<br />
<strong>Barycentric</strong> <strong>Coordinates</strong> for Planar Polygons<br />
Conclusion<br />
Transfinite interpolation<br />
Applications<br />
Future Work<br />
mean value coordinates<br />
radial basis functions<br />
Kai Hormann<br />
<strong>Generalized</strong> <strong>Barycentric</strong> <strong>Coordinates</strong>
Introduction<br />
<strong>Barycentric</strong> <strong>Coordinates</strong> for Planar Polygons<br />
Conclusion<br />
Image warping<br />
Applications<br />
Future Work<br />
original image<br />
mask<br />
warped image<br />
Kai Hormann<br />
<strong>Generalized</strong> <strong>Barycentric</strong> <strong>Coordinates</strong>
Introduction<br />
<strong>Barycentric</strong> <strong>Coordinates</strong> for Planar Polygons<br />
Conclusion<br />
Smooth shading<br />
Applications<br />
Future Work<br />
Kai Hormann<br />
<strong>Generalized</strong> <strong>Barycentric</strong> <strong>Coordinates</strong>
Introduction<br />
<strong>Barycentric</strong> <strong>Coordinates</strong> for Planar Polygons<br />
Conclusion<br />
Implementation<br />
Applications<br />
Future Work<br />
▶ efficient and robust evaluation of the function<br />
Kai Hormann<br />
<strong>Generalized</strong> <strong>Barycentric</strong> <strong>Coordinates</strong>
Introduction<br />
<strong>Barycentric</strong> <strong>Coordinates</strong> for Planar Polygons<br />
Conclusion<br />
Smooth distance function<br />
Applications<br />
Future Work<br />
Function approximates the distance function<br />
▶<br />
and along the boundary<br />
▶ smooth, except at the vertices<br />
Kai Hormann<br />
<strong>Generalized</strong> <strong>Barycentric</strong> <strong>Coordinates</strong>
Introduction<br />
<strong>Barycentric</strong> <strong>Coordinates</strong> for Planar Polygons<br />
Conclusion<br />
Open questions<br />
Applications<br />
Future Work<br />
▶ Positive coordinates inside arbitrary polygons<br />
▶ positive MV coordinates [Lipman et al. 2007]<br />
▶ only C 0 -continuous<br />
▶ harmonic coordinates [Joshi et al. 2007]<br />
▶ hard to compute<br />
▶ Relation to boundary value problems [Belyaev 2006]<br />
▶ Bijectivity of MV mappings<br />
▶ convex → convex<br />
▶ non-convex → convex<br />
▶ (non-)convex → non-convex<br />
✔<br />
✔<br />
✘<br />
✔<br />
✘<br />
Kai Hormann<br />
<strong>Generalized</strong> <strong>Barycentric</strong> <strong>Coordinates</strong>
<strong>Generalized</strong> <strong>Barycentric</strong> <strong>Coordinates</strong><br />
Kai Hormann<br />
Thank you for your attention ☺
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