Tenth SIAM Conference on Geometric Design and Computing ... · Kai Hormann Generalized Barycentric Coordinates Open questions Positive coordinates inside arbitrary polygons positive

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




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


Conclusion



Barycentric coordinates for triangles

Normalized 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 [Meyer et al. 2002]

▶ mesh deformation [Ju et al. 2005]

HistoryRelated Work


Conclusion



▶ Related Work





▶ Future Work


Conclusion

Convex PolygonsStar-Shaped PolygonsArbitrary Polygons


Arbitrary polygons

Homogeneous coordinates

Normalized coordinates

Properties

▶ partition of unity

▶ reproduction

for all

linear precision


Conclusion



Convex polygons

Theorem [FHK’06]: If all , then

▶ positivity


▶ linear along boundary

Application

▶ interpolation of data given at the vertices

▶ inside the convex hull of the

▶ direct and efficient evaluation


Conclusion



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


Conclusion



Three-point coordinates

Theorem [FHK’06]: if and only if is

▶ positive

▶ monotonic

▶ convex

▶ sub-linear

Examples

▶ WP coordinates

▶ MV coordinates

▶ DH coordinates


Conclusion



Non-convex polygons

Poles, if , because


Conclusion


Wachspress mean value discrete harmonic


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


Conclusion



Mean value coordinates

Properties

▶ well-defined everywhere in


▶ linear along boundary

▶ linear precision for

▶ smoothness at , otherwise

▶ similarity invariance for

Application

▶ direct interpolation of data


Conclusion




▶ Related Work





▶ Future Work


Conclusion

ApplicationsFuture Work


Colour interpolation


Conclusion



Vector fields


Conclusion



Transfinite interpolation

mean value coordinates radial basis functions


Conclusion



Image warping

original image warped imagemask


Conclusion



Smooth shading


Conclusion



Implementation

▶ efficient and robust evaluation of the function


Conclusion



Smooth distance function

Function approximates the distance function

▶ and along the boundary

▶ smooth, except at the vertices


Conclusion



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 ✘


Conclusion


✔

✘

Kai Hormann

Thank you for your attention ☺

Generalized Barycentric Coordinates

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Tenth SIAM Conference on Geometric Design and Computing ... · Kai Hormann Generalized Barycentric Coordinates Open questions Positive coordinates inside arbitrary polygons positive