Generalized Barycentric Coordinates
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1 Generalized Barycentric Coordinates Kai Hormann Faculty of Informatics University of Lugano
2 Cartesian coordinates y 3 2 ( 3,1) (2,2) (0,0) (1, 2) x René Descartes ( ) Appendix La Géométrie /35 Generalized Barycentric Coordinates SIAM conference GD/SPM11 Orlando 25 October 2011
3 Barycentric coordinates (1,0,0) 0) (0.5,0.5,0) (0,1,0) (0.25,0.25,0.5) (0,0,1) (0.25, 0.25,1) August Ferdinand Möbius ( ) Der barycentrische Calcul /35 Generalized Barycentric Coordinates SIAM conference GD/SPM11 Orlando 25 October 2011
4 Barycentric coordinates system of masses at positions position of the system s s barycentre : 3/35 Generalized Barycentric Coordinates SIAM conference GD/SPM11 Orlando 25 October 2011
5 4/35 Generalized Barycentric Coordinates SIAM conference GD/SPM11 Orlando 25 October 2011
6 Barycentric coordinates system of masses at positions position of the system s s barycentre : are the barycentric coordinates of not unique at least points needed to span 5/35 Generalized Barycentric Coordinates SIAM conference GD/SPM11 Orlando 25 October 2011
7 Barycentric coordinates Theorem [Möbius, 1827] : The barycentric coordinates of with respect to are unique up to a common factor example: 6/35 Generalized Barycentric Coordinates SIAM conference GD/SPM11 Orlando 25 October 2011
8 7/35 Generalized Barycentric Coordinates SIAM conference GD/SPM11 Orlando 25 October 2011
9 Barycentric coordinates for triangles normalized barycentric coordinates properties partition of unity reproduction o positivity Lagrange g property p application linear interpolation of data 8/35 Generalized Barycentric Coordinates SIAM conference GD/SPM11 Orlando 25 October 2011
10 Generalized 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 fhigher h order [Hiyoshi & Sugihara 2000] Dirichlet tessellations [Farin 1990] 9/35 Generalized Barycentric Coordinates SIAM conference GD/SPM11 Orlando 25 October 2011
11 Generalized 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] colour interpolation [Meyer et al. 2002] boundary value problems [Belyaev 2006] 10/35 Generalized Barycentric Coordinates SIAM conference GD/SPM11 Orlando 25 October 2011
12 Arbitrary polygons barycentric coordinates normalized coordinates properties partition of unity linear precision reproduction for all 11/35 Generalized Barycentric Coordinates SIAM conference GD/SPM11 Orlando 25 October 2011
![& Kós 2006] positivity Lagrange](/docs-images/76/73979881/images/13-5.jpg "property linear along boundary")
13 Theorem: If all, then Convex polygons [Floater, H. & Kós 2006] positivity Lagrange property linear along boundary application interpolation of data given at the vertices inside the convex hull of the direct and efficient evaluation 12/35 Generalized Barycentric Coordinates SIAM conference GD/SPM11 Orlando 25 October 2011
14 Normal form Theorem: All barycentric coordinates can be written as with certain real functions three-point coordinates with Theorem: Such a generating g function exists for all three-point coordinates 13/35 Generalized Barycentric Coordinates SIAM conference GD/SPM11 Orlando 25 October 2011
15 Three-point coordinates Theorem: if and only if is positive monotonic convex sub-linear examples WP coordinates MV coordinates DH coordinates 14/35 Generalized Barycentric Coordinates SIAM conference GD/SPM11 Orlando 25 October 2011
16 Non-convex polygons Wachspress mean value discrete harmonic poles, if, because 15/35 Generalized Barycentric Coordinates SIAM conference GD/SPM11 Orlando 25 October 2011
17 Star-shaped polygons Theorem: if and only if is positive super-linear examples MV coordinates DH coordinates Theorem: for some if is strictly super-linear 16/35 Generalized Barycentric Coordinates SIAM conference GD/SPM11 Orlando 25 October 2011
18 Theorem: MV coordinates have no poles in Mean value coordinates [H. & Floater 2006] 17/35 Generalized Barycentric Coordinates SIAM conference GD/SPM11 Orlando 25 October 2011
19 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 18/35 Generalized Barycentric Coordinates SIAM conference GD/SPM11 Orlando 25 October 2011
20 Implementation efficient and robust evaluation of the function 19/35 Generalized Barycentric Coordinates SIAM conference GD/SPM11 Orlando 25 October 2011
21 Colour interpolation 20/35 Generalized Barycentric Coordinates SIAM conference GD/SPM11 Orlando 25 October 2011
22 Vector fields 21/35 Generalized Barycentric Coordinates SIAM conference GD/SPM11 Orlando 25 October 2011
23 Smooth shading 22/35 Generalized Barycentric Coordinates SIAM conference GD/SPM11 Orlando 25 October 2011
24 Rendering of quadrilateral elements 23/35 Generalized Barycentric Coordinates SIAM conference GD/SPM11 Orlando 25 October 2011
25 Transfinite interpolation mean value coordinates radial basis functions 24/35 Generalized Barycentric Coordinates SIAM conference GD/SPM11 Orlando 25 October 2011
26 Mesh animation 25/35 Generalized Barycentric Coordinates SIAM conference GD/SPM11 Orlando 25 October 2011
27 Image warping original image mask warped image 26/35 Generalized Barycentric Coordinates SIAM conference GD/SPM11 Orlando 25 October 2011
![Barycentric mappings source polygon for v Ω : target polygon Ω =[v 1,, v n ] Ω =[v 1,, v n ] v j R](/docs-images/76/73979881/images/28-3.jpg "2 v j R 2 27/35 Generalized Barycentric Coordinates SIAM conference GD/SPM11 Orlando 25 October")
28 Barycentric mappings source polygon for v Ω : target polygon Ω =[v 1,, v n ] Ω =[v 1,, v n ] v j R 2 v j R 2 27/35 Generalized Barycentric Coordinates SIAM conference GD/SPM11 Orlando 25 October 2011
![1,, z n ] barycentric coordinates c j : Ω C Ω =[z 1,, z n ] z j C z](/docs-images/76/73979881/images/29-7.jpg "j C 28/35 Generalized Barycentric Coordinates SIAM conference")
29 Complex barycentric mappings [Weber et al. 2011] source polygon for z Ω : target polygon with complex Ω =[z 1,, z n ] barycentric coordinates c j : Ω C Ω =[z 1,, z n ] z j C z j C 28/35 Generalized Barycentric Coordinates SIAM conference GD/SPM11 Orlando 25 October 2011
30 Complex barycentric coordinates [Weber et al. 2009] normal form of complex barycentric coordinates edge weight functions β j : Ω C homogeneous coordinates normalized coordinates example β j = log (r j+1 /r j ) gives Cauchy Green coordinates 29/35 Generalized Barycentric Coordinates SIAM conference GD/SPM11 Orlando 25 October 2011
31 Why bother? complex barycentric mappings are more general include real-valued mappings as a special case complex barycentric mappings are more powerful coordinates are not treated separately source mean value Cauchy Green new 30/35 Generalized Barycentric Coordinates SIAM conference GD/SPM11 Orlando 25 October 2011
32 MAGIC New mappings Made-to-order Angle Guided Interpolating Coord s edge scaling edge interpolation vertex interpolation source mean value MAGIC (MV+MAGIC)/2 (MV) 31/35 Generalized Barycentric Coordinates SIAM conference GD/SPM11 Orlando 25 October 2011
33 A new perspective blend of linear edge-to-edge g similarity transforms 32/35 Generalized Barycentric Coordinates SIAM conference GD/SPM11 Orlando 25 October 2011
34 Beyond barycentric mappings blend of polynomial edge-to-edge g transforms 33/35 Generalized Barycentric Coordinates SIAM conference GD/SPM11 Orlando 25 October 2011
35 Beyond barycentric mappings blend of polynomial edge-to-edge g transforms source linear MV cubic MV 34/35 Generalized Barycentric Coordinates SIAM conference GD/SPM11 Orlando 25 October 2011
36 Do you want to know more? PS 3 Positive Gordon Wixom coordinates today, 9:50 AM 10:10 AM, Room: Jasmine MS 6 Theory and Applications of Barycentric Coordinates today, 2:15 PM 4:15 PM, Room: Oleander A Workshop Barycentric Coordinates in Graphics Processing and Finite/Boundary Element Methods honoring Dr. Eugene L. Wachspress 23 July 25 July 2012, Columbia University, New York Tutorials, Invited Lectures, Contributed Talks, and Posters organizers: N. Sukumar, G. Dasgupta, E. Grinspun, K. Hormann Web page with related material 35/35 Generalized Barycentric Coordinates SIAM conference GD/SPM11 Orlando 25 October 2011
Generalized Barycentric Coordinates
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