Generalized Barycentric Coordinates
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1 Generalized Barycentric Coordinates Kai Hormann Faculty of Informatics Università della Svizzera italiana, Lugano
2 My life in a nutshell 2009??? Associate University of Lugano 1 Generalized Barycentric Coordinates Varese 18 October 2012
3 Ticino 5-th largest canton of Switzerland Italian as official i language ~ ⅓ Million Ticinesi major cities Lugano Bellinzona Locarno mediterranean climate Italian flair 2 Generalized Barycentric Coordinates Varese 18 October 2012
4 USI Università della Svizzera italiana founded in University in Ticino 2 Campuses - Lugano - Mendrisio 4 Faculties - Architecture - Communication Sciences - Economics - Informatics Mendrisio Lugano 3 Generalized Barycentric Coordinates Varese 18 October 2012
5 Faculty of Informatics Faculty of Informatics founded in 2004 #3 in Switzerland international faculty - 23 professors - 10 countries excellent research - 5 Mio. CHF research projects in 2011 innovative teaching students (B.Sc. / M.Sc. / Ph.D.) 4 Generalized Barycentric Coordinates Varese 18 October 2012
6 Geometry processing description and manipulation of surfaces point clouds triangle meshes surface reconstruction automatic interpolation compression 5 Generalized Barycentric Coordinates Varese 18 October 2012
7 Parameterizations flattening triangle meshes texture mapping surface fitting remeshing 6 Generalized Barycentric Coordinates Varese 18 October 2012
8 Subdivision curves and surfaces limit behaviour convergence smoothness / regularity approximation order 7 Generalized Barycentric Coordinates Varese 18 October 2012
9 Barycentric rational interpolation local polynomial interpolant normalized blending function barycentric interpolant = efficient rational interpolant without poles especially good for equidistant nodes 8 Generalized Barycentric Coordinates Varese 18 October 2012
10 Cartesian coordinates point (2,2) with y x-coordinate: 2 y-coordinate: 2 3 (2,2) 2 mathematically: ( 3,1) (2,2) 2) = 2 (1,0) 1 (0,0) (1, 2) (0,1) x in general: (x,y) = x (1,0) + y (0,1) René Descartes ( ) x- and y-coordinates w.r.t. base points (1,0) and (0,1) 9 Generalized Barycentric Coordinates Varese 18 October 2012
11 Barycentric coordinates August Ferdinand Möbius ( ) (0,1,0) (1,0,0) 0) (0.5,0.5,0) (0.25,0.25,0.5) (0.25, 0.25,1) (0,0,1) point (a,b,c) with 3 coordinates w.r.t. base points A, B, C mathematically: (a,b,c) b = a A + b B + c C where A = (1,0,0) B = (0,1,0) C = (0,0,1) and a + b + c = 1 10 Generalized Barycentric Coordinates Varese 18 October 2012
12 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 11 Generalized Barycentric Coordinates Varese 18 October 2012
13 Barycentric coordinates Theorem [Möbius, 1827] : The barycentric coordinates of with respect to are unique up to a common factor example: 12 Generalized Barycentric Coordinates Varese 18 October 2012
14 Barycentric coordinates for triangles normalized barycentric coordinates properties partition of unity reproduction o positivity Lagrange g property p application linear interpolation of data 13 Generalized Barycentric Coordinates Varese 18 October 2012
15 Arbitrary polygons barycentric coordinates normalized coordinates properties partition of unity linear precision reproduction for all 14 Generalized Barycentric Coordinates Varese 18 October 2012
![& Kós 2006] positivity](/docs-images/72/66457716/images/16-4.jpg "Lagrange property linear")
16 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 15 Generalized Barycentric Coordinates Varese 18 October 2012
17 Examples Wachspress (WP) coordinates mean value (MV) coordinates discrete harmonic (DH) coordinates 16 Generalized Barycentric Coordinates Varese 18 October 2012
18 Normal form [Floater, H. & Kós 2006] 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 17 Generalized Barycentric Coordinates Varese 18 October 2012
19 Three-point coordinates Theorem: if and only if is positive monotonic convex sub-linear examples WP coordinates MV coordinates DH coordinates 18 Generalized Barycentric Coordinates Varese 18 October 2012
20 Non-convex polygons Wachspress mean value discrete harmonic poles, if, because 19 Generalized Barycentric Coordinates Varese 18 October 2012
21 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 20 Generalized Barycentric Coordinates Varese 18 October 2012
22 Theorem: MV coordinates have no poles in Mean value coordinates [H. & Floater 2006] 21 Generalized Barycentric Coordinates Varese 18 October 2012
23 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 22 Generalized Barycentric Coordinates Varese 18 October 2012
24 Implementation Mean Value coordinates 23 Generalized Barycentric Coordinates Varese 18 October 2012
25 Implementation efficient and robust evaluation of the function 24 Generalized Barycentric Coordinates Varese 18 October 2012
26 Colour interpolation 25 Generalized Barycentric Coordinates Varese 18 October 2012
27 Vector fields 26 Generalized Barycentric Coordinates Varese 18 October 2012
28 Smooth shading 27 Generalized Barycentric Coordinates Varese 18 October 2012
29 Rendering of quadrilateral elements 28 Generalized Barycentric Coordinates Varese 18 October 2012
30 Transfinite interpolation mean value coordinates radial basis functions 29 Generalized Barycentric Coordinates Varese 18 October 2012
31 Mesh animation 30 Generalized Barycentric Coordinates Varese 18 October 2012
32 Image warping original image mask warped image 31 Generalized Barycentric Coordinates Varese 18 October 2012
33 Mesh warping MV coordinates in 3D negative inside the domain MVC PMVC harmonic coordinates 32 Generalized Barycentric Coordinates Varese 18 October 2012
Generalized Barycentric Coordinates
Generalized Barycentric Coordinates Kai Hormann Faculty of Informatics University of Lugano Cartesian coordinates y 3 2 ( 3,1) 1 3 2 1 1 2 3 (2,2) (0,0) 1 2 3 (1, 2) x René Descartes (1596 1650) Appendix
Generalized Barycentric Coordinates
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