Barycentric Coordinates
Welcome	to the Barycentric Coordinates homepage! This website follows our minisymposia given at the 10th SIAM Conference on Geometric Design & Computing and the 2011 SIAM Conference on Geometric & Physical Modeling. It provides links to online resources, including slides and papers.
Contributors	Mirela Ben-Chen, Stanford University Solveig Bruvoll, University of Oslo Michael S. Floater, University of Oslo Andrew Gillette, University of California, San Diego Kai Hormann, University of Lugano Tao Ju, Washington University Jiří Kosinka, University of Cambridge Raif Rustamov, Drew University Scott Schaefer, Texas A&M University N. Sukumar, UC Davis
Slides	Minisymposium on Barycentric Coordinates and Transfinite Interpolation Lecture 1: Generalized Barycentric Coordinates by Kai Hormann Lecture 2: Barycentric Coordinates for Closed Curves by Scott Schaefer Lecture 3: Hermite Mean Value Interpolation by Michael S. Floater Lecture 4: A General, Geometric Construction of Coordinates in any Dimensions by Tao Ju Lecture 5: Transfinite Mean Value Interpolation over Volumetric Domains by Solveig Bruvoll Lecture 6: Barycentric Finite Element Methods by N. Sukumar Minisymposium on Theory and Applications of Barycentric Coordinates Lecture 1: Constructing Barycentric Coordinates on Surfaces by Raif Rustamov Lecture 2: Injective Barycentric Mappings on Convex Domains by Jiří Kosinka Lecture 3: Geometric Criteria for Generalized Barycentric Finite Elements by Andrew Gillette Lecture 4: Variational Space Deformations with Barycentric Coordinates by Mirela Ben-Chen Keynote Lecture on Generalized Barycentric Coordinates by Kai Hormann
Papers	before 1900 Der barycentrische Calcul August F. Möbius, 1827 before 2000 Continuity and convexity of projections and barycentric coordinates in convex polyhedra John A. Kalman, 1961 A rational basis for function approximation Eugene L. Wachspress, 1971 A rational basis for function approximation. II: Curved sides Eugene L. Wachspress, 1973 Pseudo-harmonic interpolation on convex domains William J. Gordon and James A. Wixom, 1974 A Rational Finite Element Basis Eugene L. Wachspress, 1975 Rational basis functions for curved elements Eugene L. Wachspress, 1979 High-order curved finite elements Eugene L. Wachspress, 1981 A pentagonal surface patch for computer aided geometric design Peter Charrot and John A. Gregory, 1984 A multisided generalization of Bézier surfaces Charles T. Loop and Tony D. DeRose, 1989 Computing discrete minimal surfaces and their conjugates Ulrich Pinkall and Konrad Polthier, 1993 Multiresolution analysis of arbitrary meshes Matthias Eck, Tony DeRose, Tom Duchamp, Hugues Hoppe, Michael Lounsbery, and Werner Stuetzle, 1995 Barycentric coordinates for convex polytopes Joe Warren, 1996 Parametrization and smooth approximation of surface triangulations Michael S. Floater, 1997 Reflection space image based rendering Brian Cabral, Marc Olano, and Philip Nemec, 1999 2002 Generalized barycentric coordinates on irregular polygons Mark Meyer, Haeyoung Lee, Alan Barr, and Mathieu Desbrun 2003 Interpolants within convex polygons: Wachspress' shape functions Gautam Dasgupta Integration within polygonal finite elements Gautam Dasgupta Mean value coordinates Michael S. Floater On the uniqueness of barycentric coordinates Joe Warren 2004 A quadrilateral rendering primitive Kai Hormann and Marco Tarini Interpolations for temperature distributions: A method for all non-concave polygons Elisabeth A. Malsch and Gautam Dasgupta Shape functions for polygonal domains with interior nodes Elisabeth A. Malsch and Gautam Dasgupta Construction of polygonal interpolants: A maximum entropy approach N. Sukumar Conforming polygonal finite elements N. Sukumar and Alireza Tabarraei 2005 Mean value coordinates in 3D Michael S. Floater, Géza Kós, and Martin Reimers Sharp error estimates for interpolatory approximation on convex polytopes Allal Guessab and Gerhard Schmeisser Algebraic construction of smooth interpolants on polygonal domains Elisabeth A. Malsch and Gautam Dasgupta Smooth two dimensional interpolants: A recipe for all polygons Elisabeth A. Malsch, John J. Lin, and Gautam Dasgupta Mean value coordinates for closed triangular meshes Tao Ju, Scott Schaefer, and Joe Warren Geometric construction of coordinates for convex polyhedra using polar duals Tao Ju, Scott Schaefer, Joe Warren, and Mathieu Desbrun 2006 Local maximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods. Marino Arroyo and Michael Ortiz On transfinite barycentric coordinates Alexander Belyaev A general construction of barycentric coordinates over convex polygons Michael S. Floater, Kai Hormann, and Géza Kós Mean value coordinates for arbitrary planar polygons Kai Hormann and Michael S. Floater Spherical barycentric coordinates Torsten Langer, Alexander Belyaev, and Hans-Peter Seidel Recent advances in the construction of polygonal finite element interpolants N. Sukumar and Elisabeth A. Malsch 2007 Barycentric rational interpolation with no poles and high rates of approximation Michael S. Floater and Kai Hormann Harmonic coordinates for character articulation Pushkar Joshi, Mark Meyer, Tony DeRose, Brian Green, and Tom Sanocki A general geometric construction of coordinates in a convex simplicial polytope Tao Ju, Peter Liepa, and Joe Warren Mean value coordinates for arbitrary spherical polygons and polyhedra in R³ Torsten Langer, Alexander Belyaev, and Hans-Peter Seidel Mean value Bézier surfaces Torsten Langer and Hans-Peter Seidel GPU-assisted positive mean value coordinates for mesh deformations Yaron Lipman, Johannes Kopf, Daniel Cohen-Or, and David Levin A unified, integral construction for coordinates over closed curves Scott Schaefer, Tao Ju, and Joe Warren Deriving the continuity of maximum-entropy basis functions via variational analysis N. Sukumar and Roger J-B Wets Overview and construction of meshfree basis functions: from moving least squares to entropy approximants N. Sukumar and Roy W. Wright Discrete Laplace operators: No free lunch Max Wardetzky, Saurabh Mathur, Felix Kälberer, and Eitan Grinspun Barycentric coordinates for convex sets Joe Warren, Scott Schaefer, Anil N. Hirani, and Mathieu Desbrun 2008 The adjoint for an algebraic finite element Gautam Dasgupta and Eugene L. Wachspress Basis functions for concave polygons Gautam Dasgupta and Eugene L. Wachspress Pointwise radial minimization: Hermite interpolation on arbitrary domains Michael S. Floater and Christian Schulz Maximum entropy coordinates for arbitrary polytopes Kai Hormann and N. Sukumar Reusable skinning templates using cage-based deformations Tao Ju, Qian-Yi Zhou, Michiel van de Panne, Daniel Cohen-Or, and Ulrich Neumann Higher order barycentric coordinates Torsten Langer and Hans-Peter Seidel Mean value Bézier maps Torsten Langer and Hans-Peter Seidel Green coordinates Yaron Lipman, David Levin, and Daniel Cohen-Or Polyhedral finite elements using harmonic basis functions Sebastian Martin, Peter Kaufmann, Mario Botsch, Martin Wicke, and Markus Gross Boundary element formulation of harmonic coordinates Raif M. Rustamov Barycentric coordinates computation in homogeneous coordinates Vaclav Skala Extended finite element method on polygonal and quadtree meshes Alireza Tabarraei and N. Sukumar Voronoi-based interpolants for fracture modelling N. Sukumar and John E. Bolander 2009 Variational harmonic maps for space deformation Mirela Ben-Chen, Ofir Weber, and Craig Gotsman Spatial deformation transfer Mirela Ben-Chen, Ofir Weber, and Craig Gotsman Transfinite mean value interpolation in general dimension Solveig Bruvoll and Michael S. Floater Transfinite mean value interpolation Christopher Dyken and Michael S. Floater Coordinates for instant image cloning Zeev Farbman, Gil Hoffer, Yaron Lipman, Daniel Cohen-Or, and Dani Lischinski On the injectivity of Wachspress and mean value mappings between convex polygons Michael S. Floater and Jiří Kosinka Poisson-based weight reduction of animated meshes Eric Landreneau and Scott Schaefer Interior distance using barycentric coordinates Raif Rustamov, Yaron Lipman, and Thomas Funkhouser Complex barycentric coordinates with applications to planar shape deformation Ofir Weber, Mirela Ben-Chen, and Craig Gotsman 2010 Barycentric interpolation and mappings on smooth convex domains Michael S. Floater and Jiří Kosinka Cage-free local deformations using Green coordinates Zheng Li, David Levin, Zhengjie Deng, Dingyuan Liu, and Xiaonan Luo Derivation and analysis of Green coordinates Yaron Lipman and David Levin Moving least squares coordinates Josiah Manson and Scott Schaefer Barycentric coordinates on surfaces Raif Rustamov A versatile framework for shape description Raif Rustamov Conversion of performance mesh animation into cage-based animation Yann Savoye and Jean-Sébastien Franco Cage-based tracking for performance animation Yann Savoye and Jean-Sébastien Franco CageIK: dual-Laplacian cage-based inverse kinematics Yann Savoye and Jean-Sébastien Franco Volumetric modeling with diffusion surfaces Kenshi Takayama, Olga Sorkine, Andrew Nealen, and Takeo Igarashi Controllable conformal maps for shape deformation and interpolation Ofir Weber and Craig Gotsman 2011 Discrete Laplacians on general polygonal meshes Marc Alexa and Max Wardetzky Error estimates for generalized barycentric interpolation Andrew Gillette, Alexander Rand, and Chandrajit Bajaj Bounded biharmonic weights for real-time deformation Alec Jacobson, Ilya Baran, Jovan Popović, and Olga Sorkine Positive Gordon–Wixom coordinates Josiah Manson, Kuiyu Li, and Scott Schaefer HOT: Hodge-optimized triangulations Patrick Mullen, Pooran Memari, Fernando de Goes, and Mathieu Desbrun Quadratic serendipity finite elements on polygons using generalized barycentric coordinates Alexander Rand, Andrew Gillette, and Chandrajit Bajaj Interpolated eigenfunctions for volumetric shape processing Raif Rustamov Video-based toon character from surface performance capture Yann Savoye Geometric computation, duality and projective space Vaclav Skala Jacobians and Hessians of mean value coordinates for closed triangular meshes Jean-Marc Thiery, Julien Tierny, and Tamy Boubekeur Affine generalised barycentric coordinates Shayne Waldron A complex view of barycentric mappings Ofir Weber, Mirela Ben-Chen, Craig Gotsman, and Kai Hormann 2012 Direct and converse results for generalized multivariate Jensen-type inequalities Allal Guessab A multivariate extension of an inequality of Brenner–Alzer Allal Guessab, Otheman Nouisser, and Josip Pečarić 2D shape manipulations with holomorphic coordinates Zheng Li and Chengming Liu PolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab Cameron Talischi, Glaucio H. Paulino, Anderson Pereira, and Ivan F. M. Menezes PolyTop: a Matlab implementation of a general topology optimization framework using unstructured polygonal finite element meshes Cameron Talischi, Glaucio H. Paulino, Anderson Pereira, and Ivan F. M. Menezes 2013 Signed Lp-distance fields Alexander Belyaev, Pierre-Alain Fayolle, and Alexander Pasko Generalized barycentric coordinates and Jensen type inequalities on convex polytopes Allal Guessab Generalized barycentric coordinates and approximations of convex functions on arbitrary convex polytopes Allal Guessab Poisson coordinates Xian-Ying Li and Shi-Min Hu Cubic mean value coordinates Xian-Ying Li, Tao Ju, and Shi-Min Hu Interpolation error estimates for mean value coordinates Alexander Rand, Andrew Gillette, and Chandrajit Bajaj Bijective composite mean value mappings Teseo Schneider, Kai Hormann, and Michael S. Floater 2014 Wachspress and mean value coordinates Michael S. Floater Gradient bounds for Wachspress coordinates on polytopes Michael S. Floater, Andrew Gillette, and N. Sukumar Local barycentric coordinates Juyong Zhang, Balin Deng, Zishun Liu, Giuseppe Patanè, Sofien Bouaziz, Kai Hormann, and Ligang Liu Geodesic based conformal mesh parameterization Qiang Zou, Jibin Zhao, Yanguo Zhao, Guangbao Liu, and Haiyang Jin 2015 On transfinite Gordon–Wixom interpolation schemes and their extensions Alexander Belyaev and Pierre-Alain Fayolle On variational and PDE-based distance function approximations Alexander Belyaev and Pierre-Alain Fayolle Generalized barycentric coordinates and applications Michael S. Floater Optimality of gradient bound for polyhedral Wachspress coordinates Michael S. Floater Convergence of Wachspress coordinates: from polygons to curved domains Jiří Kosinka and Michael Bartoň 2016 Subdividing barycentric coordinates Dmitry Anisimov, Chongyang Deng, and Kai Hormann Wasserstein barycentric coordinates: histogram regression using optimal transport Nicolas Bonneel, Gabriel Peyré, and Marco Cuturi Power coordinates: a geometric construction of barycentric coordinates on convex polytopes Max Budninskiy, Beibei Liu, Yiying Tong, and Mathieu Desbrun Complex transfinite barycentric mappings with similarity kernels Renjie Chen and Craig Gotsman On pseudo-harmonic barycentric coordinates Renjie Chen and Craig Gotsman The finite cell method for polygonal meshes: poly-FCM Sascha Duczek and Ulrich Gabbert On the monotonicity of generalized barycentric coordinates on convex polygons Michael S. Floater Convergence of barycentric coordinates to barycentric kernels Jiří Kosinka and Michael Bartoň Rational Bases and Generalized Barycentrics Eugene Wachspress 2017 Blended barycentric coordinates Dmitry Anisimov, Daniele Panozzo, and Kai Hormann The monotonicity of a family of barycentric coordinates for quadrilaterals Chongyang Deng and Feifan Shi Discretizing Wachspress coordinates is safe Kai Hormann and Jiří Kosinka Generalized Barycentric Coordinates in Computer Graphics and Computational Mechanics Kai Hormann and N. Sukumar (editors) Quadratic serendipity finite elements over convex polyhedra A. Sinu, Sundararajan Natarajan, and K. Shankar Parametric finite elements with bijective mappings Patrick Zulian, Teseo Schneider, Kai Hormann, and Rolf Krause 2018 Hermite mean value interpolation on polygons Rick Beatson, Michael S. Floater, and Carl Emil Kåshagen Path planning with divergence-based distance functions Renjie Chen, Craig Gotsman, and Kai Hormann Efficient path generation with reduced coordinates Renjie Chen, Craig Gotsman, and Kai Hormann Mean value coordinates for quad cages in 3D Jean-Marc Thiery, Pooran Memari, and Tamy Boubekeur 2019 Behaviour of exponential three-point coordinates at the vertices of convex polygon Dmitry Anisimov, Kai Hormann, and Teseo Schneider New elements with harmonic shape functions in adaptive mesh refinement Mohammad Javad Kazemzadeh-Parsi A fast numerical solver for local barycentric coordinates Jiong Tao, Bailin Deng, and Juyong Zhang Positive and smooth Gordon-Wixom coordinates Zhihao Wang, Yajuan Li, Weiyin Ma, and Chongyang Deng A family of barycentric coordinates for co-dimension 1 manifolds with simplicial facets Zhipei Yan and Scott Schaefer 2020 Iterative coordinates Chongyang Deng, Qingjun Chang, and Kai Hormann 2022 On spherical barycentric coordinates Abdellatif Aitelhad Hyperbolic barycentric coordinates and applications Alaa eddine Bensad and Aziz Ikemakhen Green coordinates for triquad cages in 3D Jean-Marc Thiery and Tamy Boubekeur 2023 A general construction of spherical barycentric coordinates and applications Alaa eddine Bensad and Aziz Ikemakhen Maximum likelihood coordinates Qingjun Chang, Chongyang Deng, and Kai Hormann Somigliana coordinates: an elasticity-derived approach for cage deformation Jiong Chen, Fernando De Goes, and Mathieu Desbrun Variational barycentric coordinates Ana Dodik, Oded Stein, Vincent Sitzmann, and Justin Solomon Polynomial 2D Green coordinates for polygonal cages Élie Michel and Jean-Marc Thiery
Links	basic facts on barycentric coordinates Barycentric coordinates from Wolfram MathWorld Barycentric coordinates from Wikipedia Barycentric coordinates from Cut-the-Knot Barycentric coordinates from Ken Joy's On-Line Computer Graphics Notes other useful online resources Ceva's Theorem Routh's Theorem Van Obel's Theorem libraries CGAL package for 2D generalized barycentric coordinates MATLAB programs for polycons and starcons