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51
Surface Parameterization: a Tutorial and Survey
 In Advances in Multiresolution for Geometric Modelling, Mathematics and Visualization
, 2005
"... Summary. This paper provides a tutorial and survey of methods for parameterizing surfaces with a view to applications in geometric modelling and computer graphics. We gather various concepts from differential geometry which are relevant to surface mapping and use them to understand the strengths and ..."
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Cited by 239 (7 self)
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Summary. This paper provides a tutorial and survey of methods for parameterizing surfaces with a view to applications in geometric modelling and computer graphics. We gather various concepts from differential geometry which are relevant to surface mapping and use them to understand the strengths and weaknesses of the many methods for parameterizing piecewise linear surfaces and their relationship to one another. 1
Mean value coordinates
 COMPUTER AIDED GEOMETRIC DESIGN
, 2003
"... We derive a generalization of barycentric coordinates which allows a vertex in a planar triangulation to be expressed as a convex combination of its neighbouring vertices. The coordinates are motivated by the Mean Value Theorem for harmonic functions and can be used to simplify and improve methods ..."
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Cited by 226 (9 self)
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We derive a generalization of barycentric coordinates which allows a vertex in a planar triangulation to be expressed as a convex combination of its neighbouring vertices. The coordinates are motivated by the Mean Value Theorem for harmonic functions and can be used to simplify and improve methods for parameterization and morphing.
Generalized Barycentric Coordinates on Irregular Polygons
 Journal of Graphics Tools
, 2002
"... In this paper we present an easy computation of a generalized form of barycentric coordinates for irregular, convex nsided polygons. Triangular barycentric coordinates have had many classical applications in computer graphics, from texture mapping to raytracing. Our new equations preserve many of ..."
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Cited by 72 (5 self)
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In this paper we present an easy computation of a generalized form of barycentric coordinates for irregular, convex nsided polygons. Triangular barycentric coordinates have had many classical applications in computer graphics, from texture mapping to raytracing. Our new equations preserve many of the familiar properties of the triangular barycentric coordinates with an equally simple calculation, contrary to previous formulations. We illustrate the properties and behavior of these new generalized barycentric coordinates through several example applications.
Mean value coordinates for arbitrary planar polygons
 ACM Transactions on Graphics
, 2006
"... Barycentric coordinates for triangles are commonly used in computer graphics, geometric modelling, and other computational sciences for various purposes, because they provide a convenient way to linearly interpolate data that is given at the corners of a triangle. The concept of barycentric coordina ..."
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Cited by 66 (14 self)
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Barycentric coordinates for triangles are commonly used in computer graphics, geometric modelling, and other computational sciences for various purposes, because they provide a convenient way to linearly interpolate data that is given at the corners of a triangle. The concept of barycentric coordinates can also be extended in several ways to convex polygons with more than three vertices, but most of these constructions break down when used in the nonconvex setting. One choice that is not limited to convex configurations are the mean value coordinates and we show that they are welldefined for arbitrary planar polygons without selfintersections. Besides many other important properties, these coordinate functions are smooth and allow an efficient and robust implementation. They are particularly useful for interpolating data that is given at the vertices of the polygons and we present several examples of their application to common problems in computer graphics and geometric modelling.
Mesh Parameterization: Theory and Practice
 SIGGRAPH ASIA 2008 COURSE NOTES
, 2008
"... Mesh parameterization is a powerful geometry processing tool with numerous computer graphics applications, from texture mapping to animation transfer. This course outlines its mathematical foundations, describes recent methods for parameterizing meshes over various domains, discusses emerging tools ..."
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Cited by 54 (5 self)
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Mesh parameterization is a powerful geometry processing tool with numerous computer graphics applications, from texture mapping to animation transfer. This course outlines its mathematical foundations, describes recent methods for parameterizing meshes over various domains, discusses emerging tools like global parameterization and intersurface mapping, and demonstrates a variety of parameterization applications.
Conforming polygonal finite elements
 International Journal for Numerical Methods in Engineering
, 2004
"... In this paper, conforming finite elements on polygon meshes are developed. Polygonal finite elements provide greater flexibility in mesh generation and are bettersuited for applications in solid mechanics which involve a significant change in the topology of the material domain. In this study, rece ..."
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Cited by 44 (11 self)
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In this paper, conforming finite elements on polygon meshes are developed. Polygonal finite elements provide greater flexibility in mesh generation and are bettersuited for applications in solid mechanics which involve a significant change in the topology of the material domain. In this study, recent advances in meshfree approximations, computational geometry, and computer graphics are used to construct different trial and test approximations on polygonal elements. A particular and notable contribution is the use of meshfree (naturalneighbour, nn) basis functions on a canonical element combined with an affine map to construct conforming approximations on convex polygons. This numerical formulation enables the construction of conforming approximation on ngons (n � 3), and hence extends the potential applications of finite elements to convex polygons of arbitrary order. Numerical experiments on secondorder elliptic boundaryvalue problems are presented to demonstrate the accuracy and convergence of the proposed method. Copyright � 2004 John Wiley & Sons, Ltd. KEY WORDS: meshfree methods; natural neighbour interpolants; natural element method; Laplace interpolant; Wachspress basis functions; mean value coordinates
Barycentric coordinates for convex sets
, 2007
"... In this paper we provide an extension of barycentric coordinates from simplices to arbitrary convex sets. Barycentric coordinates over convex 2D polygons have found numerous applications in various fields as they allow smooth interpolation of data located on vertices. However, no explicit formulatio ..."
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Cited by 40 (7 self)
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In this paper we provide an extension of barycentric coordinates from simplices to arbitrary convex sets. Barycentric coordinates over convex 2D polygons have found numerous applications in various fields as they allow smooth interpolation of data located on vertices. However, no explicit formulation valid for arbitrary convex polytopes has been proposed to extend this interpolation in higher dimensions. Moreover, there has been no attempt to extend these functions into the continuous domain, where barycentric coordinates are related to Green’s functions and construct functions that satisfy a boundary value problem. First, we review the properties and construction of barycentric coordinates in the discrete domain for convex polytopes. Next, we show how these concepts extend into the continuous domain to yield barycentric coordinates for continuous functions. We then provide a proof that our functions satisfy all the desirable properties of barycentric coordinates in arbitrary dimensions. Finally, we provide an example of constructing such barycentric functions over regions bounded by parametric curves and show how they can be used to perform freeform deformations.
Polyhedral Finite Elements Using Harmonic Basis Functions
, 2008
"... Finite element simulations in computer graphics are typically based on tetrahedral or hexahedral elements, which enables simple and efficient implementations, but in turn requires complicated remeshing in case of topological changes or adaptive refinement. We propose a flexible finite element method ..."
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Cited by 28 (4 self)
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Finite element simulations in computer graphics are typically based on tetrahedral or hexahedral elements, which enables simple and efficient implementations, but in turn requires complicated remeshing in case of topological changes or adaptive refinement. We propose a flexible finite element method for arbitrary polyhedral elements, thereby effectively avoiding the need for remeshing. Our polyhedral finite elements are based on harmonic basis functions, which satisfy all necessary conditions for FEM simulations and seamlessly generalize both linear tetrahedral and trilinear hexahedral elements. We discretize harmonic basis functions using the method of fundamental solutions, which enables their flexible computation and efficient evaluation. The versatility of our approach is demonstrated on cutting and adaptive refinement within a simulation framework for corotated linear elasticity.
Maximum Entropy Coordinates for Arbitrary Polytopes
, 2008
"... Barycentric coordinates can be used to express any point inside a triangle as a unique convex combination of the triangle’s vertices, and they provide a convenient way to linearly interpolate data that is given at the vertices of a triangle. In recent years, the ideas of barycentric coordinates and ..."
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Cited by 26 (7 self)
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Barycentric coordinates can be used to express any point inside a triangle as a unique convex combination of the triangle’s vertices, and they provide a convenient way to linearly interpolate data that is given at the vertices of a triangle. In recent years, the ideas of barycentric coordinates and barycentric interpolation have been extended to arbitrary polygons in the plane and general polytopes in higher dimensions, which in turn has led to novel solutions in applications like mesh parameterization, image warping, and mesh deformation. In this paper we introduce a new generalization of barycentric coordinates that stems from the maximum entropy principle. The coordinates are guaranteed to be positive inside any planar polygon, can be evaluated efficiently by solving a convex optimization problem with Newton’s method, and experimental evidence indicates that they are smooth inside the domain. Moreover, the construction of these coordinates can be extended to arbitrary polyhedra and higherdimensional polytopes.
A Finite Element Method on Convex Polyhedra
 EUROGRAPHICS 2007 / D. COHENOR AND P. SLAVÍK (GUEST EDITORS)
, 2007
"... We present a method for animating deformable objects using a novel finite element discretization on convex polyhedra. Our finite element approach draws upon recently introduced 3D mean value coordinates to define smooth interpolants within the elements. The mathematical properties of our basis funct ..."
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Cited by 26 (4 self)
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We present a method for animating deformable objects using a novel finite element discretization on convex polyhedra. Our finite element approach draws upon recently introduced 3D mean value coordinates to define smooth interpolants within the elements. The mathematical properties of our basis functions guarantee convergence. Our method is a natural extension to linear interpolants on tetrahedra: for tetrahedral elements, the methods are identical. For fast and robust computations, we use an elasticity model based on Cauchy strain and stiffness warping. This more flexible discretization is particularly useful for simulations that involve topological changes, such as cutting or fracture. Since splitting convex elements along a plane produces convex elements, remeshing or subdivision schemes used in simulations based on tetrahedra are not necessary, leading to less elements after such operations. We propose various operators for cutting the polyhedral discretization. Our method can handle arbitrary cut trajectories, and there is no limit on how often elements can be split.