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Fundamentals of Spherical Parameterization for 3D Meshes
 PROCEEDINGS OF THE 2006 SYMPOSIUM ON INTERACTIVE 3D GRAPHICS AND GAMES, MARCH 1417, 2006
, 2003
"... Parametrization of 3D mesh data is important for many graphics applications, in particular for texture mapping, remeshing and morphing. Closed manifold genus0 meshes are topologically equivalent to a sphere, hence this is the natural parameter domain for them. Parametrizing a triangle mesh onto the ..."
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Cited by 110 (26 self)
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Parametrization of 3D mesh data is important for many graphics applications, in particular for texture mapping, remeshing and morphing. Closed manifold genus0 meshes are topologically equivalent to a sphere, hence this is the natural parameter domain for them. Parametrizing a triangle mesh onto the sphere means assigning a 3D position on the unit sphere to each of the mesh vertices, such that the spherical triangles induced by the mesh connectivity do not overlap. Satisfying the nonoverlapping requirement is the most difficult and critical component of this process. We present a generalization of the method of barycentric coordinates for planar parametrization which solves the spherical parametrization problem, prove its correctness by establishing a connection to spectral graph theory and describe efficient numerical methods for computing these parametrizations.
A Geometric Construction of Coordinates for Convex Polyhedra using Polar Duals
, 2005
"... A fundamental problem in geometry processing is that of expressing a point inside a convex polyhedron as a combination of the vertices of the polyhedron. Instances of this problem arise often in mesh parameterization and 3D deformation. A related problem is to express a vector lying in a convex cone ..."
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Cited by 20 (2 self)
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A fundamental problem in geometry processing is that of expressing a point inside a convex polyhedron as a combination of the vertices of the polyhedron. Instances of this problem arise often in mesh parameterization and 3D deformation. A related problem is to express a vector lying in a convex cone as a nonnegative combination of edge rays of this cone. This problem also arises in many applications such as planar graph embedding and spherical parameterization. In this paper, we present a unified geometric construction for building these weighted combinations using the notion of polar duals. We show that our method yields a simple geometric construction for Wachspress’s barycentric coordinates, as well as for constructing Colin de Verdière matrices from convex polyhedra—a critical step in Lovasz’s method with applications to parameterizations.
The Stable Set Problem and the LiftandProject Ranks of Graphs
, 2002
"... We study the liftandproject procedures for solving combinatorial optimization problems, as described by Lovasz and Schrijver, in the context of the stable set problem on graphs. We investigate how the procedures' performances change as we apply fundamental graph operations. We show that the o ..."
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Cited by 5 (2 self)
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We study the liftandproject procedures for solving combinatorial optimization problems, as described by Lovasz and Schrijver, in the context of the stable set problem on graphs. We investigate how the procedures' performances change as we apply fundamental graph operations. We show that the odd subdivision of an edge and the subdivision of a star operations (as well as their common generalization, the stretching of a vertex operation) cannot decrease the N 0 , N , or N+ rank of the graph. We also provide graph classes (which contain the complete graphs) where these operations do not increase the N 0  or the N  rank. Hence we obtain the ranks for these graphs, and we also present some graphminor like characterizations for them. Despite these properties we give examples showing that in general most of these operations can increase these ranks. Finally, we provide improved bounds for N+ ranks of graphs in terms of the number of nodes in the graph and prove that the subdivision of an edge or cloning a vertex can increase the N+ rank of a graph.
On Traces of dstresses in the Skeletons of Lower Dimensions of Piecewiselinear dmanifolds
, 2001
"... We show how a dstress on a piecewiselinear realization of an oriented (nonsimplicial, in general) dmanifold in R d naturally induces stresses of lower dimensions on this manifold, and discuss implications of this construction to the analysis of selfstresses in spatial frameworks. The mappings w ..."
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We show how a dstress on a piecewiselinear realization of an oriented (nonsimplicial, in general) dmanifold in R d naturally induces stresses of lower dimensions on this manifold, and discuss implications of this construction to the analysis of selfstresses in spatial frameworks. The mappings we construct are not linear, but polynomial. In the 1860–70s J. C. Maxwell described an interesting relationship between selfstresses in planar frameworks and vertical projections of polyhedral 2surfaces. We offer a spatial analog of Maxwell’s correspondence based on our polynomial mappings. By applying our main result we derive a class of threedimensional spider webs similar to the twodimensional spider webs described by Maxwell. We also conjecture an important property of our mappings that is Author: based on the lower bound theorem (g2(d +1) = dim Stress2 ≥ 0) for dpseudomanifolds generically realized in R d+1 [12].
The Colin de Verdière number and graphs of polytopes
, 2008
"... The Colin de Verdière number µ(G) of a graph G is the maximum corank of a Colin de Verdière matrix for G (that is, of a Schrödinger operator on G with a single negative eigenvalue). In 2001, Lovász gave a construction that associated to every convex 3polytope a Colin de Verdière matrix of corank 3 ..."
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The Colin de Verdière number µ(G) of a graph G is the maximum corank of a Colin de Verdière matrix for G (that is, of a Schrödinger operator on G with a single negative eigenvalue). In 2001, Lovász gave a construction that associated to every convex 3polytope a Colin de Verdière matrix of corank 3 for its 1skeleton. We generalize the Lovász construction to higher dimensions by interpreting it as minus the Hessian matrix of the volume of the polar dual. As a corollary, µ(G) ≥ d if G is the 1skeleton of a convex dpolytope. Determination of the signature of the Hessian of the volume is based on the second Minkowski inequality for mixed volumes and on Bol’s condition for equality.
Spectral Graph Theory Lecture 26 Planar Graphs 2, the Colin de Verdière Number
, 2009
"... In this lecture, I will introduce the Colin de Verdière number of a graph, and sketch the proof that it is three for planar graphs. Along the way, I will recall two important facts about planar graphs: 1. Threeconnected planar graphs are the skeletons of threedimensional convex polytopes. 2. Plana ..."
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In this lecture, I will introduce the Colin de Verdière number of a graph, and sketch the proof that it is three for planar graphs. Along the way, I will recall two important facts about planar graphs: 1. Threeconnected planar graphs are the skeletons of threedimensional convex polytopes. 2. Planar graphs are the graphs that do not have K5 or K3,3 minors.
Realizing Planar Graphs as Convex Polytopes
"... This is a survey on methods to construct a threedimensional convex polytope with a given combinatorial structure, that is, with the edges forming a given 3connected planar graph, focusing on efforts to achieve small integer coordinates. ..."
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This is a survey on methods to construct a threedimensional convex polytope with a given combinatorial structure, that is, with the edges forming a given 3connected planar graph, focusing on efforts to achieve small integer coordinates.
Stress Matrices and M Matrices
"... In [2] a connection is made between what are called “M ” matrices, as used in Colin de Verdière’s theory of graph invariants, and stress matrices as used in rigidity theory in [1]. Following a description of stress matrices and their properties relevant to rigidity theory, it is shown how a theorem ..."
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In [2] a connection is made between what are called “M ” matrices, as used in Colin de Verdière’s theory of graph invariants, and stress matrices as used in rigidity theory in [1]. Following a description of stress matrices and their properties relevant to rigidity theory, it is shown how a theorem of László Lovász [2], using
Technion
"... Gotsman et al. (SIGGRAPH 2003) presented the first method to generate a provably bijective parameterization of a closed genus0 manifold mesh to the unit sphere. This involves the solution of a large system of nonlinear equations. However, they did not show how to solve these equations efficiently, ..."
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Gotsman et al. (SIGGRAPH 2003) presented the first method to generate a provably bijective parameterization of a closed genus0 manifold mesh to the unit sphere. This involves the solution of a large system of nonlinear equations. However, they did not show how to solve these equations efficiently, so, while theoretically sound, the method has remained impractical till now. We show why simple iterative methods to solve the equations are bound to fail, and provide an efficient numerical scheme that succeeds. Our method uses a number of optimization methods combined with an algebraic multigrid technique. With these, we are able to spherically parameterize meshes containing up to a hundred thousand vertices in a matter of minutes. 1.