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14
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 105 (25 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.
Optimal Decision Trees
 R.P.I. Math Report No. 214, Rensselaer Polytechnic Institute
, 1996
"... We propose an Extreme Point Tabu Search (EPTS) algorithm that constructs globally optimal decision trees for classification problems. Typically, decision tree algorithms are greedy. They optimize the misclassification error of each decision sequentially. Our nongreedy approach minimizes the misclas ..."
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Cited by 5 (1 self)
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We propose an Extreme Point Tabu Search (EPTS) algorithm that constructs globally optimal decision trees for classification problems. Typically, decision tree algorithms are greedy. They optimize the misclassification error of each decision sequentially. Our nongreedy approach minimizes the misclassification error of all the decisions in the tree concurrently. Using Global Tree Optimization (GTO), we can optimize existing decision trees. This capability can be used in classification and data mining applications to avoid overfitting, transfer knowledge, incorporate domain knowledge, and maintain existing decision trees. Our method works by fixing the structure of the decision tree and then representing it as a set of disjunctive linear inequalities. An optimization problem is constructed that minimizes the errors within the disjunctive linear inequalities. To reduce the misclassification error, a nonlinear error function is minimized over a polyhedral region. We show that it is suffici...
Drawing 3polytopes with good vertex resolution
 In GD’09, Proc. 17th International Symposium on Graph Drawing, 2009, Lecture Notes in Computer Science
, 2010
"... Abstract. We study the problem how to obtain a small drawing of a 3polytope with Euclidean distance between any two points at least 1. The problem can be reduced to a onedimensional problem, since it is sufficient to guarantee distinct integer xcoordinates. We develop an algorithm that yields an ..."
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Cited by 1 (1 self)
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Abstract. We study the problem how to obtain a small drawing of a 3polytope with Euclidean distance between any two points at least 1. The problem can be reduced to a onedimensional problem, since it is sufficient to guarantee distinct integer xcoordinates. We develop an algorithm that yields an embedding with the desired property such that the polytope is contained in a 2(n−2)×2×1 box. The constructed embedding can be scaled to a grid embedding whose xcoordinates are contained in [0, 2(n − 2)]. Furthermore, the point set of the embedding has a small spread, which differs from the best possible spread only by a multiplicative constant. 1
Small grid embeddings of 3polytopes
, 2009
"... We introduce an algorithm that embeds a given 3connected planar graph as a convex 3polytope with integer coordinates. The size of the coordinates is bounded by O(2 7.55n) = O(188 n). If the graph contains a triangle we can bound the integer coordinates by O(2 4.82n). If the graph contains a quadri ..."
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Cited by 1 (0 self)
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We introduce an algorithm that embeds a given 3connected planar graph as a convex 3polytope with integer coordinates. The size of the coordinates is bounded by O(2 7.55n) = O(188 n). If the graph contains a triangle we can bound the integer coordinates by O(2 4.82n). If the graph contains a quadrilateral we can bound the integer coordinates by O(2 5.54n). The crucial part of the algorithm is to find a convex plane embedding whose edges can be weighted such the sum of the weighted edges, seen as vectors, cancel at every point. It is well known that this can be guaranteed for the interior vertices by applying a technique of Tutte. We show how to extend Tutte’s ideas to construct a plane embedding where the weighted vector sums cancel also on the vertices of the boundary face.
Convex Approximation by Spherical Patches
 23RD EUROPEAN WORKSHOP ON COMPUTATIONAL GEOMETRY
, 2007
"... Given points in convex position in three dimensions, we want to find an approximating convex surface consisting of spherical patches, such that all points are within some specified tolerance bound ɛ of the approximating surface. We describe a greedy algorithm which constructs an approximating surfac ..."
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Cited by 1 (1 self)
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Given points in convex position in three dimensions, we want to find an approximating convex surface consisting of spherical patches, such that all points are within some specified tolerance bound ɛ of the approximating surface. We describe a greedy algorithm which constructs an approximating surface whose spherical patches are associated to the faces of an inscribed polytope. We show that deciding whether an approximation with not more than a given number of spherical patches exists is NPhard.
Open Problems on Polytope Reconstruction
"... Introduction. There are several dierent ways to specify convex polytopes in three dimensions. One obvious explicit representation is a list of the vertex coordinates and connectivity information between the vertices, edges, and facets. But not all this information is necessary. For example, we can r ..."
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Introduction. There are several dierent ways to specify convex polytopes in three dimensions. One obvious explicit representation is a list of the vertex coordinates and connectivity information between the vertices, edges, and facets. But not all this information is necessary. For example, we can reconstruct the explicit representation given only the vertex coordinates; this is the wellstudied ########### problem. By projective duality, we can also reconstruct the polytope from a list of halfspaces whose intersection is the polytope. Although these are the two most wellknown ways to specify polytopes, at least in the computational geometry community, they are not the only ones. Here we list several dierent theorems describing surprisingly small sets of information that are sucient to specify # dimensional convex polytopes. The purpose of this note is to pose the algorithmic versions of these theorems as intriguing open questions. That is, is there
Characterizing Graphs of Zonohedra
, 811
"... A classic theorem by Steinitz states that a graph G is realizable by a convex polyhedron if and only if G is 3connected planar. Zonohedra are an important subclass of convex polyhedra having the property that the faces of a zonohedron are parallelograms and are in parallel pairs. In this paper we g ..."
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A classic theorem by Steinitz states that a graph G is realizable by a convex polyhedron if and only if G is 3connected planar. Zonohedra are an important subclass of convex polyhedra having the property that the faces of a zonohedron are parallelograms and are in parallel pairs. In this paper we give characterization of graphs of zonohedra. We also give a linear time algorithm to recognize such a graph. In our quest for finding the algorithm, we prove that in a zonohedron P both the number of zones and the number of faces in each zone is O ( √ n), where n is the number of vertices of P.
Cauchy’s Theorem and Edge Lengths of Convex Polyhedra
, 2007
"... In this paper we explore, from an algorithmic point of view, the extent to which the facial angles and combinatorial structure of a convex polyhedron determine the polyhedron—in particular the edge lengths and dihedral angles of the polyhedron. Cauchy’s rigidity theorem of 1813 states that the dihe ..."
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In this paper we explore, from an algorithmic point of view, the extent to which the facial angles and combinatorial structure of a convex polyhedron determine the polyhedron—in particular the edge lengths and dihedral angles of the polyhedron. Cauchy’s rigidity theorem of 1813 states that the dihedral angles are uniquely determined. Finding them is a significant algorithmic problem which we express as a spherical graph drawing problem. Our main result is that the edge lengths, although not uniquely determined, can be found via linear programming. We make use of significant mathematics on convex polyhedra by Stoker, Van Heijenoort, Gale, and Shepherd.
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.