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Discrete Laplace operators: No free lunch
, 2007
"... Discrete Laplace operators are ubiquitous in applications spanning geometric modeling to simulation. For robustness and efficiency, many applications require discrete operators that retain key structural properties inherent to the continuous setting. Building on the smooth setting, we present a set ..."
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Cited by 28 (0 self)
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Discrete Laplace operators are ubiquitous in applications spanning geometric modeling to simulation. For robustness and efficiency, many applications require discrete operators that retain key structural properties inherent to the continuous setting. Building on the smooth setting, we present a set of natural properties for discrete Laplace operators for triangular surface meshes. We prove an important theoretical limitation: discrete Laplacians cannot satisfy all natural properties; retroactively, this explains the diversity of existing discrete Laplace operators. Finally, we present a family of operators that includes and extends wellknown and widelyused operators.
Infinitesimally rigid polyhedra. I. Statics of frameworks
 Transactionsofthe American Mathematical Society
, 1984
"... Abstract. From the time of Cauchy, mathematicians have studied the motions of convex polyhedra, with the faces held rigid while changes are allowed in the dihedral angles. In the 1940s Alexandrov proved that, even with additional vertices along the natural edges, and with an arbitrary triangulation ..."
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Cited by 23 (2 self)
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Abstract. From the time of Cauchy, mathematicians have studied the motions of convex polyhedra, with the faces held rigid while changes are allowed in the dihedral angles. In the 1940s Alexandrov proved that, even with additional vertices along the natural edges, and with an arbitrary triangulation of the natural faces on these vertices, such polyhedra are infinitesimally rigid. In this paper the dual (and equivalent) concept of static rigidity for frameworks is used to describe the behavior of bar and joint frameworks built around convex (and other) polyhedra. The static techniques introduced provide a new simplified proof of Alexandrov's theorem, as well as an essential extension which characterizes the static properties of frameworks built with more general patterns on the faces, including frameworks with vertices interior to the faces. The static techniques are presented and employed in a pattern appropriate to the extension of an arbitrary statically rigid framework built around any polyhedron (nonconvex, toroidal, etc.). The techniques are also applied to derive the static rigidity of tensegrity frameworks (with cables and struts in place of bars), and the static rigidity of frameworks projectively equivalent to known polyhedral frameworks. Finally, as an exercise to give an additional perspective to the results in 3space, detailed analogues of Alexandrov's theorem are presented for convex 4polytopes built as bar and joint frameworks in 4space. 1. Introduction. Over
PseudoTriangulations  a Survey
 CONTEMPORARY MATHEMATICS
"... A pseudotriangle is a simple polygon with exactly three convex vertices, and a pseudotriangulation is a facetoface tiling of a planar region into pseudotriangles. Pseudotriangulations appear as data structures in computational geometry, as planar barandjoint frameworks in rigidity theory an ..."
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Cited by 13 (4 self)
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A pseudotriangle is a simple polygon with exactly three convex vertices, and a pseudotriangulation is a facetoface tiling of a planar region into pseudotriangles. Pseudotriangulations appear as data structures in computational geometry, as planar barandjoint frameworks in rigidity theory and as projections of locally convex surfaces. This survey of current literature includes combinatorial properties and counting of special classes, rigidity theoretical results, representations as polytopes, straightline drawings from abstract versions called combinatorial pseudotriangulations, algorithms and applications of pseudotriangulations.
Overcoming Superstrictness in Line Drawing Interpretation
 IEEE Trans. on ?attern Analysis and Machine Intelli#ence
, 2002
"... AbstractÐThis paper presents a new algorithm for correcting incorrect line drawingsÐincorrect projections of a polyhedral scene. Such incorrect drawings arise, e.g., when an image of a polyhedral world is taken, the edges and vertices are extracted, and a drawing is synthesized. Along the way, the t ..."
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Cited by 9 (3 self)
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AbstractÐThis paper presents a new algorithm for correcting incorrect line drawingsÐincorrect projections of a polyhedral scene. Such incorrect drawings arise, e.g., when an image of a polyhedral world is taken, the edges and vertices are extracted, and a drawing is synthesized. Along the way, the true positions of the vertices in the 2D projection are perturbed due to digitization errors and the preprocessing. As most available algorithms for interpreting line drawings are ªsuperstrict,º theyjudge these noisyinputs as incorrect and fail to reconstruct a threedimensional scene from them. The presented method overcomes this problem bymoving the positions of all vertices until a veryclose correct drawing is found. The closeness criterion is to minimize the sum of squared distances from each vertex in the input drawing to its corrected position. With this tool, anysuperstrict method for line drawing interpretation is now practical, as it can be applied to the corrected version of the input drawing. Index TermsÐLine drawing interpretation, superstrictness, scene understanding, correction algorithms. æ 1
Analysing Spatial Realizability of Line Drawings Through . . .
, 1998
"... This work proves that the realizability of a line drawing without occluding segments can be verfred by checking the concurrence of groups of three lines to a single point. These lines are either those supporting segments in the drawing or new ones added during the test itself. Although this result w ..."
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Cited by 5 (4 self)
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This work proves that the realizability of a line drawing without occluding segments can be verfred by checking the concurrence of groups of three lines to a single point. These lines are either those supporting segments in the drawing or new ones added during the test itself. Although this result was essentially
Criteria for balance in abelian gain graphs, with an application to piecewiselinear geometry
 arXiv.org math.CO/0210052. MR 2006f:05086. Zbl. 1074.05047. Thomas Zaslavsky
"... A gain graph (Γ, g, G) is a graph Γ = (V, E) together with a group G, the gain group, and ..."
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Cited by 4 (2 self)
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A gain graph (Γ, g, G) is a graph Γ = (V, E) together with a group G, the gain group, and
Merging Multiple Formations: A MetaFormation Prospective
"... Abstract — This paper considers the problem of merging of more than two (minimally) rigid formations which do not have any common agent to obtain a single (minimally) rigid formation in ℜ 2 and ℜ 3. Following previously developed strategies for sequential merging of two rigid formations, a new set o ..."
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Cited by 3 (3 self)
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Abstract — This paper considers the problem of merging of more than two (minimally) rigid formations which do not have any common agent to obtain a single (minimally) rigid formation in ℜ 2 and ℜ 3. Following previously developed strategies for sequential merging of two rigid formations, a new set of enhanced merging operations is developed. They can be performed in a formalized metaformation framework, where the individual rigid formations are considered as metavertices and they can be merged into a metaformation. These operations for growing metaformations offer a level of control to the merging quality and optimality, in the sense of minimizing the number of metaedges (that is, edges between different metavertices) required. It is also proved that all minimally rigid metaformations in ℜ 2 can be obtained by successively merging two or more metavertices using the proposed set of metaoperations. I.
A Quantitative Steinitz' Theorem
, 1994
"... Any 3dimensional convex polytope with n vertices can be realized in Euclidean 3space with all coordinates of all vertices being integers of absolute value not exceeding n^169n³. ..."
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Cited by 3 (0 self)
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Any 3dimensional convex polytope with n vertices can be realized in Euclidean 3space with all coordinates of all vertices being integers of absolute value not exceeding n^169n³.
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.