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 well-known and widely-used operators.

A pseudo-triangle is a simple polygon with exactly three convex vertices, and a pseudo-triangulation is a face-to-face tiling of a planar region into pseudo-triangles. Pseudo-triangulations appear as data structures in computational geometry, as planar bar-and-joint 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, straight-line drawings from abstract versions called combinatorial pseudo-triangulations, algorithms and applications of pseudo-triangulations.

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 three-dimensional 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. 1

Any 3-dimensional convex polytope with n vertices can be realized in Euclidean 3-space with all coordinates of all vertices being integers of absolute value not exceeding n^169n³.

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 meta-formation framework, where the individual rigid formations are considered as metavertices and they can be merged into a meta-formation. These operations for growing meta-formations offer a level of control to the merging quality and optimality, in the sense of minimizing the number of meta-edges (that is, edges between different meta-vertices) required. It is also proved that all minimally rigid meta-formations in ℜ 2 can be obtained by successively merging two or more meta-vertices using the proposed set of meta-operations. I.

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Abstract. We study the problem how to obtain a small drawing of a 3-polytope with Euclidean distance between any two points at least 1. The problem can be reduced to a one-dimensional problem, since it is sufficient to guarantee distinct integer x-coordinates. 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 x-coordinates 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

A gain graph (Γ, g, G) is a graph Γ = (V, E) together with a group G, the gain group, and

We introduce an algorithm that embeds a given 3-connected planar graph as a convex 3-polytope 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.

We address the problem of drawing a 3-connected planar graph as a convex polyhedron in R³. We give an efficient algorithm for producing such a realization using O(n) volume under the vertex-resolution rule. Each vertex in the drawing resulting from this method is guaranteed to need no more than O(n log n) bits to represent (as a pair of rational numbers). This solves an open problem of Cohen, Eades, Lin, and Ruskey. We also show that under the angularresolution rule drawing a 3-connected planar graph as a convex polyhedron in R³ requires at least exponential volume in the worst case.