Generating spherical multiquadrangulations by restricted vertex splittings and the reducibility of equilibrium classes

A quadrangulation is a graph embedded on the sphere such that each face is bounded by a walk of length 4, parallel edges allowed. All quadrangulations can be generated by a sequence of graph operations called vertex splitting, starting from the path P_2 of length 2. We define the degree D of a splitting S and consider restricted splittings S_{i,j} with i <= D <= j. It is known that S_{2,3} generate all simple quadrangulations. Here we investigate the cases S_{1,2}, S_{1,3}, S_{1,1}, S_{2,2}, S_{3,3}. First we show that the splittings S_{1,2} are exactly the monotone ones in the sense that the resulting graph contains the original as a subgraph. Then we show that they define a set of nontrivial ancestors beyond P_2 and each quadrangulation has a unique ancestor. Our results have a direct geometric interpretation in the context of mechanical equilibria of convex bodies. The topology of the equilibria corresponds to a 2-coloured quadrangulation with independent set sizes s, u. The numbers s, u identify the primary equilibrium class associated with the body by V\'arkonyi and Domokos. We show that both S_{1,1} and S_{2,2} generate all primary classes from a finite set of ancestors which is closely related to their geometric results. If, beyond s and u, the full topology of the quadrangulation is considered, we arrive at the more refined secondary equilibrium classes. As Domokos, L\'angi and Szab\'o showed recently, one can create the geometric counterparts of unrestricted splittings to generate all secondary classes. Our results show that S_{1,2} can only generate a limited range of secondary classes from the same ancestor. The geometric interpretation of the additional ancestors defined by monotone splittings shows that minimal polyhedra play a key role in this process. We also present computational results on the number of secondary classes and multiquadrangulations.Comment: 21 pages, 11 figures and 3 table

Finding Hexahedrizations for Small Quadrangulations of the Sphere

This paper tackles the challenging problem of constrained hexahedral meshing. An algorithm is introduced to build combinatorial hexahedral meshes whose boundary facets exactly match a given quadrangulation of the topological sphere. This algorithm is the first practical solution to the problem. It is able to compute small hexahedral meshes of quadrangulations for which the previously known best solutions could only be built by hand or contained thousands of hexahedra. These challenging quadrangulations include the boundaries of transition templates that are critical for the success of general hexahedral meshing algorithms. The algorithm proposed in this paper is dedicated to building combinatorial hexahedral meshes of small quadrangulations and ignores the geometrical problem. The key idea of the method is to exploit the equivalence between quad flips in the boundary and the insertion of hexahedra glued to this boundary. The tree of all sequences of flipping operations is explored, searching for a path that transforms the input quadrangulation Q into a new quadrangulation for which a hexahedral mesh is known. When a small hexahedral mesh exists, a sequence transforming Q into the boundary of a cube is found; otherwise, a set of pre-computed hexahedral meshes is used. A novel approach to deal with the large number of problem symmetries is proposed. Combined with an efficient backtracking search, it allows small shellable hexahedral meshes to be found for all even quadrangulations with up to 20 quadrangles. All 54,943 such quadrangulations were meshed using no more than 72 hexahedra. This algorithm is also used to find a construction to fill arbitrary domains, thereby proving that any ball-shaped domain bounded by n quadrangles can be meshed with no more than 78 n hexahedra. This very significantly lowers the previous upper bound of 5396 n.Comment: Accepted for SIGGRAPH 201

A topological classification of convex bodies

The shape of homogeneous, generic, smooth convex bodies as described by the Euclidean distance with nondegenerate critical points, measured from the center of mass represents a rather restricted class M_C of Morse-Smale functions on S^2. Here we show that even M_C exhibits the complexity known for general Morse-Smale functions on S^2 by exhausting all combinatorial possibilities: every 2-colored quadrangulation of the sphere is isomorphic to a suitably represented Morse-Smale complex associated with a function in M_C (and vice versa). We prove our claim by an inductive algorithm, starting from the path graph P_2 and generating convex bodies corresponding to quadrangulations with increasing number of vertices by performing each combinatorially possible vertex splitting by a convexity-preserving local manipulation of the surface. Since convex bodies carrying Morse-Smale complexes isomorphic to P_2 exist, this algorithm not only proves our claim but also generalizes the known classification scheme in [36]. Our expansion algorithm is essentially the dual procedure to the algorithm presented by Edelsbrunner et al. in [21], producing a hierarchy of increasingly coarse Morse-Smale complexes. We point out applications to pebble shapes.Comment: 25 pages, 10 figure

Contact Representations of Graphs in 3D

We study contact representations of graphs in which vertices are represented by axis-aligned polyhedra in 3D and edges are realized by non-zero area common boundaries between corresponding polyhedra. We show that for every 3-connected planar graph, there exists a simultaneous representation of the graph and its dual with 3D boxes. We give a linear-time algorithm for constructing such a representation. This result extends the existing primal-dual contact representations of planar graphs in 2D using circles and triangles. While contact graphs in 2D directly correspond to planar graphs, we next study representations of non-planar graphs in 3D. In particular we consider representations of optimal 1-planar graphs. A graph is 1-planar if there exists a drawing in the plane where each edge is crossed at most once, and an optimal n-vertex 1-planar graph has the maximum (4n - 8) number of edges. We describe a linear-time algorithm for representing optimal 1-planar graphs without separating 4-cycles with 3D boxes. However, not every optimal 1-planar graph admits a representation with boxes. Hence, we consider contact representations with the next simplest axis-aligned 3D object, L-shaped polyhedra. We provide a quadratic-time algorithm for representing optimal 1-planar graph with L-shaped polyhedra

Loop models on random maps via nested loops: case of domain symmetry breaking and application to the Potts model

We use the nested loop approach to investigate loop models on random planar maps where the domains delimited by the loops are given two alternating colors, which can be assigned different local weights, hence allowing for an explicit Z_2 domain symmetry breaking. Each loop receives a non local weight n, as well as a local bending energy which controls loop turns. By a standard cluster construction that we review, the Q = n^2 Potts model on general random maps is mapped to a particular instance of this problem with domain-non-symmetric weights. We derive in full generality a set of coupled functional relations for a pair of generating series which encode the enumeration of loop configurations on maps with a boundary of a given color, and solve it by extending well-known complex analytic techniques. In the case where loops are fully-packed, we analyze in details the phase diagram of the model and derive exact equations for the position of its non-generic critical points. In particular, we underline that the critical Potts model on general random maps is not self-dual whenever Q \neq 1. In a model with domain-symmetric weights, we also show the possibility of a spontaneous domain symmetry breaking driven by the bending energy.Comment: 44 pages, 13 figure

Global parametrization of range image sets

We present a method to globally parameterize a surface represented by height maps over a set of planes (range images). In contrast to other parametrization techniques, we do not start with a manifold mesh. The parametrization we compute defines a manifold structure, it is seamless and globally smooth, can be aligned to geometric features and shows good quality in terms of angle and area preservation, comparable to current parametrization techniques for meshes. Computing such global seamless parametrization makes it possible to perform quad remeshing, texture mapping and texture synthesis and many other types of geometry processing operations. Our approach is based on a formulation of the Poisson equation on a manifold structure defined for the surface by the range images. Construction of such global parametrization requires only a way to project surface data onto a set of planes, and can be applied directly to implicit surfaces, nonmanifold surfaces, very large meshes, and collections of range scans. We demonstrate application of our technique to all these geometry types

On the Fiedler value of large planar graphs

The Fiedler value $\lambda_2$, also known as algebraic connectivity, is the second smallest Laplacian eigenvalue of a graph. We study the maximum Fiedler value among all planar graphs $G$ with $n$ vertices, denoted by $\lambda_{2\max}$, and we show the bounds $2+\Theta(\frac{1}{n^2}) \leq \lambda_{2\max} \leq 2+O(\frac{1}{n})$. We also provide bounds on the maximum Fiedler value for the following classes of planar graphs: Bipartite planar graphs, bipartite planar graphs with minimum vertex degree~3, and outerplanar graphs. Furthermore, we derive almost tight bounds on $\lambda_{2\max}$ for two more classes of graphs, those of bounded genus and $K_h$-minor-free graphs.Comment: 21 pages, 4 figures, 1 table. Version accepted in Linear Algebra and Its Application

Learning quadrangulated patches for 3D shape parameterization and completion

We propose a novel 3D shape parameterization by surface patches, that are oriented by 3D mesh quadrangulation of the shape. By encoding 3D surface detail on local patches, we learn a patch dictionary that identifies principal surface features of the shape. Unlike previous methods, we are able to encode surface patches of variable size as determined by the user. We propose novel methods for dictionary learning and patch reconstruction based on the query of a noisy input patch with holes. We evaluate the patch dictionary towards various applications in 3D shape inpainting, denoising and compression. Our method is able to predict missing vertices and inpaint moderately sized holes. We demonstrate a complete pipeline for reconstructing the 3D mesh from the patch encoding. We validate our shape parameterization and reconstruction methods on both synthetic shapes and real world scans. We show that our patch dictionary performs successful shape completion of complicated surface textures.Comment: To be presented at International Conference on 3D Vision 2017, 201