231 research outputs found

    HexBox: Interactive Box Modeling of Hexahedral Meshes

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    We introduce HexBox, an intuitive modeling method and interactive tool for creating and editing hexahedral meshes. Hexbox brings the major and widely validated surface modeling paradigm of surface box modeling into the world of hex meshing. The main idea is to allow the user to box-model a volumetric mesh by primarily modifying its surface through a set of topological and geometric operations. We support, in particular, local and global subdivision, various instantiations of extrusion, removal, and cloning of elements, the creation of non-conformal or conformal grids, as well as shape modifications through vertex positioning, including manual editing, automatic smoothing, or, eventually, projection on an externally-provided target surface. At the core of the efficient implementation of the method is the coherent maintenance, at all steps, of two parallel data structures: a hexahedral mesh representing the topology and geometry of the currently modeled shape, and a directed acyclic graph that connects operation nodes to the affected mesh hexahedra. Operations are realized by exploiting recent advancements in grid- based meshing, such as mixing of 3-refinement, 2-refinement, and face-refinement, and using templated topological bridges to enforce on-the-fly mesh conformity across pairs of adjacent elements. A direct manipulation user interface lets users control all operations. The effectiveness of our tool, released as open source to the community, is demonstrated by modeling several complex shapes hard to realize with competing tools and techniques

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    A Peano curve from mated geodesic trees in the directed landscape

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    A number of interesting examples of random space filling curves have been constructed in two dimensional statistical mechanics models. Examples of the above include SLE(88), arising as the scaling limit of the Peano curve between the uniform spanning tree of Z2\mathbb{Z}^2 and its dual (Lawler-Schramm-Werner '04), or SLE(16/γ216/\gamma^2), which describes the interface between two mated trees in γ\gamma-Liouville Quantum gravity (Duplantier-Miller-Sheffield '14). In this work, we construct a Peano curve from the geodesic tree in a fixed direction and its dual in the directed landscape, the putative universal space-time scaling limit object in the KPZ universality class. The geodesic tree and its dual were constructed in Bhatia '23, where it was shown that they have the same law up to reflection; indeed, the dual tree is the geodesic tree of the dual landscape as constructed there. We show that the interface between the two trees is a space filling curve that is naturally parametrized by the area and encodes the geometry of the two trees. We study the regularity and fractal properties of this Peano curve, exploiting simultaneously the symmetries of the directed landscape and probabilistic estimates obtained in the planar exponential last passage percolation, which is known to converge to the directed landscape in the scaling limit.Comment: 49 pages, 15 figures; minor edit

    On self-duality and unigraphicity for 33-polytopes

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    Recent literature posed the problem of characterising the graph degree sequences with exactly one 33-polytopal (i.e. planar, 33-connected) realisation. This seems to be a difficult problem in full generality. In this paper, we characterise the sequences with exactly one self-dual 33-polytopal realisation. An algorithm in the literature constructs a self-dual 33-polytope for any admissible degree sequence. To do so, it performs operations on the radial graph, so that the corresponding 33-polytope and its dual are modified in exactly the same way. To settle our question and construct the relevant graphs, we apply this algorithm, we introduce some modifications of it, and we also devise new ones. The speed of these algorithms is linear in the graph order

    LIPIcs, Volume 274, ESA 2023, Complete Volume

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    LIPIcs, Volume 274, ESA 2023, Complete Volum

    LIPIcs, Volume 258, SoCG 2023, Complete Volume

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    LIPIcs, Volume 258, SoCG 2023, Complete Volum

    Weakly and Strongly Fan-Planar Graphs

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    We study two notions of fan-planarity introduced by (Cheong et al., GD22), called weak and strong fan-planarity that separate two non-equivalent definitions of fan-planarity in the literature. We prove that not every weakly fan-planar graph is strongly fan-planar, while the density upper bound for both families is the same.Comment: Appears in the Proceedings of the 31st International Symposium on Graph Drawing and Network Visualization (GD 2023

    On the expected number of perfect matchings in cubic planar graphs

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    A well-known conjecture by Lov'asz and Plummer from the 1970s asserted that a bridgeless cubic graph has exponentially many perfect matchings. It was solved in the affirmative by Esperet et al. ([13]). On the other hand, Chudnovsky and Seymour ([8]) proved the conjecture in the special case of cubic planar graphs. In our work we consider random bridgeless cubic planar graphs with the uniform distribution on graphs with n vertices. Under this model we show that the expected number of perfect matchings in labeled bridgeless cubic planar graphs is asymptotically cγn, where c > 0 and γ ∼ 1.14196 is an explicit algebraic number. We also compute the expected number of perfect matchings in (not necessarily bridgeless) cubic planar graphs and provide lower bounds for unlabeled graphs. Our starting point is a correspondence between counting perfect matchings in rooted cubic planar maps and the partition function of the Ising model in rooted triangulations

    Analysis and Generation of Quality Polytopal Meshes with Applications to the Virtual Element Method

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    This thesis explores the concept of the quality of a mesh, the latter being intended as the discretization of a two- or three- dimensional domain. The topic is interdisciplinary in nature, as meshes are massively used in several fields from both the geometry processing and the numerical analysis communities. The goal is to produce a mesh with good geometrical properties and the lowest possible number of elements, able to produce results in a target range of accuracy. In other words, a good quality mesh that is also cheap to handle, overcoming the typical trade-off between quality and computational cost. To reach this goal, we first need to answer the question: ''How, and how much, does the accuracy of a numerical simulation or a scientific computation (e.g., rendering, printing, modeling operations) depend on the particular mesh adopted to model the problem? And which geometrical features of the mesh most influence the result?'' We present a comparative study of the different mesh types, mesh generation techniques, and mesh quality measures currently available in the literature related to both engineering and computer graphics applications. This analysis leads to the precise definition of the notion of quality for a mesh, in the particular context of numerical simulations of partial differential equations with the virtual element method, and the consequent construction of criteria to determine and optimize the quality of a given mesh. Our main contribution consists in a new mesh quality indicator for polytopal meshes, able to predict the performance of the virtual element method over a particular mesh before running the simulation. Strictly related to this, we also define a quality agglomeration algorithm that optimizes the quality of a mesh by wisely agglomerating groups of neighboring elements. The accuracy and the reliability of both tools are thoroughly verified in a series of tests in different scenarios
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