Tutte Embeddings of Tetrahedral Meshes

Tutte's embedding theorem states that every 3-connected graph without a $K_5$ or $K_{3,3}$ minor (i.e. a planar graph) is embedded in the plane if the outer face is in convex position and the interior vertices are convex combinations of their neighbors. We show that this result extends to simply connected tetrahedral meshes in a natural way: for the tetrahedral mesh to be embedded if the outer polyhedron is in convex position and the interior vertices are convex combination of their neighbors it is sufficient (but not necessary) that the graph of the tetrahedral mesh contains no $K_6$ and no $K_{3,3,1}$, and all triangles incident on three boundary vertices are boundary triangles

VOLMAP: a Large Scale Benchmark for Volume Mappings to Simple Base Domains

Correspondences between geometric domains (mappings) are ubiquitous in computer graphics and engineering, both for a variety of downstream applications and as core building blocks for higher level algorithms. In particular, mapping a shape to a convex or star-shaped domain with simple geometry is a fundamental module in existing pipelines for mesh generation, solid texturing, generation of shape correspondences, advanced manufacturing etc. For the case of surfaces, computing such a mapping with guarantees of injectivity is a solved problem. Conversely, robust algorithms for the generation of injective volume mappings to simple polytopes are yet to be found, making this a fundamental open problem in volume mesh processing. VOLMAP is a large scale benchmark aimed to support ongoing research in volume mapping algorithms. The dataset contains 4.7K tetrahedral meshes, whose boundary vertices are mapped to a variety of simple domains, either convex or star-shaped. This data constitutes the input for candidate algorithms, which are then required to position interior vertices in the domain to obtain a volume map. Overall, this yields more than 22K alternative test cases. VOLMAP also comprises tools to process this data, analyze the resulting maps, and extend the dataset with new meshes, boundary maps and base domains. This article provides a brief overview of the field, discussing its importance and the lack of effective techniques. We then introduce both the dataset and its major features. An example of comparative analysis between two existing methods is also present

VoroCrust: Voronoi Meshing Without Clipping

Polyhedral meshes are increasingly becoming an attractive option with particular advantages over traditional meshes for certain applications. What has been missing is a robust polyhedral meshing algorithm that can handle broad classes of domains exhibiting arbitrarily curved boundaries and sharp features. In addition, the power of primal-dual mesh pairs, exemplified by Voronoi-Delaunay meshes, has been recognized as an important ingredient in numerous formulations. The VoroCrust algorithm is the first provably-correct algorithm for conforming polyhedral Voronoi meshing for non-convex and non-manifold domains with guarantees on the quality of both surface and volume elements. A robust refinement process estimates a suitable sizing field that enables the careful placement of Voronoi seeds across the surface circumventing the need for clipping and avoiding its many drawbacks. The algorithm has the flexibility of filling the interior by either structured or random samples, while preserving all sharp features in the output mesh. We demonstrate the capabilities of the algorithm on a variety of models and compare against state-of-the-art polyhedral meshing methods based on clipped Voronoi cells establishing the clear advantage of VoroCrust output.Comment: 18 pages (including appendix), 18 figures. Version without compressed images available on https://www.dropbox.com/s/qc6sot1gaujundy/VoroCrust.pdf. Supplemental materials available on https://www.dropbox.com/s/6p72h1e2ivw6kj3/VoroCrust_supplemental_materials.pd

Sixteen space-filling curves and traversals for d-dimensional cubes and simplices

This article describes sixteen different ways to traverse d-dimensional space recursively in a way that is well-defined for any number of dimensions. Each of these traversals has distinct properties that may be beneficial for certain applications. Some of the traversals are novel, some have been known in principle but had not been described adequately for any number of dimensions, some of the traversals have been known. This article is the first to present them all in a consistent notation system. Furthermore, with this article, tools are provided to enumerate points in a regular grid in the order in which they are visited by each traversal. In particular, we cover: five discontinuous traversals based on subdividing cubes into 2^d subcubes: Z-traversal (Morton indexing), U-traversal, Gray-code traversal, Double-Gray-code traversal, and Inside-out traversal; two discontinuous traversals based on subdividing simplices into 2^d subsimplices: the Hill-Z traversal and the Maehara-reflected traversal; five continuous traversals based on subdividing cubes into 2^d subcubes: the Base-camp Hilbert curve, the Harmonious Hilbert curve, the Alfa Hilbert curve, the Beta Hilbert curve, and the Butz-Hilbert curve; four continuous traversals based on subdividing cubes into 3^d subcubes: the Peano curve, the Coil curve, the Half-coil curve, and the Meurthe curve. All of these traversals are self-similar in the sense that the traversal in each of the subcubes or subsimplices of a cube or simplex, on any level of recursive subdivision, can be obtained by scaling, translating, rotating, reflecting and/or reversing the traversal of the complete unit cube or simplex.Comment: 28 pages, 12 figures. v2: fixed a confusing typo on page 12, line

Courbure discrète : théorie et applications

International audienceThe present volume contains the proceedings of the 2013 Meeting on discrete curvature, held at CIRM, Luminy, France. The aim of this meeting was to bring together researchers from various backgrounds, ranging from mathematics to computer science, with a focus on both theory and applications. With 27 invited talks and 8 posters, the conference attracted 70 researchers from all over the world. The challenge of finding a common ground on the topic of discrete curvature was met with success, and these proceedings are a testimony of this wor

3D to 2D surface mesh parameterisation for the purposes of unstructured transmission line modelling method simulations

Small scale fabrication processes have led to the advent of very thin flexible devices such as RFID tags, flexible PCBs and smart clothing. In a geometrical sense, these present themselves as curved two dimensional surfaces embedded in a three dimensional domain. When simulating electromagnetic behaviour on these surfaces at low frequencies, a full 3D field model is not always necessary. Using 3D algorithms to solve these problems can result in a large portion of the computer memory and runtime being used to mesh and simulate areas of the domain that present little electromagnetic activity. The theme of this thesis is concerned with the improvement of the runtime and memory consumption of electromagnetic simulations of these surfaces. The main contributions of this work are presented as an investigation into the feasibility of applying a 2D Unstructured Transmission Line Modelling method (UTLM) simulation to open, curved surfaces embedded in 3D space, by providing a one-to-one mapping of the geometry to a 2D flat plane. First, an investigation into the various methods of how a computer represents unstructured meshes in its memory is presented, and how this affects the runtime of the simulation. The underlying mesh data structures used to represent the geometrical problem space can have a huge impact on the efficiency and memory consumption of the simulation. This investigation served to demonstrate that it is not just simply the optimisation of the simulation algorithms that facilitate improvements to the runtime and memory consumption of a simulation. How a computer understands the connectivity of the mesh can have far greater impacts to the computational resources available. The concepts of surface parameterisation are then introduced; a process of mapping curved surfaces embedded in a three dimensional domain to a flat two dimensional plane. By providing a one-to-one mapping of the geometry from the 3D domain to the 2D flat plane, a low frequency 2D unstructured TLM simulation can be applied, negating the need for 3D algorithms. Because this mapping is one-to-one, the results of the simulation can then be mapped back to 3D space for visualisation. Parameterisations will almost always introduce distortion to angle and area, and minimising this distortion is paramount to maintaining an accurate simulation. Test cases were used to measure the extent of this distortion, and the investigation concluded that Angle Based Flattening (ABF) and Least Squares Conformal Mapping (LSCM) methods resulted in the best quality parameterisations. Simulations were then conducted on these test cases as a demonstration of how UTLM can be performed on 2D surfaces, embedded in a 3D domain

3D to 2D surface mesh parameterisation for the purposes of unstructured transmission line modelling method simulations

Small scale fabrication processes have led to the advent of very thin flexible devices such as RFID tags, flexible PCBs and smart clothing. In a geometrical sense, these present themselves as curved two dimensional surfaces embedded in a three dimensional domain. When simulating electromagnetic behaviour on these surfaces at low frequencies, a full 3D field model is not always necessary. Using 3D algorithms to solve these problems can result in a large portion of the computer memory and runtime being used to mesh and simulate areas of the domain that present little electromagnetic activity. The theme of this thesis is concerned with the improvement of the runtime and memory consumption of electromagnetic simulations of these surfaces. The main contributions of this work are presented as an investigation into the feasibility of applying a 2D Unstructured Transmission Line Modelling method (UTLM) simulation to open, curved surfaces embedded in 3D space, by providing a one-to-one mapping of the geometry to a 2D flat plane. First, an investigation into the various methods of how a computer represents unstructured meshes in its memory is presented, and how this affects the runtime of the simulation. The underlying mesh data structures used to represent the geometrical problem space can have a huge impact on the efficiency and memory consumption of the simulation. This investigation served to demonstrate that it is not just simply the optimisation of the simulation algorithms that facilitate improvements to the runtime and memory consumption of a simulation. How a computer understands the connectivity of the mesh can have far greater impacts to the computational resources available. The concepts of surface parameterisation are then introduced; a process of mapping curved surfaces embedded in a three dimensional domain to a flat two dimensional plane. By providing a one-to-one mapping of the geometry from the 3D domain to the 2D flat plane, a low frequency 2D unstructured TLM simulation can be applied, negating the need for 3D algorithms. Because this mapping is one-to-one, the results of the simulation can then be mapped back to 3D space for visualisation. Parameterisations will almost always introduce distortion to angle and area, and minimising this distortion is paramount to maintaining an accurate simulation. Test cases were used to measure the extent of this distortion, and the investigation concluded that Angle Based Flattening (ABF) and Least Squares Conformal Mapping (LSCM) methods resulted in the best quality parameterisations. Simulations were then conducted on these test cases as a demonstration of how UTLM can be performed on 2D surfaces, embedded in a 3D domain

Advances in Discrete Differential Geometry

Differential Geometr