Delaunay Triangulation

Some basic mathematical tools such as convex sets, polytopes and combinatorial topology, are used quite heavily in applied fields such as geometric modeling, meshing, computer vision, medical imaging and robotics. This report may be viewed as a tutorial and a set of notes on convex sets, polytopes, polyhedra, combinatorial topology, Voronoi Diagrams and Delaunay Triangulations. It is intended for a broad audience of mathematically inclined readers. I have included a rather thorough treatment of the equivalence of V-polytopes and H-polytopes and also of the equivalence of V-polyhedra and H-polyhedra, which is a bit harder. In particular, the Fourier-Motzkin elimination method (a version of Gaussian elimination for inequalities) is discussed in some detail. I also included some material on projective spaces, projective maps and polar duality w.r.t. a nondegenerate quadric in order to define a suitable notion of ``projective polyhedron'' based on cones. To the best of our knowl...

It is well-known that the Voronoi diagram of points and the power diagram for weighted points, such as spheres, are cell complexes, and their respective dual structures, i.e. the Delaunay triangulation and the regular triangulation, are simplicial complexes. Hence, the topologies of these diagrams are usually stored in their dual complexes,using a very compact,data structure of arrays. The topology

This paper presents a new way to compute the Delaunay triangulation of a planar set $P$ of $n$ points, using sweep-circle technique combined with the standard recursive edge-flipping. The algorithm sweeps the plane by an increasing circle whose center is a fixed point in the convex hull of $P$.
Empirical results and comparisons show that it reduces the number of in-circle tests and edge-flips, and it is efficient in practice.

Geometry processing applications frequently rely on octree structures, since they provide simple and efficient hierarchies for discrete data. However, octrees do not guarantee direct continuous interpolation of this data inside its nodes. This motivates the use of the octree's dual structure, which is one of the simplest continuous hierarchical structures. With the emergence of pointerless representations, with their ability to reduce memory footprint and adapt to parallel architectures, the generation of duals of pointerless octrees becomes a natural challenge. This work proposes strategies for dual generation of static or dynamic pointerless octrees. Experimentally, those methods enjoy the memory reduction of pointerless representations and speed up the execution by several factors compared to the usual recursive generation.

The Voronoi diagram of a finite set of objects is a fundamental geometric structure that subdivides the embedding space into regions, each region consisting of the points that are closer to a given object than to the others. We may define many variants of Voronoi diagrams depending ...

In this paper, we propose an unstructured mesh generation method based on Lagrangian-particle fluid relaxation, imposing a global optimization strategy. With the presumption that the geometry can be described as a zero level set, an adaptive isotropic mesh is generated by three steps. First, three characteristic fields based on three modeling equations are computed to define the target mesh-vertex distribution, i.e. target feature-size function and density function. The modeling solutions are computed on a multi-resolution Cartesian background mesh. Second, with a target particle density and a local smoothing-length interpolated from the target field on the background mesh, a set of physically-motivated model equations is developed and solved by an adaptive-smoothing-length Smoothed Particle Hydrodynamics (SPH) method. The relaxed particle distribution conforms well with the target functions while maintaining isotropy and smoothness inherently. Third, a parallel fast Delaunay triangulation method is developed based on the observation that a set of neighboring particles generates a locally valid Voronoi diagram at the interior of the domain. The incompleteness near domain boundaries is handled by enforcing a symmetry boundary condition. A set of two-dimensional test cases shows the feasibility of the method. Numerical results demonstrate that the proposed method produces high-quality globally optimized adaptive isotropic meshes even for high geometric complexity.

The Poisson-Voronoi map is a natural random planar map. Being planar, a specific instance can always be colored with 4 colors with adjacent cells having distinct colors. The question we consider here is whether such a coloring can be realized in a way that would be isometry-...

Delaunay flip is an elegant, simple tool to convert a triangulation of a point set to its Delaunay triangulation. The technique has been researched extensively for full dimensional triangulations of point sets. However, an important case of triangulations which are not full dimensional is surface triangulations in three dimensions. In this paper we address the question of converting a surface triangulation to a subcomplex of the Delaunay triangulation with edge flips. We show that the surface triangulations which closely approximate a smooth surface with uniform density can be transformed to a Delaunay triangulation with a simple edge flip algorithm. The condition on uniformity becomes less stringent with increasing density of the triangulation. If the condition is dropped completely, the flip algorithm still terminates although the output surface triangulation becomes "almost Delaunay" instead of exactly Delaunay.

A simulação computacional através de métodos numéricos é uma área importante para a Engenharia, possibilitando o desenvolvimento tecnológico sem o uso direto de protótipos físicos. Para a realização de simulações são utilizadas equações diferenciais, as quais descrevem o comportamento de estruturas submetidas a fenômenos complexos. A Equação de Poisson é uma equação diferencial utilizada na Física e na Engenharia para descrever fenômenos de geração de potencial em que a fonte é conhecida. Para solucionar a equação diferencial no domínio definido para a simulação, utiliza-se o Método dos Elementos Finitos, que apresenta a vantagem de solucionar equações em domínios de geometria complexa, o qual deve ser discretizado em subdomínios, chamados elementos. Apresentam- se dois métodos de discretização de domínios ou de geração de malhas, o Método de Divisão Baricêntrica e o Método de Delaunay, que devem ser comparados em termos da qualidade na geração dos elementos. Utilizando uma estrutura quadrada como domínio, esta foi discretizada utilizando os dois métodos e submetida à simulação com a Equação de Poisson e o Método do Elementos Finitos. A partir dos resultados destas simulações, verificou-se que o Método de Delaunay gera melhores resultados que o Método de Divisão Baricêntrica, mas que a Divisão Baricêntrica serve como geradora de nós internos à malha.

Aim: Our aim is to offer and illustrates a novel meta-methodology to enhance the rigour of method selection and understanding of results in pluralist qualitative research (PQR).
Method: To do so, we make innovative use of Braun and Clarke’s (2006) articulation of four discrete dimensions characterising different forms of thematic analysis. We provide secondary analyses of an interview from the Social Media, Men who have Sex with Men and Sexual Health project using critical discursive psychology, dialogical analysis, interpretative phenomenological analysis, and psychosocial narrative analysis.
Results: All four methods identified aspects of three central foci: Compartmentalisation, Detachment, and Jouissance.
Conclusion: We discuss how our proposed meta-methodology provides a rationale for the selection of methods in a PQR, offer evidence that it can anticipate the relative similarity in focus of the methods employed, and argue that our meta-methodology reveals the possibility of identifying an ‘axial’ or ‘hub’ method’ of a PQR which might be particularly fruitful in exploring commonalities and differences in results. Finally, we examine the synergies and challenges of combining pairs of the methods we used.