There are only two nonobtuse binary triangulations of the unit nn-cube

Triangulations of the cube into a minimal number of simplices without additional vertices have been studied by several authors over the past decades. For $3\leq n\leq 7$ this so-called simplexity of the unit cube $I^n$ is now known to be $5,16,67,308,1493$, respectively. In this paper, we study triangulations of $I^n$ with simplices that only have nonobtuse dihedral angles. A trivial example is the standard triangulation into $n!$ simplices. In this paper we show that, surprisingly, for each $n\geq 3$ there is essentially only one other nonobtuse triangulation of $I^n$, and give its explicit construction. The number of nonobtuse simplices in this triangulation is equal to the smallest integer larger than $n!({\rm e}-2)$.Comment: 17 pages, 7 figure

Triangulation of Simple 3D Shapes with Well-Centered Tetrahedra

A completely well-centered tetrahedral mesh is a triangulation of a three dimensional domain in which every tetrahedron and every triangle contains its circumcenter in its interior. Such meshes have applications in scientific computing and other fields. We show how to triangulate simple domains using completely well-centered tetrahedra. The domains we consider here are space, infinite slab, infinite rectangular prism, cube and regular tetrahedron. We also demonstrate single tetrahedra with various combinations of the properties of dihedral acuteness, 2-well-centeredness and 3-well-centeredness.Comment: Accepted at the conference "17th International Meshing Roundtable", Pittsburgh, Pennsylvania, October 12-15, 2008. Will appear in proceedings of the conference, published by Springer. For this version, we fixed some typo

Distance-Sensitive Planar Point Location

Let $\mathcal{S}$ be a connected planar polygonal subdivision with $n$ edges that we want to preprocess for point-location queries, and where we are given the probability $\gamma_i$ that the query point lies in a polygon $P_i$ of $\mathcal{S}$. We show how to preprocess $\mathcal{S}$ such that the query time for a point~$p\in P_i$ depends on~$\gamma_i$ and, in addition, on the distance from $p$ to the boundary of~$P_i$---the further away from the boundary, the faster the query. More precisely, we show that a point-location query can be answered in time $O\left(\min \left(\log n, 1 + \log \frac{\mathrm{area}(P_i)}{\gamma_i \Delta_{p}^2}\right)\right)$, where $\Delta_{p}$ is the shortest Euclidean distance of the query point~$p$ to the boundary of $P_i$. Our structure uses $O(n)$ space and $O(n \log n)$ preprocessing time. It is based on a decomposition of the regions of $\mathcal{S}$ into convex quadrilaterals and triangles with the following property: for any point $p\in P_i$, the quadrilateral or triangle containing~$p$ has area $\Omega(\Delta_{p}^2)$. For the special case where $\mathcal{S}$ is a subdivision of the unit square and $\gamma_i=\mathrm{area}(P_i)$, we present a simpler solution that achieves a query time of $O\left(\min \left(\log n, \log \frac{1}{\Delta_{p}^2}\right)\right)$. The latter solution can be extended to convex subdivisions in three dimensions

Foundations of space-time finite element methods: polytopes, interpolation, and integration

The main purpose of this article is to facilitate the implementation of space-time finite element methods in four-dimensional space. In order to develop a finite element method in this setting, it is necessary to create a numerical foundation, or equivalently a numerical infrastructure. This foundation should include a collection of suitable elements (usually hypercubes, simplices, or closely related polytopes), numerical interpolation procedures (usually orthonormal polynomial bases), and numerical integration procedures (usually quadrature rules). It is well known that each of these areas has yet to be fully explored, and in the present article, we attempt to directly address this issue. We begin by developing a concrete, sequential procedure for constructing generic four-dimensional elements (4-polytopes). Thereafter, we review the key numerical properties of several canonical elements: the tesseract, tetrahedral prism, and pentatope. Here, we provide explicit expressions for orthonormal polynomial bases on these elements. Next, we construct fully symmetric quadrature rules with positive weights that are capable of exactly integrating high-degree polynomials, e.g. up to degree 17 on the tesseract. Finally, the quadrature rules are successfully tested using a set of canonical numerical experiments on polynomial and transcendental functions.Comment: 34 pages, 18 figure

An Application of Tetrahedrisation to From-Point Visibility

We propose a method for tetrahedrizing a polyhedral volume containing polyhedral holes. We use the resulting tetrahedrization to produce from-point visibility algorithms that can be either exact, conservative or aggressive. We also discuss the applications of such from-point and from-region visibility techniques, including their use in lighting and in simulating vision of artificial intelligence agents

7th International Meshing Roundtable '98

The goal of the 7th International Meshing Roundtable is to bring together researchers and developers from industry, academia, and government labs in a stimulating, open environment for the exchange of technical information related to the meshing process. In the past, the Roundtable has enjoyed significant participation from each of these groups from a wide variety of countries

Texture-Based Segmentation and Finite Element Mesh Generation for Heterogeneous Biological Image Data

The design, analysis, and control of bio-systems remain an engineering challenge. This is mainly due to the material heterogeneity, boundary irregularity, and nonlinear dynamics associated with these systems. The recent developments in imaging techniques and stochastic upscaling methods provides a window of opportunity to more accurately assess these bio-systems than ever before. However, the use of image data directly in upscaled stochastic framework can only be realized by the development of certain intermediate steps. The goal of the research presented in this dissertation is to develop a texture-segmentation method and a unstructured mesh generation for heterogeneous image data. The following two new techniques are described and evaluated in this dissertation: 1. A new texture-based segmentation method, using the stochastic continuum concepts and wavelet multi-resolution analysis, is developed for characterization of heterogeneous materials in image data. The feature descriptors are developed to efficiently capture the micro-scale heterogeneity of macro-scale entities. The materials are then segmented at a representative elementary scale at which the statistics of the feature descriptor stabilize. 2. A new unstructured mesh generation technique for image data is developed using a hierarchical data structure. This representation allows for generating quality guaranteed finite element meshes. The framework for both the methods presented in this dissertation, as such, allows them for extending to higher dimensions. The experimental results using these methods conclude them to be promising tools for unifying data processing concepts within the upscaled stochastic framework across biological systems. These are targeted for inclusion in decision support systems where biological image data, simulation techniques and artificial intelligence will be used conjunctively and uniformly to assess bio-system quality and design effective and appropriate treatments that restore system health