Random Triangle Theory with Geometry and Applications

What is the probability that a random triangle is acute? We explore this old question from a modern viewpoint, taking into account linear algebra, shape theory, numerical analysis, random matrix theory, the Hopf fibration, and much much more. One of the best distributions of random triangles takes all six vertex coordinates as independent standard Gaussians. Six can be reduced to four by translation of the center to $(0,0)$ or reformulation as a 2x2 matrix problem. In this note, we develop shape theory in its historical context for a wide audience. We hope to encourage other to look again (and differently) at triangles. We provide a new constructive proof, using the geometry of parallelians, of a central result of shape theory: Triangle shapes naturally fall on a hemisphere. We give several proofs of the key random result: that triangles are uniformly distributed when the normal distribution is transferred to the hemisphere. A new proof connects to the distribution of random condition numbers. Generalizing to higher dimensions, we obtain the "square root ellipticity statistic" of random matrix theory. Another proof connects the Hopf map to the SVD of 2 by 2 matrices. A new theorem describes three similar triangles hidden in the hemisphere. Many triangle properties are reformulated as matrix theorems, providing insight to both. This paper argues for a shift of viewpoint to the modern approaches of random matrix theory. As one example, we propose that the smallest singular value is an effective test for uniformity. New software is developed and applications are proposed

Fat 4-polytopes and fatter 3-spheres

We introduce the fatness parameter of a 4-dimensional polytope P, defined as \phi(P)=(f_1+f_2)/(f_0+f_3). It arises in an important open problem in 4-dimensional combinatorial geometry: Is the fatness of convex 4-polytopes bounded? We describe and analyze a hyperbolic geometry construction that produces 4-polytopes with fatness \phi(P)>5.048, as well as the first infinite family of 2-simple, 2-simplicial 4-polytopes. Moreover, using a construction via finite covering spaces of surfaces, we show that fatness is not bounded for the more general class of strongly regular CW decompositions of the 3-sphere.Comment: 12 pages, 12 figures. This version has minor changes proposed by the second refere

3D mesh processing using GAMer 2 to enable reaction-diffusion simulations in realistic cellular geometries

Recent advances in electron microscopy have enabled the imaging of single cells in 3D at nanometer length scale resolutions. An uncharted frontier for in silico biology is the ability to simulate cellular processes using these observed geometries. Enabling such simulations requires watertight meshing of electron micrograph images into 3D volume meshes, which can then form the basis of computer simulations of such processes using numerical techniques such as the Finite Element Method. In this paper, we describe the use of our recently rewritten mesh processing software, GAMer 2, to bridge the gap between poorly conditioned meshes generated from segmented micrographs and boundary marked tetrahedral meshes which are compatible with simulation. We demonstrate the application of a workflow using GAMer 2 to a series of electron micrographs of neuronal dendrite morphology explored at three different length scales and show that the resulting meshes are suitable for finite element simulations. This work is an important step towards making physical simulations of biological processes in realistic geometries routine. Innovations in algorithms to reconstruct and simulate cellular length scale phenomena based on emerging structural data will enable realistic physical models and advance discovery at the interface of geometry and cellular processes. We posit that a new frontier at the intersection of computational technologies and single cell biology is now open.Comment: 39 pages, 14 figures. High resolution figures and supplemental movies available upon reques

The Modified Direct Method: an Approach for Smoothing Planar and Surface Meshes

The Modified Direct Method (MDM) is an iterative mesh smoothing method for smoothing planar and surface meshes, which is developed from the non-iterative smoothing method originated by Balendran [1]. When smooth planar meshes, the performance of the MDM is effectively identical to that of Laplacian smoothing, for triangular and quadrilateral meshes; however, the MDM outperforms Laplacian smoothing for tri-quad meshes. When smooth surface meshes, for trian-gular, quadrilateral and quad-dominant mixed meshes, the mean quality(MQ) of all mesh elements always increases and the mean square error (MSE) decreases during smoothing; For tri-dominant mixed mesh, the quality of triangles always descends while that of quads ascends. Test examples show that the MDM is convergent for both planar and surface triangular, quadrilateral and tri-quad meshes.Comment: 18 page

Mesh generation by domain bisection

The research reported in this dissertation was undertaken to investigate efficient computational methods of automatically generating three dimensional unstructured tetrahedral meshes. The work on two dimensional triangular unstructured grid generation by Lewis and Robinson [LeR76] is first examined, in which a recursive bisection technique of computational order nlog(n) was implemented. This technique is then extended to incorporate new methods of geometry input and the automatic handling of multiconnected regions. The method of two dimensional recursive mesh bisection is then further modified to incorporate an improved strategy for the selection of bisections. This enables an automatic nodal placement technique to be implemented in conjunction with the grid generator. The dissertation then investigates methods of generating triangular grids over parametric surfaces. This includes a new definition of surface Delaunay triangulation with the extension of grid improvement techniques to surfaces. Based on the assumption that all surface grids of objects form polyhedral domains, a three dimensional mesh generation technique is derived. This technique is a hybrid of recursive domain bisection coupled with a min-max heuristic triangulation algorithm. This is done to achieve a computationlly efficient and reliable algorithm coupled with a fast nodal placement technique. The algorithm generates three dimensional unstructured tetrahedral grids over polyhedral domains with multi-connected regions in an average computational order of less than nlog(n)