Related papers: Clean up your Mesh! Part 1: Plane and simplex

Distributional and neural approaches to natural language semantics have been built almost exclusively on conventional linear algebra: vectors, matrices, tensors, and the operations that accompany them. These methods have achieved remarkable…

A relatively simple algebraic framework is given, in which all the compact symmetric spaces can be described and handled without distinguishing cases. We also give some applications and further results.

In this paper, we discuss some problems of elementary plane differential geometry and kinematics. Although the results are not new, the consistent use of complex-valued functions (plane curves) of a real variable (parameter) allows to…

We show there is a class of symplectic Lie algebra representations over any field of characteristic not 2 or 3 that have many of the exceptional algebraic and geometric properties of both symmetric three forms in two dimensions and…

An important problem in geometric reasoning is to find the configuration of a collection of geometric bodies so as to satisfy a set of given constraints. Recently, it has been suggested that this problem can be solved efficiently by…

A. Tarski uses in his system for the elementary geometry only the primitive concept of point, and the two primitive relations betweenness and equidistance. Another approach is the relations to be on lines instead of points. W.…

We present a new model which represents data as a mixture of simplices. Simplices are geometric structures that generalize triangles. We give a simple geometric understanding that allows us to learn a simplicial structure efficiently. Our…

Chiral de Rham complex introduced by Malikov et al. in 1998, is a sheaf of vertex algebras on any complex analytic manifold or non-singular algebraic variety. Starting from the vertex algebra of global sections of chiral de Rham complex on…

We present a new approach for computing planar hexagonal meshes that approximate a given surface, represented as a triangle mesh. Our method is based on two novel technical contributions. First, we introduce Coordinate Power Fields, which…

Planes are familiar mathematical objects which lie at the subtle boundary between continuous geometry and discrete combinatorics. A plane is geometrical, certainly, but the ways that two planes can interact break cleanly into discrete sets:…

Given five points in a three-dimensional euclidean space, one can consider five tetrahedra, using those points as vertices. We present a pentagon-like formula containing the product of three volumes of those tetrahedra in its l.h.s. and the…

One of the generalizations of the pentagon equation to higher dimensions is the so-called "six-term equation". Geometrically, it corresponds to one of the "Alexander moves", that is elementary rebuildings of simplicial complexes, namely,…

We are interested in easy geometric transformations which regularize n-polygons in the non-euclidean plane. A transformation is called easy if it can be easily implemented into an algorithm. This article is motivated by preceding work on…

We have derived an analytical formulation for estimating the volume of geometries enclosed by implicitly defined surfaces. The novelty of this work is due to two aspects. First we provide a general analytical formulation for all…

This paper is an introductory and informal exposition on the topology of polygonal meshes. We begin with a broad overview of topological notions and discuss how homeomorphisms, homotopy, and homology can be used to characterise topology. We…

The basic input for many real objects is a finite cloud of unordered points. The strongest equivalence between objects in practice is rigid motion in a Euclidean space. A recent polynomial-time classification of point clouds required a…

Towards identifying the number of minimal surfaces sharing the same boundary from the geometry of the boundary, we propose a numerical scheme with high speed and high accuracy. Our numerical scheme is based on the method of fundamental…

We expand the basic geometric elements of the simplex method to linear programs in locally convex topological vector spaces and provide conditions under which the method converges in value to optimality. This setting generalizes many…

Mesh-based learning is one of the popular approaches nowadays to learn shapes. The most established backbone in this field is MeshCNN. In this paper, we propose infusing MeshCNN with geometric reasoning to achieve higher quality learning.…

These lecture notes present a computation driven pathway from classical complex analysis to the theory of compact Riemann surfaces and their connections to algebraic geometry. The exposition follows a compute first then abstract philosophy,…