Related papers: Clean up your Mesh! Part 1: Plane and simplex
We uncover a geometric organization of the differential equations for the wavefunction coefficients of conformally coupled scalars in power-law cosmologies. To do this, we introduce a basis of functions inspired by a decomposition of the…
Some basic notions of classical algebraic geometry can be defined in arbitrary varieties of algebras $\Theta.$ For every algebra $H$ in $\Theta$ one can consider algebraic geometry in $\Theta$ over $ H.$ Correspondingly, algebras in…
A new simple geometrical interpretation of complex numbers is presented. It differs from their usual interpretation as points in the complex plane. From the new point of view the complex numbers are rather operations on vectors than points.…
This paper proves explicit formulas for the number of dissections of a convex regular polygon modulo the action of the cyclic and dihedral groups. The formulas are obtained by making use of the Cauchy-Frobenius Lemma as well as bijections…
We apply methods of nonstandard mathematics in order to regard analytic geometry in a very different way. For example, complex spaces are seen to be the "standard part" of certain algebraic nonstandard schemes. We construct a category of…
In fields ranging from computer vision to signal processing and statistics, increasing computational power allows a move from classical linear models to models that incorporate non-linear phenomena. This shift has created interest in…
String theory still remains one of the promising candidates for a unification of the theory of gravity and quantum field theory. One of its essential parts is relativistic description of moving multi-dimensional objects called membranes (or…
Point cloud registration aims to provide estimated transformations to align point clouds, which plays a crucial role in pose estimation of various navigation systems, such as surgical guidance systems and autonomous vehicles. Despite the…
The concept of number and its generalization has played a central role in the development of mathematics over many centuries and many civilizations. Noteworthy milestones in this long and arduous process were the developments of the real…
Geometric algebra was initiated by W.K. Clifford over 130 years ago. It unifies all branches of physics, and has found rich applications in robotics, signal processing, ray tracing, virtual reality, computer vision, vector field processing,…
General results on convex bodies are reviewed and used to derive an exact closed-form parametric formula for the boundary of the geometric (Minkowski) sum of $k$ ellipsoids in $n$-dimensional Euclidean space. Previously this was done…
General results on convex bodies are reviewed and used to derive an exact closed-form parametric formula for the boundary of the geometric (Minkowski) sum of $k$ ellipsoids in $n$-dimensional Euclidean space. Previously this was done…
We introduce a novel formulation for geometry on discrete points. It is based on a universal differential calculus, which gives a geometric description of a discrete set by the algebra of functions. We expand this mathematical framework so…
Let $P$ be a set of $n$ points in general position on the plane. A set of closed convex polygons with vertices in $P$, and with pairwise disjoint interiors is called a convex decomposition of $P$ if their union is the convex hull of $P$,…
Here we consider two algebras, a free unital associative complex algebra (denoted by ${\mathcal{B}}$) equiped with a multiparametric \textbf{\emph{q}}-differential structure and a twisted group algebra (denoted by ${\mathcal{A}(S_{n})}$),…
As the most common representation for 3D shapes, mesh is often stored discretely with arrays of vertices and faces. However, 3D shapes in the real world are presented continuously. In this paper, we propose to learn a continuous…
We introduce the Density Formula for (topological) drawings of graphs in the plane or on the sphere, which relates the number of edges, vertices, crossings, and sizes of cells in the drawing. We demonstrate its capability by providing…
The last two decades, since the seminal work of Selig, has seen projective geometric algebra (PGA) gain popularity as a modern coordinate-free framework for doing classical Euclidean geometry and other Cayley-Klein geometries. This…
Let $\Sigma$ be a surface with negative Euler characteristic, genus at least one and at most one boundary component. We prove that the skein algebra of $\Sigma$ over the field of rational functions can be algebraically generated by a finite…
We present a systematic calculation of the volumes of compact manifolds which appear in physics: spheres, projective spaces, group manifolds and generalized flag manifolds. In each case we state what we believe is the most natural scale or…