Related papers: A Visualization Method of Four Dimensional Polytop…

This paper considers an extension of origami geometry to the case of "folding" a three dimensional (3D) space along a plane. First, all possible incidence constraints between given points, lines and planes are analyzed by using the geometry…

The hamiltonian circuit polytope is the convex hull of feasible solutions for the circuit constraint, which provides a succinct formulation of the traveling salesman and other sequencing problems. We study the polytope by establishing its…

This article introduces the theory of Veronese polytopes, a broad generalisation of cyclic polytopes. These arise as convex hulls of points on curves with one or more connected components, obtained as the image of the rational normal curve…

In this work, we propose a new segmentation algorithm for images containing convex objects present in multiple shapes with a high degree of overlap. The proposed algorithm is carried out in two steps, first we identify the visible contours,…

Sliced optimal transport, which is basically a Radon transform followed by one-dimensional optimal transport, became popular in various applications due to its efficient computation. In this paper, we deal with sliced optimal transport on…

We present a change of basis that may allow more efficient calculation of the volumes of Birkhoff polytopes using a slicing method. We construct the basis from a special set of square matrices. We explain how to construct this basis easily…

Split-complex fields usually appear in the context of Euclidean supersymmetry. In this paper, we propose that this can be generalized to the non-Euclidean case and that, in fact, the split-complex representation may be the most natural way…

We present two algorithms for unfolding the surface of any polyhedron, all of whose faces are triangles, to a nonoverlapping, connected planar layout. The surface is cut only along polyhedron edges. The layout is connected, but it may have…

We consider polynomials of a few linear forms and show how exploit this type of sparsity for optimization on some particular domains like the Euclidean sphere or a polytope. Moreover, a simple procedure allows to detect this form of…

For any finite set $\A$ of $n$ points in $\R^2$, we define a $(3n-3)$-dimensional simple polyhedron whose face poset is isomorphic to the poset of ``non-crossing marked graphs'' with vertex set $\A$, where a marked graph is defined as a…

A two-step model for generating random polytopes is considered. For parameters $d$, $m$, and $p$, the first step is to generate a simple polytope $P$ whose facets are given by $m$ uniform random hyperplanes tangent to the unit sphere in…

While the proposal of the Tri-plane representation has advanced the development of the 3D-aware image generative models, problems rooted in its inherent structure, such as multi-face artifacts caused by sharing the same features in…

3D scatterplots are a well-established plotting technique that can be used to represent data with three or more dimensions. On paper and computer monitors they are essentially two-dimensional projections of the three-dimensional Cartesian…

Vertex splitting replaces a vertex by two copies and partitions its incident edges amongst the copies. This problem has been studied as a graph editing operation to achieve desired properties with as few splits as possible, most often…

We consider applications involving a large set of instances of projecting points to polytopes. We develop an intuition guided by theoretical and empirical analysis to show that when these instances follow certain structures, a large…

Let $P\subset\R^d$ be a $d$-dimensional polytope. The {\em realization space} of~$P$ is the space of all polytopes $P'\subset\R^d$ that are combinatorially equivalent to~$P$, modulo affine transformations. We report on work by the first…

We propose a geometric explanation for the observation that generic quadratic polynomials over split quaternions may have up to six different factorizations while generic polynomials over Hamiltonian quaternions only have two. Split…

A notion of "radially monotone" cut paths is introduced as an effective choice for finding a non-overlapping edge-unfolding of a convex polyhedron. These paths have the property that the two sides of the cut avoid overlap locally as the cut…

Creative telescoping applied to a bivariate proper hypergeometric term produces linear recurrence operators with polynomial coefficients, called telescopers. We provide bounds for the degrees of the polynomials appearing in these operators.…

We create a passive wave splitter, created purely by geometry, to engineer three-way beam splitting in electromagnetism in transverse electric polarisation. We do so by considering arrangements of Indium Phosphide dielectric pillars in air,…