Related papers: Visibility Extension via Reflective Edges to an Ex…
Let $P$ and $Q$ be simple polygons with $n$ vertices each. We wish to compute triangulations of $P$ and $Q$ that are combinatorially equivalent, if they exist. We consider two versions of the problem: if a triangulation of $P$ is given, we…
How can we tell whether an image has been mirrored? While we understand the geometry of mirror reflections very well, less has been said about how it affects distributions of imagery at scale, despite widespread use for data augmentation in…
Given a simple polygon $\mathscr{P}$, two points $x$ and $y$ within $\mathscr{P}$ are {\em visible} to each other if the line segment between $x$ and $y$ is contained in $\mathscr{P}$. The {\em visibility region} of a point $x$ includes all…
In our previous works we have classified real non-singular cubic hypersurfaces in the 5-dimensional projective space up to equivalence that includes both real projective transformations and continuous variations of coefficients preserving…
Let $P$ be a polygonal curve in $\mathbb{R}^d$ of length $n$, and $S$ be a point-set of size $k$. The Curve/Point Set Matching problem consists of finding a polygonal curve $Q$ on $S$ such that the Fr\'echet distance from $P$ is less than a…
The goal of this paper is to make the vertex operator algebra approach to mirror symmetry accessible to algebraic geometers. Compared to better-known approaches using moduli spaces of stable maps and special Lagrangian fibrations, this…
We provide an explicit geometric algorithm involving only ruler and compass constructions in order to specify the specular reflection point on the surface of a reflecting sphere of radius $r$ given two focal points $A$ and $B$ lying outside…
The Searchlight Scheduling Problem was first studied in 2D polygons, where the goal is for point guards in fixed positions to rotate searchlights to catch an evasive intruder. Here the problem is extended to 3D polyhedra, with the guards…
We study the complexity of computing the projection of an arbitrary $d$-polytope along $k$ orthogonal vectors for various input and output forms. We show that if $d$ and $k$ are part of the input (i.e. not a constant) and we are interested…
Electromagnetic waves propagating in the background provided by a spacetime hosting a strong curvature, naked singularity, are fully studied. The analysis is performed not only in the realm of geometrical optics -- which, not surprisingly,…
Deciding whether a graph can be embedded in a grid using only unit-length edges is NP-complete, even when restricted to binary trees. However, it is not difficult to devise a number of graph classes for which the problem is polynomial, even…
We study the problem of placing a set $T$ of broadcast towers in a simple polygon $P$ in order for any point to locate itself in the interior of $P$. Let $V(p)$ denote the visibility polygon of a point $p$, as the set of all points $q \in…
Symmetry is ubiquitous throughout nature and can often give great insights into the formation, structure and stability of objects studied by mathematicians, physicists, chemists and biologists. However, perfect symmetry occurs rarely so…
Specular reflections pose a significant challenge for object segmentation, as their sharp intensity transitions often mislead both conventional algorithms and deep learning based methods. However, as the specular reflection must lie on the…
Orthogonal weight matrices are used in many areas of deep learning. Much previous work attempt to alleviate the additional computational resources it requires to constrain weight matrices to be orthogonal. One popular approach utilizes…
We introduce two explicit examples of polynomials orthogonal on the unit circle. Moments and the reflection coefficients are expressed in terms of Jacobi elliptic functions. We find explicit expression for these polynomials in terms of a…
The first order equation relating object and image location for a mirror of arbitrary conic-sectional shape is derived. It is also shown that the parabolic reflecting surface is the only one free of aberration and only in the limiting case…
Many objects are naturally symmetric, and this symmetry can be exploited to infer unseen 3D properties from a single 2D image. Recently, NeRD is proposed for accurate 3D mirror plane estimation from a single image. Despite the unprecedented…
In this survey paper, we present open problems and conjectures on visibility graphs of points, segments and polygons along with necessary backgrounds for understanding them.
We study the complexity of symmetric assembly puzzles: given a collection of simple polygons, can we translate, rotate, and possibly flip them so that their interior-disjoint union is line symmetric? On the negative side, we show that the…