Related papers: Snell meets Fagnano. Path optimization through an …
The billiard systems within quadrics, playing the role of discrete analogues of geodesics on ellipsoids, are incorporated into the theory of integrable quad-graphs. An initial observation is that the Six-pointed star theorem, as the…
This paper explores the number of parallelograms that appear in a billiard path that enters one corner of a rectangle and leaves a second corner of a rectangle as a function of the normalized dimensions of the rectangle.
In this work we present a general introduction to the Signorini problem (or thin obstacle problem). It is a self-contained survey that aims to cover the main currently known results regarding the thin obstacle problem. We present the theory…
The distant graph $G = G(\mathbb{P}(Z),\triangle)$ of the projective line over the ring of integers is considered. The shortest path problem in this graph is solved by use of Klein's geometric interpretation of Euclidean continued…
We investigate the problem of computing a minimum set of solutions that approximates within a specified accuracy $\epsilon$ the Pareto curve of a multiobjective optimization problem. We show that for a broad class of bi-objective problems…
We study the effect of edge diffraction on the semiclassical analysis of two dimensional quantum systems by deriving a trace formula which incorporates paths hitting any number of vertices embedded in an arbitrary potential. This formula is…
We present a $O(n^{\frac{3}{2}})$-time algorithm for the \emph{shortest (diagonal) flip path problem} for \emph{lattice} triangulations with $n$ points, improving over previous $O(n^2)$-time algorithms. For a large, natural class of inputs,…
Suppose we are given an environment consisting of axis-parallel and diagonal line segments with integer endpoints, each of which may be reflective or non-reflective, with integer endpoints, and an initial position for a light ray passing…
Let ${\cal T}$ be a triangulation of a set ${\cal P}$ of $n$ points in the plane, and let $e$ be an edge shared by two triangles in ${\cal T}$ such that the quadrilateral $Q$ formed by these two triangles is convex. A {\em flip} of $e$ is…
We consider a non-minimal billiard trajectory inside the cube. We study the language of the associated orbit when the map is coded with three letters associated to three non-parallel faces of the cube.
Given a polygon $P$ in the triangular grid, we obtain a permutation $\pi_P$ via a natural billiards system in which beams of light bounce around inside of $P$. The different cycles in $\pi_P$ correspond to the different trajectories of…
We provide explicit lower estimates on the complexity growth in typical directions for a class of irrational triangle billiards
The optimization problem that arises out of the least median of squared residuals method in linear regression is analyzed. To simplify the analysis, the problem is replaced by an equivalent one of minimizing the median of absolute…
Flips in triangulations of convex polygons arise in many different settings. They are isomorphic to rotations in binary trees, define edges in the 1-skeleton of the Associahedron and cover relations in the Tamari Lattice. The complexity of…
We revisit the minimum-link path problem: Given a polyhedral domain and two points in it, connect the points by a polygonal path with minimum number of edges. We consider settings where the vertices and/or the edges of the path are…
The principal angles between binary collision subspaces in an $N$-billiard system in $d$-dimensional Euclidean space are computed. These angles are computed for equal masses and arbitrary masses. We then provide a bound on the number of…
This work studies path planning in two-dimensional space, in the presence of polygonal obstacles. We specifically address the problem of building a roadmap graph, that is, an abstract representation of all the paths that can potentially be…
Given a strictly convex domain $\Omega$ in $\R^2$, there is a natural way to define a billiard map in it: a rectilinear path hitting the boundary reflects so that the angle of reflection is equal to the angle of incidence. In this paper we…
Science about optimization methods is rapidly developing today. In machine learning, computer vision, biology, medicine, construction and in many other different areas optimization methods have vast popularity and they appear as important…
The study of polygonal billiards, particularly those in the regular pentagon, has been the subject of two recent papers. One of these papers approaches the problem of discovering the periodic trajectories on the pentagon by identifying…