arXiv subjects

Auguste Gezalyan

Publications and source records attributed to Auguste Gezalyan.

3 recordsLinked to original sources

arXiv2026

The Hyperbolic Surface Distance, Diameter, and Dirichlet Problems

Despite the prominence of hyperbolic surfaces in mathematics, basic algorithmic questions about them, even computing the distance between two points, have remained open, leaving many features of these surfaces inaccessible. The classical machinery assumes a polyhedral structure absent on a smooth surface. We remove these obstacles. We begin with an efficient $O(g^2)$ algorithm for the distance between two points, where $g$ is the genus of the surface. Building on it, we obtain an $O(g^2 \log g)$ method for answering distance queries from a fixed source and, as a consequence, for recentering a Dirichlet domain around an arbitrary point. This understanding of distances on the surface then lets us approximate the diameter to within any $\eps$ in time $O(g^3 \log g / \eps^2)$. We further show that the diameter, a single real number encoding a great deal about the surface, is exactly computable. Its hyperbolic cosine is an algebraic number over the field encoding the coefficients of the hyperbolic isometries defining the surface.

cs.CG↗

arXiv2024

Ipelets for the Convex Polygonal Geometry

There are many structures, both classical and modern, involving convex polygonal geometries whose deeper understanding would be facilitated through interactive visualizations. The Ipe extensible drawing editor, developed by Otfried Cheong, is a widely used software system for generating geometric figures. One of its features is the capability to extend its functionality through programs called Ipelets. In this media submission, we showcase a collection of new Ipelets that construct a variety of geometric objects based on polygonal geometries. These include Macbeath regions, metric balls in the forward and reverse Funk distance, metric balls in the Hilbert metric, polar bodies, the minimum enclosing ball of a point set, and minimum spanning trees in both the Funk and Hilbert metrics. We also include a number of utilities on convex polygons, including union, intersection, subtraction, and Minkowski sum (previously implemented as a CGAL Ipelet). All of our Ipelets are programmed in Lua and are freely available.

cs.CG↗

arXiv2023

Delaunay Triangulations in the Hilbert Metric

The Hilbert metric is a distance function defined for points lying within the interior of a convex body. It arises in the analysis and processing of convex bodies, machine learning, and quantum information theory. In this paper, we show how to adapt the Euclidean Delaunay triangulation to the Hilbert geometry defined by a convex polygon in the plane. We analyze the geometric properties of the Hilbert Delaunay triangulation, which has some notable differences with respect to the Euclidean case, including the fact that the triangulation does not necessarily cover the convex hull of the point set. We also introduce the notion of a Hilbert ball at infinity, which is a Hilbert metric ball centered on the boundary of the convex polygon. We present a simple randomized incremental algorithm that computes the Hilbert Delaunay triangulation for a set of $n$ points in the Hilbert geometry defined by a convex $m$-gon. The algorithm runs in $O(n (\log n + \log^3 m))$ expected time. In addition we introduce the notion of the Hilbert hull of a set of points, which we define to be the region covered by their Hilbert Delaunay triangulation. We present an algorithm for computing the Hilbert hull in time $O(n h \log^2 m)$, where $h$ is the number of points on the hull's boundary.

cs.CG↗