Showing 1–11 of 11 results for author: Gezalyan, A

Proximity Alert: Ipelets for Neighborhood Graphs and Clustering

Authors: Gitan Balogh, June Cagan, Bea Fatima, Auguste H. Gezalyan, Danesh Sivakumar, Arushi Srinivasan, Yixuan Sun, Vahe Zaprosyan, David M. Mount

Abstract: Neighborhood graphs and clustering algorithms are fundamental structures in both computational geometry and data analysis. Visualizing them can help build insight into their behavior and properties. The Ipe extensible drawing editor, developed by Otfried Cheong, is a widely used software system for generating figures. One particular aspect of Ipe is the ability to add Ipelets, which extend its fun… ▽ More Neighborhood graphs and clustering algorithms are fundamental structures in both computational geometry and data analysis. Visualizing them can help build insight into their behavior and properties. The Ipe extensible drawing editor, developed by Otfried Cheong, is a widely used software system for generating figures. One particular aspect of Ipe is the ability to add Ipelets, which extend its functionality. Here we showcase a set of Ipelets designed to help visualize neighborhood graphs and clustering algorithms. These include: $\eps$-neighbor graphs, furthest-neighbor graphs, Gabriel graphs, $k$-nearest neighbor graphs, $k^{th}$-nearest neighbor graphs, $k$-mutual neighbor graphs, $k^{th}$-mutual neighbor graphs, asymmetric $k$-nearest neighbor graphs, asymmetric $k^{th}$-nearest neighbor graphs, relative-neighbor graphs, sphere-of-influence graphs, Urquhart graphs, Yao graphs, and clustering algorithms including complete-linkage, DBSCAN, HDBSCAN, $k$-means, $k$-means++, $k$-medoids, mean shift, and single-linkage. Our Ipelets are all programmed in Lua and are freely available. △ Less

Submitted 27 March, 2026; originally announced March 2026.

Visualizing Higher Order Structures, Overlap Regions, and Clustering in the Hilbert Geometry

Authors: Hridhaan Banerjee, Soren Brown, June Cagan, Auguste H. Gezalyan, Megan Hunleth, Veena Kailad, Chaewoon Kyoung, Rowan Shigeno, Yasmine Tajeddin, Andrew Wagger, Kelin Zhu, David M. Moun

Abstract: Higher-order Voronoi diagrams and Delaunay mosaics in polygonal metrics have only recently been studied, yet no tools exist for visualizing them. We introduce a tool that fills this gap, providing dynamic interactive software for visualizing higher-order Voronoi diagrams and Delaunay mosaics along with clustering and tools for exploring overlap and outer regions in the Hilbert polygonal metric. We… ▽ More Higher-order Voronoi diagrams and Delaunay mosaics in polygonal metrics have only recently been studied, yet no tools exist for visualizing them. We introduce a tool that fills this gap, providing dynamic interactive software for visualizing higher-order Voronoi diagrams and Delaunay mosaics along with clustering and tools for exploring overlap and outer regions in the Hilbert polygonal metric. We prove that $k^{th}$ order Voronoi cells are not always star-shaped and establish complexity bounds for our algorithm, which generates all order Voronoi diagrams at once. Our software unifies and extends previous tools for visualizing the Hilbert, Funk, and Thompson geometries. △ Less

Submitted 27 March, 2026; originally announced March 2026.

On Voronoi diagrams in the Funk Conical Geometry

Authors: Aditya Acharya, Auguste Henry Gezalyan, David M. Mount, Danesh Sivakumar

Abstract: The forward and reverse Funk weak metrics are fundamental distance functions on convex bodies that serve as the building blocks for the Hilbert and Thompson metrics. In this paper we study Voronoi diagrams under the forward and reverse Funk metrics in polygonal and elliptical cones. We establish several key geometric properties: (1) bisectors consist of a set of rays emanating from the apex of the… ▽ More The forward and reverse Funk weak metrics are fundamental distance functions on convex bodies that serve as the building blocks for the Hilbert and Thompson metrics. In this paper we study Voronoi diagrams under the forward and reverse Funk metrics in polygonal and elliptical cones. We establish several key geometric properties: (1) bisectors consist of a set of rays emanating from the apex of the cone, and (2) Voronoi diagrams in the $d$-dimensional forward (or reverse) Funk metrics are equivalent to additively-weighted Voronoi diagrams in the $(d-1)$-dimensional forward (or reverse) Funk metrics on bounded cross sections of the cone. Leveraging this, we provide an $O\big(n^{\ceil{\frac{d-1}{2}}+1}\big)$ time algorithm for creating these diagrams in $d$-dimensional elliptical cones using a transformation to and from Apollonius diagrams, and an $O(mn\log(n))$ time algorithm for 3-dimensional polygonal cones with $m$ facets via a reduction to abstract Voronoi diagrams. We also provide a complete characterization of when three sites have a circumcenter in 3-dimensional cones. This is one of the first algorithmic studies of the Funk metrics. △ Less

Submitted 21 February, 2026; originally announced February 2026.

Classifiers in High Dimensional Hilbert Metrics

Authors: Aditya Acharya, Auguste H. Gezalyan, David M. Mount

Abstract: Classifying points in high dimensional spaces is a fundamental geometric problem in machine learning. In this paper, we address classifying points in the $d$-dimensional Hilbert polygonal metric. The Hilbert metric is a generalization of the Cayley-Klein hyperbolic distance to arbitrary convex bodies and has a diverse range of applications in machine learning and convex geometry. We first present… ▽ More Classifying points in high dimensional spaces is a fundamental geometric problem in machine learning. In this paper, we address classifying points in the $d$-dimensional Hilbert polygonal metric. The Hilbert metric is a generalization of the Cayley-Klein hyperbolic distance to arbitrary convex bodies and has a diverse range of applications in machine learning and convex geometry. We first present an efficient LP-based algorithm in the metric for the large-margin SVM problem. Our algorithm runs in time polynomial to the number of points, bounding facets, and dimension. This is a significant improvement on previous works, which either provide no theoretical guarantees on running time, or suffer from exponential runtime. We also consider the closely related Funk metric. We also present efficient algorithms for the soft-margin SVM problem and for nearest neighbor-based classification in the Hilbert metric. △ Less

Submitted 19 January, 2026; originally announced January 2026.

Software for the Thompson and Funk Polygonal Geometry

Authors: Hridhaan Banerjee, Carmen Isabel Day, Auguste H. Gezalyan, Olga Golovatskaia, Megan Hunleth, Sarah Hwang, Nithin Parepally, Lucy Wang, David M. Mount

Abstract: Metric spaces defined within convex polygons, such as the Thompson, Funk, reverse Funk, and Hilbert metrics, are subjects of recent exploration and study in computational geometry. This paper contributes an educational piece of software for understanding these unique geometries while also providing a tool to support their research. We provide dynamic software for manipulating the Funk, reverse Fun… ▽ More Metric spaces defined within convex polygons, such as the Thompson, Funk, reverse Funk, and Hilbert metrics, are subjects of recent exploration and study in computational geometry. This paper contributes an educational piece of software for understanding these unique geometries while also providing a tool to support their research. We provide dynamic software for manipulating the Funk, reverse Funk, and Thompson balls in convex polygonal domains. Additionally, we provide a visualization program for traversing the Hilbert polygonal geometry. △ Less

Submitted 3 March, 2025; originally announced March 2025.

French Onion Soup, Ipelets for Points and Polygons

Authors: Klint Faber, Auguste H. Gezalyan, Adam Martinson, Aniruddh Mutnuru, Nithin Parepally, Ryan Parker, Mihil Sreenilayam, Aram Zaprosyan, David M. Mount

Abstract: There are many structures, both classical and modern, involving point-sets and polygons whose deeper understanding can 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.… ▽ More There are many structures, both classical and modern, involving point-sets and polygons whose deeper understanding can 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 based structures based on point sets and polygons. These include quadtrees, trapezoidal maps, beta skeletons, floating bodies of convex polygons, onion graphs, fractals (Sierpiński triangle and carpet), simple polygon triangulations, and random point sets in simple polygons. All of our Ipelets are programmed in Lua and are freely available. △ Less

Submitted 3 March, 2025; originally announced March 2025.

On The Heine-Borel Property and Minimum Enclosing Balls

Authors: Hridhaan Banerjee, Carmen Isabel Day, Megan Hunleth, Sarah Hwang, Auguste H. Gezalyan, Olya Golovatskaia, Nithin Parepally, Lucy Wang, David M. Mount

Abstract: In this paper, we contribute a proof that minimum radius balls over metric spaces with the Heine-Borel property are always LP type. Additionally, we prove that weak metric spaces, those without symmetry, also have this property if we fix the direction in which we take their distances from the centers of the balls. We use this to prove that the minimum radius ball problem is LP type in the Hilbert… ▽ More In this paper, we contribute a proof that minimum radius balls over metric spaces with the Heine-Borel property are always LP type. Additionally, we prove that weak metric spaces, those without symmetry, also have this property if we fix the direction in which we take their distances from the centers of the balls. We use this to prove that the minimum radius ball problem is LP type in the Hilbert and Thompson metrics and Funk weak metric. In doing so, we contribute a proof that the topology induced by the Thompson metric coincides with the Hilbert. We provide explicit primitives for computing the minimum radius ball in the Hilbert metric. △ Less

Submitted 3 March, 2025; v1 submitted 22 December, 2024; originally announced December 2024.

Ipelets for the Convex Polygonal Geometry

Authors: Nithin Parepally, Ainesh Chatterjee, Auguste Gezalyan, Hongyang Du, Sukrit Mangla, Kenny Wu, Sarah Hwang, David Mount

Abstract: 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 Ipe… ▽ More 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. △ Less

Submitted 15 March, 2024; originally announced March 2024.

Delaunay Triangulations in the Hilbert Metric

Authors: Auguste Gezalyan, Soo Kim, Carlos Lopez, Daniel Skora, Zofia Stefankovic, David M. Mount

Abstract: 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 Hilb… ▽ More 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. △ Less

Submitted 10 December, 2023; originally announced December 2023.

Analysis of Dynamic Voronoi Diagrams in the Hilbert Metric

Authors: Madeline Bumpus, Xufeng Caesar Dai, Auguste H. Gezalyan, Sam Munoz, Renita Santhoshkumar, Songyu Ye, David M. Mount

Abstract: The Hilbert metric is a projective metric defined on a convex body which generalizes the Cayley-Klein model of hyperbolic geometry to any convex set. In this paper we analyze Hilbert Voronoi diagrams in the Dynamic setting. In addition we introduce dynamic visualization software for Voronoi diagrams in the Hilbert metric on user specified convex polygons. The Hilbert metric is a projective metric defined on a convex body which generalizes the Cayley-Klein model of hyperbolic geometry to any convex set. In this paper we analyze Hilbert Voronoi diagrams in the Dynamic setting. In addition we introduce dynamic visualization software for Voronoi diagrams in the Hilbert metric on user specified convex polygons. △ Less

Submitted 1 July, 2024; v1 submitted 5 April, 2023; originally announced April 2023.

Voronoi Diagrams in the Hilbert Metric

Authors: Auguste H. Gezalyan, David M. Mount

Abstract: The Hilbert metric is a distance function defined for points lying within a convex body. It generalizes the Cayley-Klein model of hyperbolic geometry to any convex set, and it has numerous applications in the analysis and processing of convex bodies. In this paper, we study the geometric and combinatorial properties of the Voronoi diagram of a set of point sites under the Hilbert metric. Given any… ▽ More The Hilbert metric is a distance function defined for points lying within a convex body. It generalizes the Cayley-Klein model of hyperbolic geometry to any convex set, and it has numerous applications in the analysis and processing of convex bodies. In this paper, we study the geometric and combinatorial properties of the Voronoi diagram of a set of point sites under the Hilbert metric. Given any convex polygon $K$ bounded by $m$ sides, we present two algorithms (one randomized and one deterministic) for computing the Voronoi diagram of an $n$-element point set in the Hilbert metric induced by $K$. Our randomized algorithm runs in $O(m n + n (\log n)(\log m n))$ expected time, and our deterministic algorithm runs in time $O(m n \log n)$. Both algorithms use $O(m n)$ space. We show that the worst-case combinatorial complexity of the Voronoi diagram is $Θ(m n)$. △ Less

Submitted 6 December, 2021; originally announced December 2021.