12 papers
The Hyperbolic Surface Distance, Diameter, and Dirichlet Problems
Vincent Despre, Auguste Gezalyan, Marc Pouget
Despite the prominence of hyperbolic surfaces in mathematics, basic algorithmic questions about them, even computing the distance between two points, have remained open, leaving ma…
Proximity Alert: Ipelets for Neighborhood Graphs and Clustering
Gitan Balogh, June Cagan, Bea Fatima +6
Neighborhood graphs and clustering algorithms are fundamental structures in both computational geometry and data analysis. Visualizing them can help build insight into their behavi…
Visualizing Higher Order Structures, Overlap Regions, and Clustering in the Hilbert Geometry
Hridhaan Banerjee, Soren Brown, June Cagan +9
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 thi…
On Voronoi diagrams in the Funk Conical Geometry
Aditya Acharya, Auguste Henry Gezalyan, David M. Mount +1
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…
Classifiers in High Dimensional Hilbert Metrics
Aditya Acharya, Auguste H. Gezalyan, David M. Mount
Classifying points in high dimensional spaces is a fundamental geometric problem in machine learning. In this paper, we address classifying points in the -dimensional Hilbert po…
Software for the Thompson and Funk Polygonal Geometry
Hridhaan Banerjee, Carmen Isabel Day, Auguste H. Gezalyan +6
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.…
French Onion Soup, Ipelets for Points and Polygons
Klint Faber, Auguste H. Gezalyan, Adam Martinson +6
There are many structures, both classical and modern, involving point-sets and polygons whose deeper understanding can be facilitated through interactive visualizations. The Ipe ex…
On The Heine-Borel Property and Minimum Enclosing Balls
Hridhaan Banerjee, Carmen Isabel Day, Megan Hunleth +6
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,…
Ipelets for the Convex Polygonal Geometry
Nithin Parepally, Ainesh Chatterjee, Auguste Gezalyan +5
There are many structures, both classical and modern, involving convex polygonal geometries whose deeper understanding would be facilitated through interactive visualizations. The…
Delaunay Triangulations in the Hilbert Metric
Auguste Gezalyan, Soo Kim, Carlos Lopez +3
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…
Analysis of Dynamic Voronoi Diagrams in the Hilbert Metric
Madeline Bumpus, Xufeng Caesar Dai, Auguste H. Gezalyan +4
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 Hilbe…
Voronoi Diagrams in the Hilbert Metric
Auguste H. Gezalyan, David M. Mount
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…