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Documents authored by Gezalyan, Auguste H.

Document
Media Exposition
DOI: 10.4230/LIPIcs.SoCG.2024.92
Ipelets for the Convex Polygonal Geometry (Media Exposition)

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

Published in: LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

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 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).
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Nithin Parepally, Ainesh Chatterjee, Auguste H. Gezalyan, Hongyang Du, Sukrit Mangla, Kenny Wu, Sarah Hwang, and David M. Mount. Ipelets for the Convex Polygonal Geometry (Media Exposition). In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 92:1-92:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{parepally_et_al:LIPIcs.SoCG.2024.92,
  author =	{Parepally, Nithin and Chatterjee, Ainesh and Gezalyan, Auguste H. and Du, Hongyang and Mangla, Sukrit and Wu, Kenny and Hwang, Sarah and Mount, David M.},
  title =	{{Ipelets for the Convex Polygonal Geometry}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{92:1--92:7},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.92},
  URN =		{urn:nbn:de:0030-drops-200375},
  doi =		{10.4230/LIPIcs.SoCG.2024.92},
  annote =	{Keywords: Hilbert metric, Macbeath Regions, Polar Bodies, Convexity}
}
Document
DOI: 10.4230/LIPIcs.SWAT.2024.25
Delaunay Triangulations in the Hilbert Metric

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

Published in: LIPIcs, Volume 294, 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)

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 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 + log³ 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 log² m), where h is the number of points on the hull’s boundary.
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Auguste H. Gezalyan, Soo H. Kim, Carlos Lopez, Daniel Skora, Zofia Stefankovic, and David M. Mount. Delaunay Triangulations in the Hilbert Metric. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{gezalyan_et_al:LIPIcs.SWAT.2024.25,
  author =	{Gezalyan, Auguste H. and Kim, Soo H. and Lopez, Carlos and Skora, Daniel and Stefankovic, Zofia and Mount, David M.},
  title =	{{Delaunay Triangulations in the Hilbert Metric}},
  booktitle =	{19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
  pages =	{25:1--25:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-318-8},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{294},
  editor =	{Bodlaender, Hans L.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.25},
  URN =		{urn:nbn:de:0030-drops-200657},
  doi =		{10.4230/LIPIcs.SWAT.2024.25},
  annote =	{Keywords: Delaunay Triangulations, Hilbert metric, convexity, randomized algorithms}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2023.35
Voronoi Diagrams in the Hilbert Metric

Authors: Auguste H. Gezalyan and David M. Mount

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

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 m-sided convex polygon Ω in the plane, we present two randomized incremental algorithms and one deterministic algorithm. The first randomized algorithm and the deterministic algorithm compute the Voronoi diagram of a set of n point sites. The second randomized algorithm extends this to compute the Voronoi diagram of the set of n sites, each of which may be a point or a line segment. Our algorithms all run in expected time O(m n log n). The algorithms use O(m n) storage, which matches the worst-case combinatorial complexity of the Voronoi diagram in the Hilbert metric.
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Auguste H. Gezalyan and David M. Mount. Voronoi Diagrams in the Hilbert Metric. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 35:1-35:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{gezalyan_et_al:LIPIcs.SoCG.2023.35,
  author =	{Gezalyan, Auguste H. and Mount, David M.},
  title =	{{Voronoi Diagrams in the Hilbert Metric}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{35:1--35:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.35},
  URN =		{urn:nbn:de:0030-drops-178851},
  doi =		{10.4230/LIPIcs.SoCG.2023.35},
  annote =	{Keywords: Voronoi diagrams, Hilbert metric, convexity, randomized algorithms}
}
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