Mathematics > Metric Geometry
[Submitted on 26 Jun 2019 (v1), last revised 18 Aug 2023 (this version, v3)]
Title:Voronoi Cells in Metric Algebraic Geometry of Plane Curves
View PDFAbstract:Voronoi cells of varieties encode many features of their metric geometry. We prove that each Voronoi or Delaunay cell of a plane curve appears as the limit of a sequence of cells obtained from point samples of the curve. We use this result to study metric features of plane curves, including the medial axis, curvature, evolute, bottlenecks, and reach. In each case, we provide algebraic equations defining the object and, where possible, give formulas for the degrees of these algebraic varieties. We show how to identify the desired metric feature from Voronoi or Delaunay cells, and therefore how to approximate it by a finite point sample from the variety.
Submission history
From: Madeline Brandt [view email][v1] Wed, 26 Jun 2019 20:29:17 UTC (3,840 KB)
[v2] Fri, 1 Nov 2019 21:00:06 UTC (3,840 KB)
[v3] Fri, 18 Aug 2023 14:04:47 UTC (4,043 KB)
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