The size of spanning disks for polygonal curves

1999/06/30 by Joel Hass, Jack Snoeyink, William P. Thurston
Mathematics · #math.GT #msc:57M25 #msc:57R05 #msc:68U05

paper · pdf

published as Discrete and Computational Geometry 29 (2003) 1--17. · 17 pages, 16 figures

arxiv created 2002/03/23 · arxiv updated 2009/11/30

Abstract

Let K be a closed polygonal curve in \RR³ consisting of n line segments. Assume that K is unknotted, so that it is the boundary of an embedded disk in \RR³. This paper considers the question: How many triangles are needed to triangulate a Piecewise-Linear (PL) spanning disk of K? The main result exhibits a family of unknotted polygons with n edges, n → ∞, such that the minimal number of triangles needed in any triangulated spanning disk grows exponentially with n. For each integer n ≥ 0, there is a closed, unknotted, polygonal curve K_n in R³ having less than 10n+9 edges, with the property that any Piecewise-Linear triangulated disk spanning the curve contains at least 2^n-1 triangles.

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