The size of spanning disks for polygonal curves
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Open Access Publications from the University of California

## The size of spanning disks for polygonal curves

• Author(s): Hass, Joel
• Snoeyink, Jack
• Thurston, William P.
• et al.

## Published Web Location

https://arxiv.org/pdf/math/9906197.pdf
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Abstract

Let $K$ be a closed polygonal curve in $\RR^3$ consisting of $n$ line segments. Assume that $K$ is unknotted, so that it is the boundary of an embedded disk in $\RR^3$. 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, $n \to \infty$, such that the minimal number of triangles needed in any triangulated spanning disk grows exponentially with $n$. For each integer $n \ge 0$, there is a closed, unknotted, polygonal curve $K_n$ in $R^3$ 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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