The size of spanning disks for polygonal curves
Skip to main content
Open Access Publications from the University of California

Department of Mathematics

Faculty bannerUC Davis

The size of spanning disks for polygonal curves

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

Published Web Location
No data is associated with this publication.

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.

Item not freely available? Link broken?
Report a problem accessing this item