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Fat Triangles Determine Linearly Many Holes

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Emo Welzl (joint work with Jiri Matousek, Janos Pach, Micha Sharir, Shmuel Sifrony)
Institut für Informatik
Freie Universität Berlin
email: emo@inf.fu-berlin.de
Report B 92-13
June 92

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Server: fubinf.inf.fu-berlin.de
File: pub/reports/tr-b-92-13.ps.gz

We show that for every fixed $\delta>0$ the following holds:
If $F$ is a union of $n$ triangles, all of whose angles are at least
$\delta$, then the complement of $F$ has $O(n)$ connected components,
and the boundary of $F$ consists of $O(n \log \log n)$ straight
segments (where the constants of proportionality depend on $\delta$).
This latter complexity becomes linear if all triangles are
of roughly the same size or if they are all infinite wedges.