Fat triangles determine linearly many holes

Jiri Matousek; Janos Pach; Micha Sharir; Shmuel Sifrony; Emo Welzl

doi:10.1137/S009753979018330X

Fat triangles determine linearly many holes

Jiri Matousek^*, Janos Pach, Micha Sharir, Shmuel Sifrony, Emo Welzl

^*Corresponding author for this work

Charles University

Research output: Contribution to journal › Article › peer-review

99 Scopus citations

Abstract

The authors show that for every fixed δ > 0 the following holds: if F is a union of n triangles, all of whose angles are at least δ, 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 δ). This latter complexity becomes linear if all triangles are of roughly the same size or if they are all infinite wedges.

Original language	English
Pages (from-to)	154-169
Number of pages	16
Journal	SIAM Journal on Computing
Volume	23
Issue number	1
DOIs	https://doi.org/10.1137/S009753979018330X
State	Published - 1994
Externally published	Yes

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title = "Fat triangles determine linearly many holes",

abstract = "The authors show that for every fixed δ > 0 the following holds: if F is a union of n triangles, all of whose angles are at least δ, 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 δ). This latter complexity becomes linear if all triangles are of roughly the same size or if they are all infinite wedges.",

author = "Jiri Matousek and Janos Pach and Micha Sharir and Shmuel Sifrony and Emo Welzl",

year = "1994",

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AU - Sifrony, Shmuel

AU - Welzl, Emo

PY - 1994

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N2 - The authors show that for every fixed δ > 0 the following holds: if F is a union of n triangles, all of whose angles are at least δ, 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 δ). This latter complexity becomes linear if all triangles are of roughly the same size or if they are all infinite wedges.

AB - The authors show that for every fixed δ > 0 the following holds: if F is a union of n triangles, all of whose angles are at least δ, 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 δ). This latter complexity becomes linear if all triangles are of roughly the same size or if they are all infinite wedges.

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Fat triangles determine linearly many holes

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