A note on lenses in arrangements of pairwise intersecting circles in the plane
The electronic journal of combinatorics, Tome 31 (2024) no. 2
Let ${\cal{F}}$ be a family of $n$ pairwise intersecting circles in the plane. We show that the number of lenses, that is convex digons, in the arrangement induced by ${\cal{F}}$ is at most $2n-2$. This bound is tight. Furthermore, if no two circles in ${\cal{F}}$ touch, then the geometric graph $G$ on the set of centers of the circles in ${\cal{F}}$ whose edges correspond to the lenses generated by ${\cal{F}}$ does not contain pairs of avoiding edges. That is, $G$ does not contain pairs of edges that are opposite edges in a convex quadrilateral. Such graphs are known to have at most $2n-2$ edges.
DOI :
10.37236/12054
Classification :
05C30, 05C75, 05C10
Mots-clés : digon, unit distance problem
Mots-clés : digon, unit distance problem
Affiliations des auteurs :
Rom Pinchasi 1
@article{10_37236_12054,
author = {Rom Pinchasi},
title = {A note on lenses in arrangements of pairwise intersecting circles in the plane},
journal = {The electronic journal of combinatorics},
year = {2024},
volume = {31},
number = {2},
doi = {10.37236/12054},
zbl = {1543.05087},
url = {http://geodesic.mathdoc.fr/articles/10.37236/12054/}
}
Rom Pinchasi. A note on lenses in arrangements of pairwise intersecting circles in the plane. The electronic journal of combinatorics, Tome 31 (2024) no. 2. doi: 10.37236/12054
Cité par Sources :