ISSN:
1432-0444
Source:
Springer Online Journal Archives 1860-2000
Topics:
Computer Science
,
Mathematics
Notes:
Abstract. Let B be a finite pseudodisk collection in the plane. By the principle of inclusion—exclusion, the area or any other measure of the union is $$ \mu\left( {{\bigcup}{\}}, B \right) = \sum_{{\sigma} \in 2^B-\{\emptyset\}} (-1)^{{\rm card\,}{\simplex}-1} \mu\left(\bigcap {\sigma}\right). $$ We show the existence of a two-dimensional abstract simplicial complex, ${{\cal X}} \subseteq 2^B$ , so the above relation holds even if ${\cal X}$ is substituted for 2 B . In addition, ${\cal X}$ can be embedded in R 2 so its underlying space is homotopy equivalent to ${\rm int\,}{\union\,{B}}$ , and the frontier of ${\cal X}$ is isomorphic to the nerve of the set of boundary contributions.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/PL00009295
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