Discrete & Computational Geometry

, Volume 17, Issue 3, pp 287–306

Inclusion-exclusion complexes for pseudodisk collections

Authors

  • H. Edelsbrunner
    • Department of Computer ScienceUniversity of Illinois
  • E. A. Ramos
    • Department of Computer ScienceUniversity of Illinois
Article

DOI: 10.1007/PL00009295

Cite this article as:
Edelsbrunner, H. & Ramos, E.A. Discrete Comput Geom (1997) 17: 287. doi:10.1007/PL00009295

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
$$\mu \left( { \cup B} \right) = \sum\limits_{\sigma \in 2^B - \left\{ {\not 0} \right\}} {( - 1)^{card \sigma - 1} \mu \left( { \cap \sigma } \right)} .$$
.

We show the existence of a two-dimensional abstract simplicial complex, χ ⊆ 2B, so the above relation holds even if χ is substituted for 2B. In addition, χ can be embedded in ℝ2 so its underlying space is homotopy equivalent to int ⋃ B, and the frontier of χ is isomorphic to the nerve of the set of boundary contributions.

Copyright information

© Springer-Verlag 1997