|
Organizers |
On the Intersection of Simple Curves
by
Ivan Gotchev
University of Maine
Coauthors: Robert Franzosa (University of Maine)
Definition 1. Any curve in R2 topologically equivalent to a closed line segment or to a 1-sphere is called simple.
Definition 2. A pair of curves has finite configuration if their intersection has finitely many components.
Theorem (Shöenflies). Let L be a 1-sphere in R2. Then every homeomorphism of L into R2 can be extended to give a homeomorphism of R2 onto R2.
Let (L1, M1) and (L2, M2) be two pairs of simple curves, each having finite configuration. Using generalizations and modifications of the Shöenflies theorem we find necessary and sufficient conditions on the nature of the intersections L1 \cap M1 and L2 \cap M2, for there to be a homeomorphism f:R2 --> R2 such that f(L1)=L2 and f(M1)=M2.
Our work has been motivated by the need in the field of Geographic Information Systems to be able to qualitatively distinguish and efficiently model the different ways that geographic regions can lie in relation to each other.
Date received: May 15, 2001
Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagw-51.