Transitions between 4-intersection values of planar regions

Kathleen Bell

United States

Western Kentucky University

Department of Mathematics

Thomas A. Richmond

https://orcid.org/0000-0003-1883-8146

United States

Western Kentucky University

Department of Mathematics

Submitted: 2016-10-13

|

Accepted: 2017-01-25

|

Published: 2017-04-03

DOI: https://doi.org/10.4995/agt.2017.6716
Funding Data

Downloads

Keywords:

upper semicontinuous, lower semicontinuous, Vietoris topology, spatial region, 4-intersection value

Supporting agencies:

This research was not funded

Abstract:

If A(t) and B(t) are subsets of the Euclidean plane which are continuously morphing, we investigate the question of whether they may morph directly from being disjoint to overlapping so that the boundary and interior of A(t) both intersect the boundary and interior of B(t) without first passing through a state in which only their boundaries intersect.  More generally, we consider which 4-intersection values---binary 4-tuples specifying whether the boundary and interior of A(t) intersect the boundary and interior of B(t)---are adjacent to which in the sense that one may morph into the other without passing through a third value.  The answers depend on what forms the regions A(t) and B(t) are allowed to assume and on the definition of continuous morphing of the sets.
Show more Show less

References:

C. Adams and R. Franzosa, Introduction to Topology: Pure and Applied, Pearson Prentice Hall, Upper Saddle River, NJ, 2008.

J. Chen, C. Li, Z. Li and C. Gold, A Voronoi-based 9-intersection model for spatial relations, International Journal for Geographical Information Science 15, no. 3 (2001), 201-220.

https://doi.org/10.1080/13658810151072831

E. Clementini, J. Sharma and M. Egenhofer, Modeling topological spatial relations: strategies for query processing, Computers and Graphics 18, no. 6 (1994), 815-822.

https://doi.org/10.1016/0097-8493(94)90007-8

M. Egenhofer and K. Al-Taha, Reasoning about gradual changes of topological relationships, in: A Frank, I. Campari, and U. Valueentini (Eds.), Theories and Models of Spatio-Temporal Reasoing in Geographic Space, Pisa, Italy, Lecture Notes in Computer Science, 639. New York: Springer-Veralg, 1992, pp. 196-219.

https://doi.org/10.1007/3-540-55966-3_12

M. Egenhofer, E. Clementini and P. di Felice, Topological relations between regions with holes, International Journal for Geographical Information Systems 8, no. 2 (1994), 129-144.

https://doi.org/10.1080/02693799408901990

M. Egenhofer and R. Franzosa, Point-set topological spatial relations, International Journal for Geographical Information Systems 5, no. 2 (1991) 161-174.

https://doi.org/10.1080/02693799108927841

M. Egenhofer and R. Franzosa, On equivalence of topological relations, International Journal for Geographical Information Systems 8, no. 6 (1994), 133-152.

S. Francaviglia, A. Lechicki and S. Levi, Quasi-uniformization of hyperspaces and convergence of nets of semicontinuous multifunctions, J. Math. Anal. Appl. 112 (1985), 347-370.

https://doi.org/10.1016/0022-247X(85)90246-X

N. M. Gotts, An axiomatic approach to topology for spatial information systems, Research Report 96.25, University of Leeds, School of Computer Science, 1996.

E. Klein and A. C. Thompson, Theory of Correspondences: Including Applications to Economics. Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley and Sons, New York, 1984.

Y. Kurata and M. Egenhofer, The e 9+-intersectionintersection for topological relations between a directed line segment and a region, in: Proceedings of the 1st Workshop on Behavioral Monitoring and Interpretation. TZI-Bericht, Technologie-Zentrum Informatik, Universität Bremen, Germany, Vol. 42, 2007, pp. 62-76.

D. Mark and M. Egenhofer, An evaluation of the 9-intersection for region-line relations, San Jose, CA: GIS/LIS '92, (1992) 513-521.

K. Nedas, M. Egenhofer and D. Wilmsen, Metric details of topological line-line relations, International Journal for Geographical Information Science 21, no. 1 (2007), 21-48.

https://doi.org/10.1080/13658810600852164

A. J. Roy and J. G. Stell, Indeterminate regions, Internat. J. Approximate Reasoning 27 (2001), 205-234.

https://doi.org/10.1016/S0888-613X(01)00033-0

T. Smith and K. Park, Algebraic approach to spatial reasoning, International Journal for Geographical Information Systems 6, no. 3 (1992), 177-192.

https://doi.org/10.1080/02693799208901904

J. Wu, C. Claramunt and M. Deng, Modelling movement patterns using topological relations between a directed line and a region, IWGS '14 Proceedings of the 5th ACM SGISPATIAL International Workshop on GeoStreaming. New York: ACM, 2014, 43-52.

Show more Show less