Properties of Conflict Sets in the Plane

Dirk Siersma

Banach Center Publications (1999)

  • Volume: 50, Issue: 1, page 267-276
  • ISSN: 0137-6934

Abstract

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This paper studies the smoothness and the curvature of conflict sets of the distance function in the plane. Conflict sets are also well known as 'bisectors'. We prove smoothness in the case of two convex sets and give a formula for the curvature. We generalize moreover to weighted distance functions, the so-called Johnson-Mehl model.

How to cite

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Siersma, Dirk. "Properties of Conflict Sets in the Plane." Banach Center Publications 50.1 (1999): 267-276. <http://eudml.org/doc/209013>.

@article{Siersma1999,
abstract = {This paper studies the smoothness and the curvature of conflict sets of the distance function in the plane. Conflict sets are also well known as 'bisectors'. We prove smoothness in the case of two convex sets and give a formula for the curvature. We generalize moreover to weighted distance functions, the so-called Johnson-Mehl model.},
author = {Siersma, Dirk},
journal = {Banach Center Publications},
keywords = {conflict set; bisector; Johnson-Mehl model},
language = {eng},
number = {1},
pages = {267-276},
title = {Properties of Conflict Sets in the Plane},
url = {http://eudml.org/doc/209013},
volume = {50},
year = {1999},
}

TY - JOUR
AU - Siersma, Dirk
TI - Properties of Conflict Sets in the Plane
JO - Banach Center Publications
PY - 1999
VL - 50
IS - 1
SP - 267
EP - 276
AB - This paper studies the smoothness and the curvature of conflict sets of the distance function in the plane. Conflict sets are also well known as 'bisectors'. We prove smoothness in the case of two convex sets and give a formula for the curvature. We generalize moreover to weighted distance functions, the so-called Johnson-Mehl model.
LA - eng
KW - conflict set; bisector; Johnson-Mehl model
UR - http://eudml.org/doc/209013
ER -

References

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  1. [Be] M. Berger, Geometry I and II, Universitext, Springer, Berlin, 1987. 
  2. [BG] J. W. Bruce, P. J. Giblin, Curves and Singularities, Cambridge Univ. Press, Cambridge, 1984. 
  3. [Cox] H. S. M. Coxeter, Introduction to Geometry, Wiley, New York, 1961. 
  4. [Go] A. Goddijn, Smoothness Properties of Conflict Sets (in Dutch), Nascholingscursus Achtergronden van de Meetkunde, Utrecht, 1997 (translation in preparation). 
  5. [Ma] M. van Manen, Thesis Project on Conflict Sets (in progress), Utrecht University. 
  6. [Mi] J. Milnor, Morse Theory, Ann. of Math. Stud. 51, Princeton Univ. Press, Princeton, 1963. 
  7. [Ok] A. Okabe, B. Boots, K. Sugihara, Spatial Tesselations: Concepts and Applications of Voronoi Diagrams, Wiley Ser. Probab. Math. Statist. Appl. Probab. Statist., Wiley, Chichester, 1992. Zbl0877.52010
  8. [Po] I. R. Porteous, Geometric Differentiation for the Intelligence of Curves and Surfaces, Cambridge Univ. Press, Cambridge, 1994. Zbl0806.53001
  9. [SSG] J. Sotomayor, D. Siersma, R. Garcia, Curvatures of conflict surfaces in Euclidean 3-space, this volume. Zbl0983.53003

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