# Curvatures of conflict surfaces in Euclidean 3-space

Jorge Sotomayor; Dirk Siersma; Ronaldo Garcia

Banach Center Publications (1999)

- Volume: 50, Issue: 1, page 277-285
- ISSN: 0137-6934

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topSotomayor, Jorge, Siersma, Dirk, and Garcia, Ronaldo. "Curvatures of conflict surfaces in Euclidean 3-space." Banach Center Publications 50.1 (1999): 277-285. <http://eudml.org/doc/209014>.

@article{Sotomayor1999,

abstract = {This article extends to three dimensions results on the curvature of the conflict curve for pairs of convex sets of the plane, established by Siersma [3]. In the present case a conflict surface arises, equidistant from the given convex sets. The Gaussian, mean curvatures and the location of umbilic points on the conflict surface are determined here. Initial results on the Darbouxian type of umbilic points on conflict surfaces are also established. The results are expressed in terms of the principal directions and on the curvatures of the borders of the given convex sets.},

author = {Sotomayor, Jorge, Siersma, Dirk, Garcia, Ronaldo},

journal = {Banach Center Publications},

keywords = {conflict curve; conflict surface; convex sets; curvatures; umbilic points},

language = {eng},

number = {1},

pages = {277-285},

title = {Curvatures of conflict surfaces in Euclidean 3-space},

url = {http://eudml.org/doc/209014},

volume = {50},

year = {1999},

}

TY - JOUR

AU - Sotomayor, Jorge

AU - Siersma, Dirk

AU - Garcia, Ronaldo

TI - Curvatures of conflict surfaces in Euclidean 3-space

JO - Banach Center Publications

PY - 1999

VL - 50

IS - 1

SP - 277

EP - 285

AB - This article extends to three dimensions results on the curvature of the conflict curve for pairs of convex sets of the plane, established by Siersma [3]. In the present case a conflict surface arises, equidistant from the given convex sets. The Gaussian, mean curvatures and the location of umbilic points on the conflict surface are determined here. Initial results on the Darbouxian type of umbilic points on conflict surfaces are also established. The results are expressed in terms of the principal directions and on the curvatures of the borders of the given convex sets.

LA - eng

KW - conflict curve; conflict surface; convex sets; curvatures; umbilic points

UR - http://eudml.org/doc/209014

ER -

## References

top- [1] G. Darboux, Sur la forme des lignes de courbure dans le voisinage d'un ombilic, in: Leçons sur la théorie générale des surfaces, Vol. IV, Gauthier-Villars, Paris, 1896, 448-465.
- [2] I. R. Porteous, Geometric Differentiation for the Intelligence of Curves and Surfaces, Cambridge Univ. Press, Cambridge, 1994. Zbl0806.53001
- [3] D. Siersma, Properties of conflict sets in the plane, this volume. Zbl0959.53003
- [4] J. Sotomayor and C. Gutiérrez, Structurally stable configurations of lines of principal curvature, Astérisque 98-99 (1982), 195-215. Zbl0521.53003
- [5] J. Sotomayor and C. Gutiérrez, Lines of Curvature and Umbilic Points on Surfaces. Text of Course delivered at the XVIII Brazilian Mathematics Colloquium, IMPA, Rio de Janeiro, 1991. Zbl1160.53304
- [6] M. Spivak, A Comprehensive Introduction to Differential Geometry, vols. 1, 3, Publish or Perish, Wilmington, 1979.
- [7] J. B. Wilker, Equidistant sets and their connectivity properties, Proc. Amer. Math. Soc. 47 (1975), 446-452. Zbl0295.50017

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