Curvatures of conflict surfaces in Euclidean 3-space

• Autorzy: Jorge Sotomayor , Dirk Siersma , Ronaldo Garcia
• Języki publikacji: EN

Abstrakty

EN

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.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 1999
• Tom: 50
• Numer: 1
• Strony: 277-285

Daty

wydano

1999

Twórcy

autor

Jorge Sotomayor

• Instituto de Matemática e Estatí stica, Universidade de São Paulo, Caixa Postal 66281, São Paulo, SP, Cep 05315-970, Brazil

autor

Dirk Siersma

• Mathematisch Instituut, Universiteit Utrecht, Postbus 80.010, 3508 TA Utrecht, The Netherlands

autor

Ronaldo Garcia

• Instituto de Matemática e Estatística, Universidade Federal de Goiás, Caixa Postal 131, Goiânia, GO, Cep 74001-970, Brazil

Bibliografia

• [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.
• [3] D. Siersma, Properties of conflict sets in the plane, this volume.
• [4] J. Sotomayor and C. Gutiérrez, Structurally stable configurations of lines of principal curvature, Astérisque 98-99 (1982), 195-215.
• [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.
• [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.