Download PDF - The rectifying developable and the spherical Darboux image of a space curveAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

The rectifying developable and the spherical Darboux image of a space curve

Authors 1, 2, 2

Affiliations

  1. Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
  2. Department of Mathematics, Ochanomizu University, Bunkyou-ku Otsuka Tokyo 112-8610, Japan

Abstract

In this paper we study singularities of certain surfaces and curves associated with the family of rectifying planes along space curves. We establish the relationships between singularities of these subjects and geometric invariants of curves which are deeply related to the order of contact with helices.

Bibliography

  1. J. W. Bruce, P. J. Giblin, Curves and Singularities, 2nd ed., Cambridge Univ. Press, Cambridge, 1992.
  2. M. P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, Englewood Cliffs, 1976.
  3. J. P. Cleave, The form of the tangent developable at points of zero torsion on space curves, Math. Proc. Cambridge Philos. Soc. 88 (1980), 403-407.
  4. G. Ishikawa, Determinacy of envelope of the osculating hyperplanes to a curve, Bull. London Math. Soc. 25 (1993), 603-610.
  5. G. Ishikawa, Developable of a curve and its determinacy relative to the osculation-type, Quart. J. Math. Oxford Ser. (2) 46 (1995), 437-451.
  6. G. Ishikawa, Topological classification of the tangent developable of space curves, Hokkaido Univ. Preprint Series 341 (1996).
  7. J. Koenderink, Solid Shape, MIT Press, Cambridge, MA, 1990.
  8. D. Mond, On the tangent developable of a space curve, Math. Proc. Cambridge Philos. Soc. 91 (1982), 351-355.
  9. D. Mond, Singularities of the tangent developable surface of a space curve, Quart. J. Math. Oxford Ser. (2) 40 (1989), 79-91.
  10. I. R. Porteous, The normal singularities of submanifold, J. Differential Geom. 5 (1971), 543-564.
  11. I. R. Porteous, Geometric Differentiation for the Intelligence of Curves and Surfaces, Cambridge Univ. Press, Cambridge, 1994.
  12. O. P. Shcherbak, Projectively dual space curves and Legendre singularities (in Russian), Trudy Tbiliss. Univ. 232-233 (1982), 280-336; English translation: Selecta Math. Soviet. 5 (1986), 391-421.
Banach Center Publications
1999/50/1
Export to PBN, Opens in new tab
Pages:
137-149
Main language of publication
English
Published
1999
Exact and natural sciences