Flat approximations of surfaces along curves

• Autorzy: Izumiya S. , Otani S.
• Języki publikacji: EN

Abstrakty

EN

We consider a developable surface tangent to a surface along a curve on the surface. We call it an osculating developable surface along the curve on the surface. We investigate the uniqueness and the singularities of such developable surfaces. We discover two new invariants of curves on a surface which characterize these singularities. As a by-product, we show that a curve is a contour generator with respect to an orthogonal projection or a central projection if and only if one of these invariants constantly equal to zero.

Słowa kluczowe

EN

flat approximations; curves on surfaces; contour generators; developable surface

PL

płaszczyzna przybliżona; krzywizna powierzchni

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2015
• Tom: Vol. 48, nr 2
• Strony: 217--241
• Opis fizyczny: Bibliogr. 8 poz., rys.

Twórcy

autor

Izumiya S.

• Department f Mathematics, Hokkaido University, Sapporo 060-0810, Japan

autor

Otani S.

• Kyoritsu Shuppan Co. Ltd, Kohinata Bunkyo-Ku, Tokyo 112-8700, Japan

Bibliografia

• [1] J. W. Bruce, P. J. Giblin, Curves and Singularities (second edition), Cambridge Univ. Press, 1992.
• [2] R. Cipolla, P. J. Giblin, Visual Motion of Curves and Surfaces, Cambridge Univ. Press, 2000.
• [3] P. Hartman, L. Nirenberg, On spherical image maps whose Jacobians do not change sign, Amer. J. Math. 81 (1959), 901–920.
• [4] G. Ishikawa, Singularities of flat extensions from generic surfaces with boundaries, Differ. Geom. Appl. 28 (2010), 341–354.
• [5] S. Izumiya, N. Takeuchi, Geometry of ruled surfaces, Applicable Mathematics in the Golden Age, Narosa Publishing House, New Delhi, 2003, 305–338.
• [6] S. Izumiya, K. Saji, M. Takahashi, Horospherical flat surfaces in hyperbolic 3-space, J. Math. Soc. Japan 62 (2010), 789–849.
• [7] D. Mond, Singularities of the tangent developable surface of a space curve, Quart. J. Math. 40 (1989), 79–91.
• [8] I. Vaisman, A First Course in Differential Geometry, Pure and Applied Mathematics, A Series of Monograph and Textbooks, Marcel Dekker, 1984.