Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-24T08:51:47.506Z Has data issue: false hasContentIssue false
  • English
  • Français

Evolutes and focal surfaces of framed immersions in the Euclidean space

Part of: Classical differential geometry Differential topology Theory of singularities and catastrophe theory

Published online by Cambridge University Press:  26 January 2019

Shun'ichi Honda  and
Masatomo Takahashi
Shun'ichi Honda
Affiliation:
Department of Mathematics, Hokkaido University Sapporo 060-0810, Japan ([email protected])
Masatomo Takahashi
Affiliation:
Muroran Institute of Technology, Muroran050-8585, Japan ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a smooth curve with singular points in the Euclidean space. As a smooth curve with singular points, we have introduced a framed curve or a framed immersion. A framed immersion is a smooth curve with a moving frame and the pair is an immersion. We define an evolute and a focal surface of a framed immersion in the Euclidean space. The evolutes and focal surfaces of framed immersions are generalizations of each object of regular space curves. We give relationships between singularities of the evolutes and of the focal surfaces. Moreover, we consider properties of the evolutes, focal surfaces and repeated evolutes.

Keywords

evolutefocal surfaceframed immersionsingular point

MSC classification

Primary: 57R45: Singularities of differentiable mappings
Secondary: 58K05: Critical points of functions and mappings 53A04: Curves in Euclidean space 53A05: Surfaces in Euclidean space
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 1 , February 2020 , pp. 497 - 516
Copyright
Copyright © Royal Society of Edinburgh 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Arnol'd, V. I.. Singularities of caustics and wave fronts. Mathematics and its applications,vol. 62 (Dordrecht: Kluwer Academic Publishers: 1990).CrossRefGoogle Scholar
2Bishop, R. L.. There is more than one way to frame a curve. Amer. Math. Monthly 82 (1975), 246–251.CrossRefGoogle Scholar
3Bruce, J. W. and Giblin, P. J.. Curves and singularities. A geometrical introduction to singularity theory, 2nd edn (Cambridge: Cambridge University Press, 1992).CrossRefGoogle Scholar
4Fuchs, D.. Evolutes and involutes of spatial curves. Amer. Math. Monthly 120 (2013), 217–231.CrossRefGoogle Scholar
5Fukunaga, T. and Takahashi, M.. Existence and uniqueness for Legendre curves. J. Geom. 104 (2013), 297–307.CrossRefGoogle Scholar
6Fukunaga, T. and Takahashi, M.. Existence conditions of framed curves for smooth curves. J. Geom. 108 (2017), 763–774.CrossRefGoogle Scholar
7Gray, A., Abbena, E. and Salamon, S.. Modern differential geometry of curves and surfaces with Mathematica, 3rd edn, Studies in Advanced Mathematics (Boca Raton, FLChapman and Hall/CRC, 2006).Google Scholar
8Honda, S.. Rectifying developable surfaces of framed base curves and framed helices. to appear in Advanced Studies in Pure Mathematics (2017).Google Scholar
9Honda, S. and Takahashi, M.. Framed curves in the Euclidean space. Adv. Geom. 16 (2016), 265–276.CrossRefGoogle Scholar
10Honda, S. and Takahashi, M.. Evolutes of framed immersions in the Euclidean space. Hokkaido Univ. Preprint Series in Math. 1095, URL: https://eprints.lib.hokudai.ac.jp/dspace/handle/2115/69899.Google Scholar
11Izumiya, S. and Otani, S.. Flat approximations of surfaces along curves. Demonst. Math. 48 (2015), 217–241.Google Scholar
12Kokubu, M., Rossman, W., Saji, K., Umehara, M. and Yamada, K.. Singularities of flat fronts in hyperbolic space. Pacific J. Math. 221 (2005), 303–351.CrossRefGoogle Scholar
13Porteous, I. R.. Geometric differentiation for the intelligence of curves and surfaces, 2nd edn (Cambridge: Cambridge University Press, 2001).Google Scholar
14Romero-Fuster, M. C. and Sanabria-Codesal, E.. Generalized evolutes, vertices and conformal invariants of curves in ℝn+1. Indag. Math. N. S. 10 (1999), 297–305.CrossRefGoogle Scholar
15Uribe-Vargas, R.. On vertices, focal curvatures and differential geometry of space curves. Bull. Braz. Math. Soc. (N.S.) 36 (2005), 285–307.CrossRefGoogle Scholar