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Evolutes and focal surfaces of framed immersions in the Euclidean space

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

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 (s-honda@math.sci.hokudai.ac.jp)
Masatomo Takahashi
Affiliation:
Muroran Institute of Technology, Muroran050-8585, Japan (masatomo@mmm.muroran-it.ac.jp)
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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

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