Evolutes of fronts in the Euclidean plane

T. Fukunaga and M. Takahashi

Journal of Singularities
volume 10 (2014), 92-107

Received 26 December 2012. Received in revised form 30 December 2013.

DOI: 10.5427/jsing.2014.10f

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The evolute of a regular curve in the Euclidean plane is given by not only the caustics of the regular curve, envelope of normal lines of the regular curve, but also the locus of singular points of parallel curves. In general, the evolute of a regular curve has singularities, since such points correspond to vertices of the regular curve and there are at least four vertices for simple closed curves. If we repeat an evolute, we cannot define the evolute at a singular point. In this paper, we define an evolute of a front and give properties of such an evolute by using a moving frame along a front and the curvature of the Legendre immersion. As applications, repeated evolutes are useful to recognize the shape of curves.


evolute, parallel curve, front, Legendre immersion

Mathematical Subject Classification:

58K05, 53A04, 57R45

Author(s) information:

T. Fukunaga M. Takahashi
Kyushu Sangyo University Muroran Institute of Technology
Fukuoka 813-8503, Japan. Muroran 050-8585, Japan
email: tfuku@ip.kyusan-u.ac.jp email: masatomo@mmm.muroran-it.ac.jp