On Legendre curves in normed planes

Vitor Balestro, Horst Martini and Ralph Teixeira

Vol. 297 (2018), No. 1, 1–27

DOI: 10.2140/pjm.2018.297.1

Abstract

Legendre curves are smooth plane curves which may have singular points, but still have a well defined smooth normal (and corresponding tangent) vector field. Because of the existence of singular points, the usual curvature concept for regular curves cannot be extended to these curves. However, Fukunaga and Takahashi defined and studied functions that play the role of curvature functions of a Legendre curve, and whose ratio extends the curvature notion in the usual sense. In the same direction, our paper is devoted to the extension of the concept of circular curvature from regular to Legendre curves, but additionally referring not only to the Euclidean plane. For the first time we will extend the concept of Legendre curves to normed planes. Generalizing in such a way the results of the mentioned authors, we define new functions that play the role of circular curvature of Legendre curves, and tackle questions concerning existence, uniqueness, and invariance under isometries for them. Using these functions, we study evolutes, involutes, and pedal curves of Legendre curves for normed planes, and the notion of contact between such curves is correspondingly extended, too. We also provide new ways to calculate the Maslov index of a front in terms of our new curvature functions. It becomes clear that an inner product is not necessary in developing the theory of Legendre curves. More precisely, only a fixed norm and the associated orthogonality (of Birkhoff type) are necessary.

Keywords

Birkhoff orthogonality, cusps, evolutes, fronts, immersions, involutes, Legendre curves, Maslov index, Minkowski geometry, normed planes, singularities

Mathematical Subject Classification 2010

Primary: 46B20, 51L10, 52A21, 53A04, 53A35

Milestones

Received: 17 April 2017

Revised: 2 March 2018

Accepted: 25 May 2018

Published: 7 October 2018