Classical curve theory in normed planes

Martini, Horst; Wu, Senlin

doi:10.1016/j.cagd.2014.03.003

Cited by 15 publications

(11 citation statements)

References 60 publications

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“…The geometry of finite dimensional real Banach (or normed) spaces, also called Minkowski geometry, is studied in the basic references [1], [6], [7], [8], [9] and [10], and recently it has strong relations to fields like optimization, discrete and computational geometry, convexity, convex and functional analysis, approximation theory and so on. Also from the viewpoint of differential geometry it is natural to develop geometric concepts for norms.…”

Section: Introductionmentioning

confidence: 99%

Curvature types of planar curves for gauges

2020

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In this paper results from the differential geometry of curves are extended from normed planes to gauge planes which are obtained by neglecting the symmetry axiom. Based on the gauge analogue of the notion of Birkhoff orthogonality from Banach space theory, we study all curvature types of curves in gauge planes, thus generalizing their complete classification for normed planes. We show that (as in the subcase of normed planes) there are four such types, and we call them analogously Minkowski, normal, circular, and arc-length curvature. We study relations between them and extend, based on this, also the notions of evolutes and involutes to gauge planes.Mathematics Subject Classification (2010). 46B20, 52A10, 52A21, 53A04, 53A35

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Section: Introductionmentioning

confidence: 99%

Curvature types of planar curves for gauges

2020

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“…In the first case comprehensive surveys exist (see, e.g., [3]), in the latter case these concepts are only widespread and partially hidden in the literature, and a systematic representation of their variety and relations to each other (as good starting point for further research) is still missing. We refer to the paper [31], where this unsatisfying situation regarding curve theory in normed planes is already described, and another related and comprehensive source (even referring to non-symmetric metrics, called gauges) is [24]. It is our goal to fill the gap outlined here for the particular viewpoint of curvature notions.…”

Section: Introductionmentioning

confidence: 99%

Concepts of curvatures in normed planes

Balestro¹,

Martini

Shonoda

2019

Expositiones Mathematicae

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“…The concept of curvature of regular curves in the Euclidean plane can be extended to normed planes in several ways (see [2] for an exposition of the topic, and [17] refers, more generally, to classical curve theory in such planes). One of the curvature types obtained by these extensions, namely the circular curvature, can be regarded as the inverse of the radius of a 2nd-order contact circle at the respective point of the curve.…”

Section: Introductionmentioning

confidence: 99%

On Legendre curves in normed planes

2018

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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 straightforwardly 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 extend the curvature notion in the usual sense. Going into 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.

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Classical curve theory in normed planes

Cited by 15 publications

References 60 publications

Curvature types of planar curves for gauges

Curvature types of planar curves for gauges

Concepts of curvatures in normed planes

On Legendre curves in normed planes

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