Curvature types of planar curves for gauges

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...

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Bibliographic Details
Published in:Journal of geometry 2020, Vol.111 (1), Article 12
Main Authors: Balestro, Vitor, Martini, Horst, Sakaki, Makoto
Format: Article
Language:English
Subjects:
Banach spaces
Curvature
Curves
Differential geometry
Gauges
Geometry
Involutes
Mathematics
Mathematics and Statistics
Orthogonality
Planes
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Summary: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.
ISSN:0047-2468
1420-8997
DOI:10.1007/s00022-020-0526-7