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Minkowski Symmetry Sets of Plane Curves

Part of: Differential topology Classical differential geometry Local differential geometry

Published online by Cambridge University Press:  19 May 2016

Graham Mark Reeve  and
Farid Tari
Graham Mark Reeve
Affiliation:
Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Avenida Trabalhador São-carlense, 400 – Centro, CEP: 13566–590, São Carlos, São Paulo, Brazil (grahamreeve86@gmail.com; faridtari@icmc.usp.br)
Farid Tari
Affiliation:
Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Avenida Trabalhador São-carlense, 400 – Centro, CEP: 13566–590, São Carlos, São Paulo, Brazil (grahamreeve86@gmail.com; faridtari@icmc.usp.br)
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Abstract

We study the Minkowski symmetry set of a closed smooth curve γ in the Minkowski plane. We answer the following question, which is analogous to one concerning curves in the Euclidean plane that was treated by Giblin and O’Shea (1990): given a point p on γ, does there exist a bi-tangent pseudo-circle that is tangent to γ both at p and at some other point q on γ? The answer is yes, but as pseudo-circles with non-zero radii have two branches (connected components) it is possible to refine the above question to the following one: given a point p on γ, does there exist a branch of a pseudo-circle that is tangent to γ both at p and at some other point q on γ? This question is motivated by the earlier quest of Reeve and Tari (2014) to define the Minkowski Blum medial axis, a counterpart of the Blum medial axis of curves in the Euclidean plane.

Keywords

curvesMinkowski planesymmetry setsevolutescausticssingularities

MSC classification

Secondary: 53A35: Non-Euclidean differential geometry 53B30: Lorentz metrics, indefinite metrics 57R45: Singularities of differentiable mappings
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 60 , Issue 2 , May 2017 , pp. 461 - 480
Copyright
Copyright © Edinburgh Mathematical Society 2017 

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