Generalized Evolutes of Planar Curves and its Properties

Document Type : Original Paper

Authors

1 University of Guilan

2 Departamento de Matematica, Universidade Federal de Vicosa, Brazil

Abstract

In this ‎paper‎, we present a generalization of the evolute of a curve in the plane and study its geometry. Consider a curve in the plane, denoted by ‎‎$‎\gamma‎$‎, with curvature ‎$‎\kappa‎$‎. If ‎‎f‎‎ is a smooth real-valued function, we define the spatial curve ‎$‎\gamma_{f}‎$‎ in such a way that it ‎is the ‎generalization of the evolute of ‎$‎\gamma‎$‎. The process of obtaining this curve is through the introduction of an angular ‎surface‎. Theorem 2 shows that the singular points of ‎$‎\gamma_{f}‎$‎ correspond to the vertices of ‎$‎\gamma‎$‎ and are independent of the choice of the function ‎$‎f‎$‎. In such points, the generalized evolute has a cusp singularity if and only if the curve ‎$‎\gamma‎$‎ has a regular vertex at ‎$‎s=s_0‎$‎. Furthermore, we investigate the contact of the generalized evolute ‎$‎\gamma_f‎$‎ with a sphere and a plane. Additionally, by introducing a certain type of parallel curve, we study its geometric properties.

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History

  • Receive Date: 09 February 2024
  • Accept Date: 02 December 2024

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