Abstract
In this paper, we examine the Lorentzian similar plane curvesusing the hyperbolic structure and spherical arc length parameter. We classifyall self-similar Lorentzian plane curves and give formulas for pseudo shapecurvatures of evolute, involute and parallel curves of a nonnull plane curve
Highlights
A similarity transformation, which consists of a rotation, a translation and an isotropic scaling, is an automorphism preserving the angles and ratios between lengths
The similarity transformations are used in many areas of the pure and applied mathematics
Alcazar et al [14] presented a novel and deterministic algorithm to detect whether two given rational plane curves are related by means of a similarity transformation, which is a central question in Pattern Recognition
Summary
A similarity transformation (or similitude), which consists of a rotation, a translation and an isotropic scaling, is an automorphism preserving the angles and ratios between lengths. Encheva and Georgiev [20] studied the di¤erential geometric invariants of Frenet curves under a similarity map in 2-dimensional Euclidean space. Schwenk-Schellscmidt et al [3] characterized conic sections by using their spherical image in terms of appropriate eigenvalue equations of second order in the Euclidean plane They investigated the evolutes and involutes and their geometric properties in relation to the eigenvalue equations considered in [21]. Simsek and Özdemir [10] introduced the hyperbolic structure in the Cli¤ord algebra Cl1;1 and gave a formula for the curvature function of Lorentzian plane curves by means of the hyperbolic structure They [11] investigated the Lorentzian similarity geometry of nonnull Frenet curves in any dimensional space. We determine all nonnull self-similar plane curves and show that the evolutes and parallel curves of hyperbolic logarithmic spirals are self-similar and similar curves, respectively
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