Sturmian Spirals in Lorentz-Minkowski Plane

Abstract

The Sturm spirals which can be introduced as those plane curves whose curvature radius is equal to the distance from the origin are embedded in to one parameter family of curves. In this paper, we consider the spacelike and timelike Sturmian spirals in Lorentz-Minkowski plane.

Curves in the Lorentz-Minkowski plane with curvature depending on their position

“…We call this curve the Lorentzian Norwich spiral ( Figure 5). In [7], the authors studied the Lorentzian curves satisfying the geometric condition:…”

“…The Norwich (or Sturm) spiral obviously corresponds to the case = μ 1. Making use of the co-moving frame associated with a curve in 2 , the cases > μ 1 and < < μ 0 1 (using the parameter t given by = s ρ t d d ) were completely determined in [7] in terms of elementary functions.…”

“…But now, concerning the Lorentzian version of Singer's problem, our knowledge is much more restricted in comparison with the Euclidean case. In fact, we can only mention the articles [7,8] in this line, both devoted to Sturmian spiral curves. The authors initiated in [9] the study of the spacelike and timelike curves in 2 satisfying = ( ) κ κ y or = ( ) κ κ x .…”

“…In this way, we are successful in Section 4 by studying the most regular curves in the Lorentz-Minkowski plane satisfying the geometric condition ( ) = + / κ ρ λ μ ρ 2 , ∈ λ μ , , generalizing the Sturmian spirals studied in [7] corresponding to = λ 0. Moreover, we find out the Lorentzian versions of some classical interesting curves in the Euclidean context corresponding to the sinusoidal spirals, characterized in…”

Elastic Sturmian spirals in the Lorentz-Minkowski plane

“…By going to the three-dimensional pseudo-Euclidean space one can also study the third class of elastic Sturmian spirals on the lightlike cone in the spirit of [8], but this task is beyond the present study. The description of the generalized Sturmian spirals in the Lorentz-Minkowski plane can be found in [9].…”