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formerly Central European Journal of Mathematics

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Volume 14, Issue 1


Volume 13 (2015)

Elastic Sturmian spirals in the Lorentz-Minkowski plane

Ali Uçum / Kazım İlarslan / Ivaïlo M. Mladenov
  • Corresponding author
  • Institute of Biophysics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Block 21, 1113, Sofia, Bulgaria
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Published Online: 2016-12-30 | DOI: https://doi.org/10.1515/math-2016-0103


In this paper we consider some elastic spacelike and timelike curves in the Lorentz-Minkowski plane and obtain the respective vectorial equations of their position vectors in explicit analytical form. We study in more details the generalized Sturmian spirals in the Lorentz-Minkowski plane which simultaneously are elastics in this space.

Keywords: Elastic curves; Lorentz-Minkowski plane; Sturmian spirals

MSC 2010: 53A35; 53B30; 53B50

1 Introduction

The curvature κ, of a given curve γ : IMn in a Riemannian manifold, can be interpreted as the tension that γ receives at each point as a result of the way it is immersed in the surrounding space. In 1740, Bernoulli proposed a simple geometric model for an elastic curve in E2, according to which an elastic curve or elastica is a critical point of the elastic energy functional γκ2.. Elastic curves in E2 were already classified by Euler in 1743 but it was not until 1928 that they were also studied in E3 by Radon, who derived the Euler-Lagrange equations and showed that they can be integrated explicitly. The elastica problem in real space forms has been recently considered using different approaches (see [15] and [6]).

Are there other interesting elastic curves? This question has been answered affirmatively by Marinov et al. in [7] where the Sturmian spirals in the Euclidean plane were described explicitly. There they have studied also the curves which belong to the class of the so called generalized Sturmian spirals which obey to the elastica equation. Eventually, they found analytical formulas for their parameterizations and presented a few illustrative plots.

Here, we are interested in elastic spacelike and timelike curves with tension in the Lorentz-Minkowski plane and derive the explicit vectorial equations of the respective generalized Sturmian spirals which belong to this class. 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].

2 Preliminaries

The Lorentz-Minkowski plane E12 is the Euclidean plane R2 equipped with an indefinite flat metric g given by the infinitesimal distance


where (x1, x2) are the rectangular coordinates in E12. Recall that any given vector vE12{0} can be either spacelike if g(v, v) > 0, timelike if g(v, v) < 0 or null (lightlike) if g(v, v) = 0. The norm of a vector v is provided by the formula v=|g(v,v)| Two vectors v and w are said to be orthogonal, if g(v, w) = 0. An arbitrary curve γ(s) in E12, can locally be spacelike or timelike, if all its velocity vectors γ′(s) are, respectively, spacelike or timelike. Any spacelike or timelike curve γ can be parametrized by the so called arc-length parameter s for which g(γ′(s), γ′(s)) = ±1 (see [10]). In particular, every spacelike curve α(s) in the Lorentz-Minkowski plane can be represented in the form (cf. [11, 12] and [13])


and the corresponding Frenet vector fields along it are given by the formulas


It is an easy task to see that they obey to the relations


in which the function


is the curvature of the curve.

The intrinsic equation of the spacelike elastic curves in the Lorentz-Minkowski plane reads


in which the overdots mean derivatives with respect to s and λ is the tension constant.

Similarly, any timelike curve β(s) in the Lorentz-Minkowski plane can be parameterized as follows


This time the Frenet vector fields


and the curvature


satisfy the relations


The respective intrinsic equation of the timelike elastic curves is


with the same notation.

3 Elastic curves in the Lorentz-Minkowski plane

Below we will find the explicit parameterizations (up to quadratures) of the spacelike and timelike elastic curves in the Lorentz-Minkowski plane.

Let α:IE12 be a spacelike elastic curve in the Lorentz-Minkowski plane. Then its position vector x(s)=(x(s),z(s)) is given by the formulas


where λ is the tension and the constants c1, c2 ∈ ℝ are such that c12c22.

Let β : IE12 be a timelike elastic curve in Lorentz-Minkowski plane. Then the position vector x(s)=(x(s),z(s)) is given by the formulas


in which λ is the tension and the constants c1, c2 ∈ ℝ are such that c12c22.

4 Spacelike elastic Sturmian spirals

Let α : IE12,α(s)=(x(s),z(s)) be a spacelike elastic Sturmian spiral in Lorentz-Minkowski plane which means that its curvature is given by the function κ = σ/r, where r=z2x2 and σ ∈ ℝ+. Before making use of the intrinsic equation (5) let us integrate it with respect to s. This produces the equation


in which E denotes the integration constant which can be interpreted as energy. For the special case of the Sturmian spirals it can be rewritten in the form


As the tension λ is a physical property it can be assumed to be a positive constant. Therefore, there are only two cases to be considered, depending on the sign of E.

Case 1. Let us start by assuming that both E and λ are positive. We will write this fact formally as


with a and c being non-zero positive constants which comply with the condition a > c. Substituting the above expressions of E and λ in (20) we obtain


As a consequence, the solutions of the differential equation (21) are either




where the first slot in the Jacobian sinus elliptic function sn ) is reserved for the argument, the second one for the so called elliptic modulus which is a real number between zero and one.

Now, when (22) holds we have


The equality on the right hand side is satisfied for s0 = 2K(k)/a, where K(k) is the complete elliptic integral of the first kind and κ(s0) = a. Then using (24) in (13) we can conclude that (c1, c2) = ((c2-a2})/4,0) . Taking into account all above and Theorem 3.1 we end up with the parameterization


where (.,.) is the Jacobian amplitude function, cn and dn are the remaining Jacobian elliptic functions and F(.,.), E(.,.) denote the incomplete elliptic inegrals of the first, respectively second kind (for more details see [15]).

Now, let us switch to the solution presented in (23). Proceeding in the same manner we obtain


for s0 = 2(K(k))/a complemented by κ(s0) = c. Entering with (26) in (13) we obtain that (c1, c2) = ((a2}-c2)/4,0) . Substituting (26) in the formulas of Theorem 1, we find


Case 2. When E is negative, we can write respectively


Inserting E and λ in (20) we obtain the equation


The solution of the above differential equation is


and therefore


All of this, along with (13), produces (c1, c2) = (-(a2 + c2)/4,0) . Relying on Theorem 3.1 we obtain finally the parameterization of the curve, i.e.,


5 Timelike elastic Sturmian spirals

Let β : IE12,β(s)=(x(s),z(s)) be a timelike elastic Sturmian spiral in the Lorentz-Minkowski plane. Integrating the intrinsic equation (9) we have

The left figure is obtained by formulas (25) with a = 2 and c = 1, this one in the middle via formulas (27) with a = 3 and c = 1, and that one at most right by formulas (32) with a = 1 and c = 2. In all these cases σ ≡ 2.
Fig. 1

The left figure is obtained by formulas (25) with a = 2 and c = 1, this one in the middle via formulas (27) with a = 3 and c = 1, and that one at most right by formulas (32) with a = 1 and c = 2. In all these cases σ ≡ 2.

where E as before is the constant of integration (the energy). Since β is a Sturmian spiral, the curvature function κ is given by the function κ=σr, where r=x2z2 and σ ∈ ℝ+. Rewriting appropriately the equation (33), we find


Again, depending on the sign of E there are two cases.

Case 1. Both E and λ are positive which is ensured by writing


Substituting these values of E and λ in (34), we obtain


The solution of the above differential equation is of the form


Therefore we get


Substituting (37) in Theorem 3.2, we find


Case 2. Assuming that E is negative, that is


we find that the intrinsic equation of the sought curves is


Regarding the solution of the above differential equation we have


From (41) we have also


A similar method as in the spacelike case produces (c1, c2) = (0, -(a2 + c2)/4). By using (42) in Theorem 3.2 we have finally the explicit parameterization


Both cases of timelike elastic Sturmian spirals in the Lorentz-Minkowski plane are pictured in Figure 2 for a specific choice of the parameters.

The left hand side figure is obtained by formulas (38), and that one on the right via formulas (43). In both cases a = 2, c = 1 and σ = 1.
Fig. 2

The left hand side figure is obtained by formulas (38), and that one on the right via formulas (43). In both cases a = 2, c = 1 and σ = 1.

6 Conclusions

The Lorentz-Minkowski plane is equipped with an indefinite metric and one should expect different types of curves to those in the Euclidean case. Elsewhere Marinov et al [7] have studied the elastic Sturmian spirals in the Euclidean plane. By drawing inspiration from this work we have considered here the spacelike and timelike elastic Sturmian spirals in the Lorentz-Minkowski plane E12. Our main results are the derivation of the explicit parametric equations for spacelike and timelike elastic Sturmian spirals in E12. We have presented also in Figure 1 and Figure 2 some graphical illustrations of these curves that are realized using MathematicaO.

Last but not least, we believe that the results presented in this paper suggest that some other curves like the elastic Serret's curves, elastic Bernoulli's Lemniscate and the generalized elastic curves whose curvature depend on the distance from the origin deserve to be studied in some details in the Lorentz-Minkowski plane E12 as well. We are planning to report on realization of this program elsewhere.


The first author would like to thank TUBITAK for the financial support during his PhD study. The third named author is partially supported by TUBITAK (The Scientific and Technological Research Council of Turkey) within the frame of Programme 2221.


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About the article

Received: 2016-10-16

Accepted: 2016-11-22

Published Online: 2016-12-30

Published in Print: 2016-01-01

Citation Information: Open Mathematics, Volume 14, Issue 1, Pages 1149–1156, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0103.

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© Uçum Ali et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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