- Research
- Open Access
Curvature of curves parameterized by a time scale
- Sibel Paşalı Atmaca^{1}Email author and
- Ömer Akgüller^{1}
https://doi.org/10.1186/s13662-015-0384-z
© Paşalı Atmaca and Akgüller; licensee Springer. 2015
- Received: 28 October 2014
- Accepted: 20 January 2015
- Published: 20 February 2015
Abstract
Curvature is a fundamental characteristic of curves in differential geometry, as well as in discrete geometry. In this paper we present time scales analogy of the curvature defined by the concept of symmetric derivative on time scales. The goal of our paper is to define this intrinsic characteristic accurately. For this purpose, we consider tangent spaces via symmetric differentiation.
Keywords
- symmetric differentiation
- time scales calculus
- curvature
1 Introduction
The time scales calculus, which is introduced by Hilger [4], is the theory to unify discrete and continuous calculus. Geometric aspect of the theory of time scales has been extensively studied after the introduction of partial derivatives on time scales [5–9]. However, an intrinsic characteristic such as curvature of a curve parameterized by a time scale is still an open question. In this paper, we present the concept of curvature via symmetric derivative on time scales. This approach involves both characteristics of discrete and classical differential geometry, and it is accurately applicable to globally discrete settings.
In [10], authors briefly introduced the symmetric derivative on time scales and its relation to forward and backward dynamic derivatives. The main purpose of that study is to aim differentiability of the functions such as \(f(t)=|t|\) at \(t=0\). Beside the differentiability, this calculus comes up with a more accurate definition for tangent lines of curves parameterized by time scales. As it is stated in [11], delta (and respectively nabla) derivatives of functions lead us to the concept of ‘complete differentiability’. To speak of one variable case, σ-complete differentiability needs the equality of right- and left-hand side derivatives, which makes the strong geometric restrictions for curves involving left dense-right scattered or right dense-left scattered points. Besides, tangent lines are not well defined at isolated points. The main disadvantage of this approach can be seen in [12], where the curvature can only be defined at dense and scattered points separately.
This paper is organized as follows. In Section 2, we introduce symmetric partial differentiation on time scales. We also present the relationship between symmetric differentiation and delta-nabla differentiation. Since the change of tangent spaces is a fundamental characteristic to define curvature, we present tangent lines and tangent planes of curves and surfaces parameterized by time scales in Section 3. In this section, the accuracy of a new tangent space definition via symmetric differentiation can be seen throughout the illustrative examples. Finally, in Section 4, we study the curvature of curves parameterized by an arbitrary time scale. Throughout the study we use the notion such as \(f^{\sigma}(t)=f(\sigma (t))\) and \(f^{\sigma\sigma}(t)=f(\sigma(\sigma(t)))\) to increase the readability of the paper.
2 Symmetric partial derivative on time scales
Definition 1
Since we restrict our interest to curves and surfaces on time scales, we will consider a two-dimensional case, i.e., \(n=2\), throughout the study. Interested readers can simply extend this idea to a higher-dimensional case.
Definition 2
Definition 3
Proposition 4
Proof
If \(t_{0}\) is left dense and right scattered, then \(\gamma_{1}(t_{0}) = \frac {\sigma_{1}(t_{0})-t_{0}}{\sigma_{1}(t_{0})-t_{0}}=1 \) and \(f^{\Diamond _{1}}(t_{0},s_{0})=f^{\Delta_{1}}(t_{0},s_{0})\).
If \(t_{0}\) is right dense and left scattered, then \(\sigma_{1}(t_{0})=t_{0}\) implies \(\gamma_{1}(t_{0})=0\) and therefore \(f^{\Diamond _{1}}(t_{0},s_{0})=f^{\nabla_{1}}(t_{0},s_{0})\). Also if \(t_{0}\) is dense \(\gamma_{1}(t_{0})= \lim_{t \to t_{0}}\frac {t_{0}-t}{2t_{0}-t}=\frac{1}{2}\) and \(f^{\Diamond_{1}}(t_{0},s_{0})=\frac {1}{2}f^{\Delta_{1}}(t_{0},s_{0})+\frac{1}{2}f^{\nabla_{1}}(t_{0},s_{0})\).
Moreover, if \(t_{0}\) is an isolated point, then \(\gamma_{1}(t_{0})=\frac {\sigma_{1}(t_{0})-t_{0}}{\sigma_{1}(t_{0})-\rho_{1}(t_{0})}\) and \(f^{\Diamond _{1}}(t_{0},s_{0})=\frac{\sigma_{1}(t_{0})-t_{0}}{\sigma_{1}(t_{0})-\rho_{1}(t_{0})} f^{\Delta _{1}}(t_{0},s_{0})+\frac{t_{0}-\rho_{1}(t_{0})}{\sigma_{1}(t_{0})-\rho_{1}(t_{0})}f^{\nabla _{1}}(t_{0},s_{0})\).
The same procedure can be followed to obtain a similar result \(\frac {\partial f(t,s)}{\Diamond_{2} s}=\gamma_{2}(s_{0}) \frac{\partial f(t_{0},s_{0})}{\Delta_{2} s}+(1-\gamma_{2}(s_{0}))\frac{\partial f(t_{0},s_{0})}{\nabla_{2} s}\). □
3 Tangent spaces
The geometric theory of the curves and surfaces parameterized by time scales and their analysis with the delta derivative can be found in [8, 9, 15]. It is possible to define curves and surfaces with the idea of symmetric differentiation in the same fashion. One can also extend the previous results to the symmetric differentiation.
Definition 5
A ◊-regular curve α is defined as a vector-valued mapping from \([a,b] \subset\mathbb{T}\) to \(\mathbb{R}^{3}\) with the non-zero norm \(\|\alpha^{\Diamond}(t_{0})\|\) for all \(t_{0} \in[a,b]\). Moreover, if \(\|\alpha^{\Delta}(t_{0})\|=1\) for all \(t_{0} \in[a,b]\), then α is called ‘arc length parameterized curve’.
Definition 6
- (i)\(\varphi: U \to\mathbb{R}^{3}\) is ◊-differentiable and for all \((t,s) \in U\)i.e., φ is ◊-regular.$$\frac{\partial\varphi(t,s)}{\Diamond_{1} t} \times \frac{\partial\varphi(t,s)}{\Diamond_{2} s} \neq0, $$
- (ii)
\(\varphi(U)=\mathcal{S}\cap A\) and \(\varphi: U \to\varphi (U)\) is a homeomorphism.
Definition 7
Let \(\alpha: \mathbb{T} \to\mathbb{R}^{n}\) be a ◊-regular curve and \(t_{0} \in\mathbb{T}^{\kappa}_{\kappa}\). The line with the slope \(\alpha^{\Diamond}(t_{0})\) passing at the point \(\alpha(t_{0})\) is called the diamond-tangent line of α at \(t_{0}\).
Remark 8
Definition 9
Let\(\mathcal{S}\) be a surface with the patch \(\varphi: U \to\mathcal {S}\), where \(U \subset\mathbb{T}_{1} \times\mathbb{T}_{2}\) and \((t_{0},s_{0}) \in{\mathbb{T}_{1}}^{\kappa}_{\kappa}\times{\mathbb{T}_{2}}^{\kappa }_{\kappa}\). The plane with the normal vector \(\varphi^{\Diamond_{1}}\times\varphi ^{\Diamond_{2}}\) passing at the point \(\varphi(t_{0},s_{0})\) is called a diamond-tangent plane of \(\mathcal{S}\) at \((t_{0},s_{0})\).
Remark 10
4 Curvature of curves on time scales
Definition 11
Remark 12
Declarations
Acknowledgements
The authors thank the organizing and scientific committee of the International Congress in Honour of Professor Ravi P Agarwal, Bursa-TURKEY June 23-26, 2014.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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