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Curvature of curves parameterized by a time scale
Advances in Difference Equations volume 2015, Article number: 49 (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.
Introduction
Curvature of a curve measures how sharply a curve bends; that is the second order amount by which a curve deviates from being a straight line. In classical differential geometry, curvature of an arclength parameterized curve \(\alpha: I \to\mathbb{R}^{3}\) can be computed by \(\kappa(s)=\\alpha''(s)\\) [1]. However, in a discrete case there are several approaches to define the curvature of a discrete curve. A discrete curve \(\alpha_{d}\) is composed of a series of sequential discrete points, see [2], or can be defined as the line segments \(v_{i1}v_{i}\) given by an ordered list of points \(v_{0}, \ldots, v_{N} \in\mathbb{R}^{2}\), i.e., polygons [3]. The most direct definition of the curvature of a discrete curve as a polygon is the turning angle curvature. Turning angle curvature \(\kappa_{a}\) can be computed by
where the vertice vector \(\vec{v}_{i}=\vec{v}_{i+1}\vec{v}_{i}\), \(i \in\{ 0,\ldots,N\}\), see [3]. A curvature model for a discrete curve as a composition of a series of sequential discrete points can be computed as
where ε is the sampling parameter on underlying continuous curve to determine the sequential discrete points. This approach is mostly concerned with second order finite differences.
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, σcomplete differentiability needs the equality of right and lefthand side derivatives, which makes the strong geometric restrictions for curves involving left denseright scattered or right denseleft 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 deltanabla 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.
Symmetric partial derivative on time scales
Let n be fixed and for all \(\mathbb{T}_{i}\) be time scales where \(i \in I=\{1,2,\ldots,n \}\). An ndimensional time scale can be defined by the Cartesian product as follows:
For \(u \in\mathbb{T}_{i}\), the forward and backward jump operators can be defined as \(\sigma_{i}(u)=\inf\{v \in\mathbb{T}_{i} \mid v>u\}\) and \(\rho_{i}(u)=\sup\{v \in\mathbb{T} \mid v< u\}\), respectively. If \(\mathbb{T}_{i}\) has a left scattered maximum M and right scattered minimum m, then \((\mathbb{T}_{i} )^{\kappa}_{\kappa}=\mathbb{T}_{i} \backslash\{M,m\}\), \((\mathbb{T}_{i} )^{\kappa}_{\kappa}=\mathbb {T}_{i}\); otherwise, see [11].
Let \(f: \Lambda^{n} \to\mathbb{R}\) be a realvalued function. The symmetric partial derivative of f can be defined as
existing as a finite number, and is denoted by \(f^{\Diamond_{i}}(t)\) or \(\frac{\partial f(t)}{\Diamond_{i} t_{i}}\), where \(f^{\sigma_{i}}(s_{i})=f(t_{1},\ldots,\sigma_{i}(s_{i}), \ldots,t_{n})\), \(f^{\rho _{i}}(s_{i})=f(t_{1},\ldots,\rho_{i}(s_{i}),\ldots,t_{n})\), and \(t = (t_{1},\ldots,t_{n}) \in (\mathbb{T}_{1} )^{\kappa}_{\kappa }\times\cdots\times (\mathbb{T}_{n} )^{\kappa}_{\kappa}\).
Definition 1
A function \(f : \Lambda^{n} \to\mathbb{R}\) is symmetric differentiable at a point \(t^{0}=(t^{0}_{1},\ldots,t^{0}_{n})\in (\mathbb{T}_{1} )^{\kappa}_{\kappa}\times\cdots\times (\mathbb{T}_{n} )^{\kappa }_{\kappa}\) if there exist numbers \(A_{1},\ldots,A_{n}\) independent of \(t=(t_{1},\ldots,t_{n})\in\Lambda^{n}\) such that for all \(t \in U_{\delta}(t^{0})\) and \(i \in\{1,\ldots,n\}\),
where δ is a sufficiently small positive number, \(U_{\delta}(t^{0})\) is the δneighborhood of \(t^{0}\), and \(\alpha_{i}=\alpha_{i}(t^{0},t)\) are defined on \(U_{\delta}(t^{0})\) such that it is equal to zero for \(t=t^{0}\) and \(\lim_{t \to t^{0}}\alpha_{i} = 0\) for all \(i \in \{1,\ldots,n\}\).
If \(\mathbb{T}_{1}=\cdots=\mathbb{T}_{n}=\mathbb{R}\), then Definition 1 coincides with the classical symmetric differentiability, see [13, 14]. To point out why we do not need a restriction such as ‘complete’ in the definition of symmetric differentiation, let us consider a onedimensional case. If \(t_{0}\) is left dense and right scattered, i.e., \(\sigma(t_{0})>t_{0}\) and \(\rho(t_{0})=t_{0}\), then
This equation leads us to
where \(\lim_{t \to t^{0}}\alpha= 0\) and \(\gamma (t)=\lim_{t \to t_{0}} \frac{\sigma(t_{0})t}{\sigma(t_{0})+t_{0}2t}\) as in [10, Proposition 4.2]. For right dense and left scattered and isolated points, the A values become
and
respectively.
Since we restrict our interest to curves and surfaces on time scales, we will consider a twodimensional case, i.e., \(n=2\), throughout the study. Interested readers can simply extend this idea to a higherdimensional case.
Definition 2
Let \(f : \mathbb{T}_{1} \times\mathbb{T}_{2} \to\mathbb{R}\) be a realvalued function and \((t_{0},s_{0}) \in (\mathbb{T}_{1} )^{\kappa}_{\kappa}\times (\mathbb{T}_{2} )^{\kappa}_{\kappa}\). For all \(\varepsilon_{1} > 0\), there is an open (relative to the topology of \(\mathbb{T}_{1} \times\mathbb{T}_{2}\)) neighborhood \(U_{1}\) of \((t_{0},s)\) such that for all \((t,s) \in U_{1}\),
Definition 3
Let \(f : \mathbb{T}_{1} \times\mathbb{T}_{2} \to\mathbb{R}\) be a realvalued function and \((t_{0},s_{0}) \in (\mathbb{T}_{1} )^{\kappa}_{\kappa}\times (\mathbb{T}_{2} )^{\kappa}_{\kappa}\). For all \(\varepsilon_{2} > 0\), there is an open (relative to the topology of \(\mathbb{T}_{1} \times\mathbb{T}_{2}\)) neighborhood \(U_{2}\) of \((t,s_{0})\) such that for all \((t,s) \in U_{2}\),
Note that higher order or mixed symmetric partial derivatives of a function defined on \(\mathbb{T}_{1} \times\mathbb{T}_{2}\) can be defined in the same sense as
and
Proposition 4
If \(f: \mathbb{T}_{1}\times\mathbb{T}_{2} \to\mathbb{R}\) is delta and nabla differentiable, then f is symmetric differentiable for each \((t,s) \in (\mathbb{T}_{1} )^{\kappa}_{\kappa}\times (\mathbb{T}_{2} )^{\kappa}_{\kappa}\) with
and
where
Proof
where \(\gamma_{1}(t_{0})=\lim_{t\to t_{0}} \frac{\sigma_{1}(t_{0})t}{\sigma _{1}(t_{0})+2t_{0}2t\rho_{1}(t_{0})}\) and \(\bar{\gamma}_{1}(t_{0})=\lim_{t\to t_{0}}\frac{2t_{0}t\rho_{1}(t_{0})}{\sigma_{1}(t_{0})+2t_{0}2t\rho_{1}(t_{0})}\) for all \(t_{0} \in\mathbb{T}_{1}\). It is straightforward that \(\gamma_{1} + \bar {\gamma}_{1} := 1\).
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 \(\frac {\partial f(t,s)}{\Diamond_{2} s}=\gamma_{2}(s_{0}) \frac{\partial f(t_{0},s_{0})}{\Delta_{2} s}+(1\gamma_{2}(s_{0}))\frac{\partial f(t_{0},s_{0})}{\nabla_{2} s}\). □
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 vectorvalued mapping from \([a,b] \subset\mathbb{T}\) to \(\mathbb{R}^{3}\) with the nonzero 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
Let \(\mathcal{S}\) be a closed subset of \(\mathbb{R}^{3}\). \(\mathcal{S}\) is a surface if for each point P in \(\mathcal{S}\), there is a neighborhood A of P and a function \(\varphi: U \to\mathcal{S}\) where U is a closed set in \(\mathbb{R}^{2}\) and an open set in time scale topology satisfying the following conditions:

(i)
\(\varphi: U \to\mathbb{R}^{3}\) is ◊differentiable and for all \((t,s) \in U\)
$$\frac{\partial\varphi(t,s)}{\Diamond_{1} t} \times \frac{\partial\varphi(t,s)}{\Diamond_{2} s} \neq0, $$i.e., φ is ◊regular.

(ii)
\(\varphi(U)=\mathcal{S}\cap A\) and \(\varphi: U \to\varphi (U)\) is a homeomorphism.
The function \(\varphi: U \to\mathcal{S}\) is called a surface patch. \(\mathcal{S}\) is called a smooth surface if, for all points P in \(\mathcal{S}\), there exists a surface patch such that \(P \in\varphi(U)\).
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 diamondtangent line of α at \(t_{0}\).
Remark 8
Bythis definition, it is clear that we do not need to have the equality of the left and right side dynamic derivatives. Hence, the tangent line defined with the symmetric derivative on time scales is more sensitive to the geometric change of the curve.In Figure 1 and Figure 2, \(\mathcal{L}_{0}\) is the diamondtangent line of the α at \(t_{0}\) with the slope \(\alpha^{\Diamond}(t_{0})\). It is clear that the translations on the right or left continuous arcs of the curves will lead to new tangent lines. Hence, geometric changes in the curve affect the tangent line.
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 diamondtangent plane of \(\mathcal{S}\) at \((t_{0},s_{0})\).
Remark 10
As in the definition of diamondtangent line for a curve, diamondtangent plane for a surface is also defined on isolated points. In Figure 3, \(\Omega_{0}\) is the diamondtangent plane of the \(\mathcal{S}\) at \((t_{0},s_{0})\) with the normal \(\varphi^{\Diamond _{1}}(t_{0},s_{0})\times\varphi^{\Diamond_{2}}(t_{0},s_{0})\). It is clear that the geometric changes in the surface lead to new tangent planes.
Curvature of curves on time scales
Definition 11
Let \(\alpha: \mathbb{T} \to\mathbb{R}^{n}\) be an arc length parameterized regular curve. The curvature of α at \(t_{0} \in \mathbb{T}^{\kappa}_{\kappa}\) is the norm of the second order symmetric derivative of α, i.e.,
Remark 12
If \(\mathbb{T}=\mathbb{R}\), then
If the time scale \(\mathbb{T}\) is completely discrete, then
Hence \(\kappa_{\Diamond}(t_{0}) = \\alpha^{\Diamond\Diamond}(t_{0})\\) can be computed very accurately. Moreover, if \(\mathbb{T}\) is with a nonzero constant graininess h
and the norm of the second order symmetric derivative leads us to \(\kappa_{s} (t_{0})\).
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Acknowledgements
The authors thank the organizing and scientific committee of the International Congress in Honour of Professor Ravi P Agarwal, BursaTURKEY June 2326, 2014.
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SPA and ÖA worked together in the derivation of the mathematical results. Both authors read and approved the final manuscript.
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Paşalı Atmaca, S., Akgüller, Ö. Curvature of curves parameterized by a time scale. Adv Differ Equ 2015, 49 (2015). https://doi.org/10.1186/s136620150384z
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Keywords
 symmetric differentiation
 time scales calculus
 curvature