AMC Brito da Cruz

CE Aull

D Uçar

E Özyılmaz

GS Guseinov

JL Cieśliński

M Bohner

MP do Carmo

S Colvin

S Hilger

Sibel Paşalı Atmaca

SP Atmaca

Y An

Ömer Akgüller

Advances in Difference Equations

English

Crossref

Pas¸alı Atmaca and Akgüller Advances in Diﬀerence Equations  (2015) 2015:49 DOI 10.1186/s13662-015-0384-zRESEARCH Open AccessCurvature of curves parameterized by a timescaleSibel Pas¸alı Atmaca* and Ömer Akgüller*Correspondence: sibela@mu.edu.trDepartment of Mathematics,Faculty of Science, Mugˇla SıtkıKoçman University, Mentes¸e, Mugˇla,48000, TurkeyAbstractCurvature is a fundamental characteristic of curves in diﬀerential geometry, as well asin discrete geometry. In this paper we present time scales analogy of the curvaturedeﬁned by the concept of symmetric derivative on time scales. The goal of our paperis to deﬁne this intrinsic characteristic accurately. For this purpose, we considertangent spaces via symmetric diﬀerentiation.Keywords: symmetric diﬀerentiation; time scales calculus; curvature1 IntroductionCurvature of a curvemeasures how sharply a curve bends; that is the second order amountby which a curve deviates from being a straight line. In classical diﬀerential geometry, cur-vature of an arc-length parameterized curve α : I →R can be computed by κ(s) = ‖α′′(s)‖[]. However, in a discrete case there are several approaches to deﬁne the curvature of adiscrete curve. A discrete curve αd is composed of a series of sequential discrete points,see [], or can be deﬁned as the line segments |vi–vi| given by an ordered list of pointsv, . . . , vN ∈R, i.e., polygons []. The most direct deﬁnition of the curvature of a discretecurve as a polygon is the turning angle curvature. Turning angle curvature κa can be com-puted byκa(t) = sgn|∠(vi–, vi)|‖vi–‖ + ‖vi‖ ,where the vertice vector vi = vi+–vi, i ∈ {, . . . ,N}, see []. A curvaturemodel for a discretecurve as a composition of a series of sequential discrete points can be computed asκs(s) = sgn‖α(s) – α(s – ε) + α(s – ε)‖ε,where ε is the sampling parameter on underlying continuous curve to determine the se-quential discrete points. This approach is mostly concerned with second order ﬁnite dif-ferences.The time scales calculus, which is introduced by Hilger [], is the theory to unify dis-crete and continuous calculus. Geometric aspect of the theory of time scales has beenextensively studied after the introduction of partial derivatives on time scales [–]. How-ever, an intrinsic characteristic such as curvature of a curve parameterized by a time scale© 2015 Pas¸alı Atmaca and Akgüller; licensee Springer. This is an Open Access article distributed under the terms of the CreativeCommons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and re-production in any medium, provided the original work is properly credited.Pas¸alı Atmaca and Akgüller Advances in Diﬀerence Equations  (2015) 2015:49 Page 2 of 7is still an open question. In this paper, we present the concept of curvature via sym-metric derivative on time scales. This approach involves both characteristics of discreteand classical diﬀerential geometry, and it is accurately applicable to globally discrete set-tings.In [], authors brieﬂy introduced the symmetric derivative on time scales and its rela-tion to forward and backward dynamic derivatives. The main purpose of that study is toaim diﬀerentiability of the functions such as f (t) = |t| at t = . Beside the diﬀerentiability,this calculus comes up with a more accurate deﬁnition for tangent lines of curves param-eterized by time scales. As it is stated in [], delta (and respectively nabla) derivatives offunctions lead us to the concept of ‘complete diﬀerentiability’. To speak of one variablecase, σ -complete diﬀerentiability needs the equality of right- and left-hand side deriva-tives, which makes the strong geometric restrictions for curves involving left dense-rightscattered or right dense-left scattered points. Besides, tangent lines are not well deﬁnedat isolated points. The main disadvantage of this approach can be seen in [], where thecurvature can only be deﬁned at dense and scattered points separately.This paper is organized as follows. In Section , we introduce symmetric partial diﬀeren-tiation on time scales.We also present the relationship between symmetric diﬀerentiationand delta-nabla diﬀerentiation. Since the change of tangent spaces is a fundamental char-acteristic to deﬁne curvature, we present tangent lines and tangent planes of curves andsurfaces parameterized by time scales in Section . In this section, the accuracy of a newtangent space deﬁnition via symmetric diﬀerentiation can be seen throughout the illustra-tive examples. Finally, in Section , we study the curvature of curves parameterized by anarbitrary time scale. Throughout the study we use the notion such as f σ (t) = f (σ (t)) andf σσ (t) = f (σ (σ (t))) to increase the readability of the paper.2 Symmetric partial derivative on time scalesLet n be ﬁxed and for allTi be time scales where i ∈ I = {, , . . . ,n}. An n-dimensional timescale can be deﬁned by the Cartesian product as follows:n = T × · · · ×Tn ={(t, . . . , tn) | ti ∈ Ti,∀i ∈ I}.For u ∈ Ti, the forward and backward jump operators can be deﬁned as σi(u) = inf{v ∈ Ti |v > u} and ρi(u) = sup{v ∈ T | v < u}, respectively. If Ti has a left scattered maximum Mand right scattered minimumm, then (Ti)κκ = Ti\{M,m}, (Ti)κκ = Ti; otherwise, see [].Let f : n → R be a real-valued function. The symmetric partial derivative of f can bedeﬁned aslimsi→tiρi(ti) =siσi(ti) =sif σi (si) – f (t, . . . , ti, . . . , tn) + f (t, . . . , si – ti, . . . , tn) – f ρi (si)σ (ti) + si – ti – ρ(ti)existing as a ﬁnite number, and is denoted by f ♦i (t) or ∂f (t)♦iti , where fσi (si) = f (t, . . . ,σi(si),. . . , tn), f ρi (si) = f (t, . . . ,ρi(si), . . . , tn), and t = (t, . . . , tn) ∈ (T)κκ × · · · × (Tn)κκ .Deﬁnition  A function f :n →R is symmetric diﬀerentiable at a point t = (t , . . . , tn) ∈(T)κκ ×· · ·×(Tn)κκ if there exist numbersA, . . . ,An independent of t = (t, . . . , tn) ∈ n suchPas¸alı Atmaca and Akgüller Advances in Diﬀerence Equations  (2015) 2015:49 Page 3 of 7that for all t ∈Uδ(t) and i ∈ {, . . . ,n},f(t , . . . ,σi(ti), . . . , tn)– f (t, . . . , tn) + f(t – t, . . . , ti – ti, . . . , tn – tn)– f(t , . . . ,ρi(ti), . . . , tn)=n∑i=Ai[σi(ti)+ ti – ti – ρi(ti)]+n∑i=αi[σi(ti)+ ti – ti – ρi(ti)],where δ is a suﬃciently small positive number, Uδ(t) is the δ-neighborhood of t, andαi = αi(t, t) are deﬁned on Uδ(t) such that it is equal to zero for t = t and limt→t αi = for all i ∈ {, . . . ,n}.If T = · · · = Tn = R, then Deﬁnition  coincides with the classical symmetric diﬀeren-tiability, see [, ]. To point out why we do not need a restriction such as ‘complete’ inthe deﬁnition of symmetric diﬀerentiation, let us consider a one-dimensional case. If t isleft dense and right scattered, i.e., σ (t) > t and ρ(t) = t, thenf(σ (t))– f (t) + f (t – t) – f (t) = A[σ (t) + t – t]+ α[σ (t) + t – t].This equation leads us toA = f ♦ = γ (t)f (t) +( – γ (t))f ′–(t),where limt→t α =  and γ (t) = limt→t σ (t)–tσ (t)+t–t as in [, Proposition .]. For right denseand left scattered and isolated points, the A values becomeA = f ♦ = γ (t)f ′+ +( – γ (t))f ∇ (t)andA = f ♦ = γ (t)f (t) +( – γ (t))f ∇ (t),respectively.Since we restrict our interest to curves and surfaces on time scales, we will considera two-dimensional case, i.e., n = , throughout the study. Interested readers can simplyextend this idea to a higher-dimensional case.Deﬁnition  Let f : T × T → R be a real-valued function and (t, s) ∈ (T)κκ × (T)κκ .For all ε > , there is an open (relative to the topology of T × T) neighborhood U of(t, s) such that for all (t, s) ∈U,∣∣[f(σ(t), s)– f (t, s) + f (t – t, s) – f(ρ(t), s)]– f ♦[σ(t) + t – t – ρ(t)]∣∣≤ ε∣∣σ(t) + t – t – ρ(t)∣∣.Deﬁnition  Let f : T × T → R be a real-valued function and (t, s) ∈ (T)κκ × (T)κκ .For all ε > , there is an open (relative to the topology of T × T) neighborhood U ofPas¸alı Atmaca and Akgüller Advances in Diﬀerence Equations  (2015) 2015:49 Page 4 of 7(t, s) such that for all (t, s) ∈U,∣∣[f(t,σ(s))– f (t, s) + f (t, s – s) – f(t,ρ(s))]– f ♦[σ(s) + s – s – ρ(s)]∣∣≤ ε∣∣σ(s) + s – s – ρ(s)∣∣.Note that higher order or mixed symmetric partial derivatives of a function deﬁned onT ×T can be deﬁned in the same sense as∂f (t, s)♦t= ∂♦t(∂f (t, s)♦t), ∂f (t, s)♦s= ∂♦s(∂f (t, s)♦s)and∂f (t, s)♦t♦s= ∂♦s(∂f (t, s)♦t), ∂f (t, s)♦s♦t= ∂♦t(∂f (t, s)♦s).Proposition  If f : T × T → R is delta and nabla diﬀerentiable, then f is symmetricdiﬀerentiable for each (t, s) ∈ (T)κκ × (T)κκ with∂f (t, s)♦t= γ(t)∂f (t, s)t+( – γ(t))∂f (t, s)∇tand∂f (t, s)♦s= γ(s)∂f (t, s)s+( – γ(s))∂f (t, s)∇s ,whereγ(t) = limt→tσ(t) – tσ(t) + t – t – ρ(t)and γ(s) = lims→sσ(s) – sσ(s) + s – s – ρ(s).Proof∂f♦t= lim(t,s)→(t,s)f (σ(t), s) – f (t, s) + f (t – t, s) – f (ρ(t), s)σ(t) + t – t – ρ(t)= lim(t,s)→(t,s)σ(t) – tσ(t) + t – t – ρ(t)f (σ(t), s) – f (t, s)σ(t) – t+ lim(t,s)→(t,s)t – t – ρ(t)σ(t) + t – t – ρ(t)f (t – t, s) – f (ρ(t), s)t – t – ρ(t)= lim(t,s)→(t,s)σ(t) – tσ(t) + t – t – ρ(t)∂f (t, s)t+ lim(t,s)→(t,s)t – t – ρ(t)σ(t) + t – t – ρ(t)+ ∂f (t, s)∇t= γ(t)∂f (t, s)t+ γ¯(t) +∂f (t, s)∇t ,where γ(t) = limt→t σ(t)–tσ(t)+t–t–ρ(t) and γ¯(t) = limt→tt–t–ρ(t)σ(t)+t–t–ρ(t) for all t ∈ T.It is straightforward that γ + γ¯ := .Pas¸alı Atmaca and Akgüller Advances in Diﬀerence Equations  (2015) 2015:49 Page 5 of 7If t is left dense and right scattered, then γ(t) = σ(t)–tσ(t)–t =  and f♦ (t, s) = f  (t, s).If t is right dense and left scattered, then σ(t) = t implies γ(t) =  and there-fore f ♦ (t, s) = f ∇ (t, s). Also if t is dense γ(t) = limt→t t–tt–t = and f ♦ (t, s) = f  (t, s) + f ∇ (t, s).Moreover, if t is an isolated point, then γ(t) = σ(t)–tσ(t)–ρ(t) and f♦ (t, s) = σ(t)–tσ(t)–ρ(t) ×f  (t, s) + t–ρ(t)σ(t)–ρ(t) f∇ (t, s).The same procedure can be followed to obtain a similar result ∂f (t,s)♦s = γ(s)∂f (t,s)s +(–γ(s)) ∂f (t,s)∇s . 3 Tangent spacesThe geometric theory of the curves and surfaces parameterized by time scales and theiranalysis with the delta derivative can be found in [, , ]. It is possible to deﬁne curvesand surfaces with the idea of symmetric diﬀerentiation in the same fashion. One can alsoextend the previous results to the symmetric diﬀerentiation.Deﬁnition  A ♦-regular curve α is deﬁned as a vector-valued mapping from [a,b] ⊂ Tto R with the non-zero norm ‖α♦(t)‖ for all t ∈ [a,b]. Moreover, if ‖α(t)‖ =  for allt ∈ [a,b], then α is called ‘arc length parameterized curve’.Deﬁnition  Let S be a closed subset of R. S is a surface if for each point P in S , thereis a neighborhood A of P and a function ϕ :U → S where U is a closed set in R and anopen set in time scale topology satisfying the following conditions:(i) ϕ :U →R is ♦-diﬀerentiable and for all (t, s) ∈U∂ϕ(t, s)♦t× ∂ϕ(t, s)♦s = ,i.e., ϕ is ♦-regular.(ii) ϕ(U) = S ∩A and ϕ :U → ϕ(U) is a homeomorphism.The function ϕ : U → S is called a surface patch. S is called a smooth surface if, for allpoints P in S , there exists a surface patch such that P ∈ ϕ(U).Deﬁnition  Let α : T → Rn be a ♦-regular curve and t ∈ Tκκ . The line with the slopeα♦(t) passing at the point α(t) is called the diamond-tangent line of α at t.Remark  By this deﬁnition, it is clear that we do not need to have the equality of the leftand right side dynamic derivatives. Hence, the tangent line deﬁned with the symmetricderivative on time scales ismore sensitive to the geometric change of the curve. In Figure and Figure , L is the diamond-tangent line of the α at t with the slope α♦(t). It is clearthat the translations on the right or left continuous arcs of the curves will lead to newtangent lines. Hence, geometric changes in the curve aﬀect the tangent line.Deﬁnition LetS be a surfacewith the patchϕ :U → S , whereU ⊂ T×T and (t, s) ∈Tκκ × Tκκ . The plane with the normal vector ϕ♦ × ϕ♦ passing at the point ϕ(t, s) iscalled a diamond-tangent plane of S at (t, s).Remark  As in the deﬁnition of diamond-tangent line for a curve, diamond-tangentplane for a surface is also deﬁned on isolated points. In Figure ,  is the diamond-Pas¸alı Atmaca and Akgüller Advances in Diﬀerence Equations  (2015) 2015:49 Page 6 of 7Figure 1 Diamond-tangent line at the left denseand right scattered point of the curve α.Therefore, γ (t0) = 1 and α♦(t0) = α(t0).Figure 2 Diamond-tangent line at the isolatedpoint of the curve α. Therefore, γ (t0) =σ (t0)–t0σ (t0)–ρ(t0),and α♦(t0) =σ (t0)–t0σ (t0)–ρ(t0)α(t0) +t0–ρ(t0)σ (t0)–ρ(t0)α∇ (t0).Figure 3 Diamond-tangent plane with thenormal vector ϕ♦1 (t0, s0)× ϕ♦2 (t0, s0) at thepoint ϕ(t0, s0) of the surface S .tangent plane of the S at (t, s) with the normal ϕ♦ (t, s) × ϕ♦ (t, s). It is clear thatthe geometric changes in the surface lead to new tangent planes.4 Curvature of curves on time scalesDeﬁnition  Let α : T → Rn be an arc length parameterized regular curve. The curva-ture of α at t ∈ Tκκ is the norm of the second order symmetric derivative of α, i.e.,κ♦(t) =∥∥α♦♦(t)∥∥.Remark  If T =R, thenκ♦(t) =∥∥α♦♦(t)∥∥ =∥∥α′′(t)∥∥ = κ(t).Pas¸alı Atmaca and Akgüller Advances in Diﬀerence Equations  (2015) 2015:49 Page 7 of 7If the time scale T is completely discrete, thenα♦♦(t) =(σ (t) – t)(σ (t) – ρ(t))(ασσ (t) – α(t))– (t – ρ(t))(σ (t) – ρ(t))(α(t) – αρρ(t)).Hence κ♦(t) = ‖α♦♦(t)‖ can be computed very accurately. Moreover, if T is with anonzero constant graininess hα♦♦(t) =h(α(t + h) – α(t) + α(t – h))= ε(α(s) – α(s – ε) + α(s – ε))and the norm of the second order symmetric derivative leads us to κs(t).Competing interestsThe authors declare that they have no competing interests.Authors’ contributionsSPA and ÖA worked together in the derivation of the mathematical results. Both authors read and approved the ﬁnalmanuscript.AcknowledgementsThe authors thank the organizing and scientiﬁc committee of the International Congress in Honour of Professor Ravi PAgarwal, Bursa-TURKEY June 23-26, 2014.Received: 28 October 2014 Accepted: 20 January 2015References1. do Carmo, MP: Diﬀerential Geometry of Curves and Surfaces. Prentice Hall, Englewood Cliﬀs (1976)2. An, Y, Shao, C, Wang, X, Li, Z: Estimating principal curvatures and principal directions from discrete surfaces usingdiscrete curve model. J. Inf. Comput. Sci. 8(2), 296-311 (2011)3. Bobenko, AI, Schröder, P, Sullivan, JM, Ziegler, GM (eds): Discrete Diﬀerential Geometry. Oberwolfach Seminars,vol. 38, pp. 137-161 (2008)4. Hilger, S: Analysis on measure chains - a uniﬁed approach to continuous and discrete calculus. Results Math. 18,18-56 (1990)5. Özyılmaz, E: Directional derivative of vector ﬁeld and regular curves on time scales. Appl. Math. Mech. 27(10),1349-1360 (2006)6. 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Colvin, S, Debnath, L: On symmetric partial derivatives and symmetric diﬀerentiability. Gac. Mat. 25(3-4), 100-106(1973)15. Guseinov, GS, Özyılmaz, E: Tangent lines of generalized regular curves parametrized by time scales. Turk. J. Math.25(4), 553-562 (2001)

Curvature of curves parameterized by a time scale

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Curvature of curves parameterized by a time scale

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