# Forward Curvatures on Time Scales

Authors: M. S. Seyyidoglu; Y. Tunçer; D. Uçar; M. K. Berktaş; V. F. Hatipoğlu
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We also introduce forward curvature of a curve and give some formulas to calculate forward curvature of a curve on time scales which may be an arbitrary closed subsets of the set of all real numbers. We also introduce the length of a curve parametrized by a time scale parameter inℝ3.


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## Body

## 1. Introduction

The study of dynamic equations on time scales is an area of mathematics that recently has received a lot of attention. The calculus on time scales has been introduced in order to unify the theories of continuous and discrete processes and in order to extend those theories to a more general class of so-called dynamic equations.In recent years there have been a few research activities concerning the application of differential geometry on time scales. In [1] Guseinov and Ozyılmaz have defined the notions of forward tangent line, Δ-regular curve, and natural Δ-parametrization. Furthermore, in [2] Bohner and Guseinov, have introduced the concept of a curve parametrized by a time scale parameter and they have given integral formulas for computation of its length in plane. They have established a version of the classical Green formula suitable to time scales. In [3] Ozyılmaz has introduced the directional derivative according to the vector fields.The general idea of this paper is to study forward curvature of curves where in the parametric equations the parameter varies in a time scale. We present the “differential” part of classical differential geometry on time scale calculus. The new results generalize the well known formulas stated in classical differential geometry. We illustrate our results by applying them to various kinds of time scales.


## 2. Basic Definitions

A time scale𝕋 is an arbitrary nonempty closed subset of the real numbers ℝ. The time scale 𝕋 is a complete metric space with the usual metric. We assume throughout that a time scale 𝕋 has the topology that it inherits from the real numbers with the standard topology.Fort∈𝕋 we define the forward jump operatorσ:𝕋→𝕋 by (2.1)σ(t):=inf⁡{s∈T:s>t},while thebackward jump operatorρ:𝕋→𝕋 is defined by(2.2)ρ(t):=sup⁡{s∈T:s<t}.Ifσ(t)>t, we say that t is right-scattered, while, if ρ(t)<t, we say that t is left-scattered. Also, if σ(t)=t, then t is called right-dense, and, if ρ(t)=t, then t is called left-dense. The graininess functionμ:𝕋→[0,∞) is defined by (2.3)μ(t):=σ(t)-t.We introduce the set𝕋κ which is derived from the time scale 𝕋 as follows. If 𝕋 has left-scattered maximum m, then 𝕋κ=𝕋-{m}, otherwise 𝕋κ=𝕋. For a,b∈𝕋 with a≤b we define the interval [a,b] in 𝕋 by (2.4)[a,b]={t∈T:a≤t≤b}.We will let[a,b]κ denote [a,ρ(b)] if b is left-scattered and [a,b] if b is left-dense.Definition 2.1 (see [4]).
Assume thatf:𝕋→ℝ is a function, and let t∈𝕋κ. Then we define fΔ(t) to be the number (provided it exists) with the property that, given any ε>0, there is a neighborhood U of t such that
(2.5)|f(σ(t))-f(s)-fΔ(t)[σ(t)-s]|≤ε|σ(t)-s|,∀s∈U.
We call fΔ(t) the delta(or Hilger) derivativeof f at t. Moreover, we say that f is delta(or Hilger) differentiableon 𝕋κ provided fΔ(t) exists for all t∈𝕋κ.Theorem 2.2 (see [4]).
Forf:𝕋→ℝ and t∈𝕋κ the following hold.        (i)
Iff is Δ-differentiable at t, then f is continuous at t.(ii)
Iff is continuous at t and t is right-scattered, then f is Δ-differentiable at t and(2.6)fΔ(t)=f(σ(t))-f(t)σ(t)-t(iii)
Ift is right-dense, then f is Δ-differentiable at  t if and only if the limit(2.7)lim⁡s→tf(t)-f(s)t-s
exists as a finite number. In this case fΔ(t) is equal to this limit.(iv)
Iff is Δ-differentiable at  t, then(2.8)f(σ(t))=f(t)+μ(t)fΔ(t).Theorem 2.3 (see [4]).
Iff,g is Δ-differentiable at t∈𝕋κ, then (i)
f+g is Δ-differentiable at t and(2.9)(f+g)Δ(t)=fΔ(t)+gΔ(t).(ii)
For any constantc, cf is Δ-differentiable at t and(2.10)(cf)Δ(t)=cfΔ(t).(iii)
fg is Δ-differentiable at t and(2.11)(fg)Δ(t)=fΔ(t)g(t)+f(σ(t))gΔ(t)=f(t)gΔ(t)+fΔ(t)g(σ(t)).(iv)
Ifg(t)g(σ(t))≠0, then f/g is Δ-differentiable at t and(2.12)(fg)Δ(t)=fΔ(t)g(t)-gΔ(t)f(t)g(t)g(σ(t)).Theorem 2.4 (chain rule [4]).
Assume thatν:𝕋→ℝ is strictly increasing and 𝕋̃≔ν(𝕋) is a time scale. Let w:𝕋̃→ℝ. If νΔ(t) and wΔ̃(ν(t)) exists for t∈𝕋κ, then
(2.13)(w∘ν)Δ=(wΔ̃∘ν)νΔ,
where one denotes the derivative on 𝕋̃ by Δ̃.Theorem 2.5 (derivative of the inverse [4]).
Assume thatν:𝕋→ℝ is strictly increasing and 𝕋̃:=ν(𝕋) is a time scale. Then
(2.14)1νΔ=(ν-1)Δ̃∘ν
at points where νΔ is different from zero.Definition 2.6 (see [4]).
A functionf:𝕋→ℝ is called rd-continuous provided it is continuous at right-dense points in 𝕋 and its left-sided limits exist (finite) at left-dense points in 𝕋.Definition 2.7 (see [1]).
AΔ-regular curve (or an arc of aΔ-regular curve)γ is defined as a mapping γ:[a,b]⊂𝕋→ℝ3 that is Δ-differentiable on [a,b]κ with rd-continuous Δ-derivatives and
(2.15)‖γΔ(t)‖≠0,fort∈[a,b]κ.Definition 2.8 (see [1]).
Letγ:[a,b]⊂𝕋→ℝ3 be a curve, Q0 a point on γ, and L a line through Q0σ, where
(2.16)Q0=(γ1(t0),γ2(t0),γ3(t0)),Q0σ=(γ1(σ(t0)),γ2(σ(t0)),γ3(σ(t0))),t0∈[a,b]κ.
Take on γ any point Q. Denote by d the distance of the point Q from the point Q0σ, and by δ the distance of Q from the line L. If δ/d→0 as Q→Q0,Q≠Q0σ, then we say that L is the forward tangent line to the curve γ at the point Q0.Theorem 2.9 (see [1]).
EveryΔ-regular curve γ:[a,b]⊂𝕋→ℝ3 has at any point Q0=(γ1(t0),γ2(t0),γ3(t0)),t0∈[a,b]κ, the forward tangent line that has the vector γΔ(t0) as its direction vector.Definition 2.10 (see [1]).
Letγ be a Δ-regular curve in ℝ3 given by the equation
(2.17)γ=γ(t),t∈[a,b]⊂T.
We define the function p by
(2.18)p(t)=∫at‖γΔ(s)‖Δs,t∈[a,b].
The variable p can be used as a parameter for the curve γ. Such a parametrization of a curve we call natural Δ-parametrization.Theorem 2.11 (see [1]).
In the case of naturalΔ-parametrization of the curve γ the forward tangent vector is a unit vector.Definition 2.12 (see [2]).
Letγ be a continuous curve with equation γ(t)=(γ1(t),γ2(t)). A partition of [a,b] is any finite ordered set
(2.19)P={t0,t1,…,tn}⊂[a,b],wherea=t0<t1<⋯<tn=b.
Let us set
(2.20)l(γ,P)=∑i=1n[γ1(ti)-γ1(ti-1)]2+[γ2(ti)-γ2(ti-1)]2.
The curve γ is rectifiable if
(2.21)sup⁡P⁡{l(γ,P):Pisapartitionof[a,b]}=:l(γ)<∞.
The nonnegative number l(γ) is called the length of the curve γ. If the supremum does not exist, the curve is said to be nonrectifiable.Theorem 2.13 (see [2]).
Let the functionsγ1 and γ2 be continuous on [a,b] and Δ-differentiable on [a,b). If their Δ-derivatives γ1Δ and γ2Δ are bounded and Δ-integrable over [a,b), then the curve γ(t)=(γ1(t),γ2(t)) is rectifiable and its length l(γ) can be evaluated by the formula
(2.22)l(γ)=∫ab[γ1Δ(t)]2+[γ2Δ(t)]2Δt.


## 3. Forward Curvatures on Time Scales

It is easy to see that the notion of rectifiable curve in Definition2.12 for ℝ2 can be adapted to ℝ3.Definition 3.1.
Letγ be a continuous curve in ℝ3. Let P be a partition of [a,b] as in (2.19), and set
(3.1)l(γ,P)=∑i=1n∑j=13[γj(ti)-γj(ti-1)]2,
where γ(t)=(γ1(t),γ2(t),γ3(t)). We say that curve γ is rectifiable if
(3.2)sup⁡P⁡{l(γ,P):Pisapartitionof[a,b]}=:l(γ)<∞.
In Theorem 2.13, the length of the curve in plane is given. We introduce the length of a curve parametrized by a time scale parameter in ℝ3 in the following lemma.Lemma 3.2.
Let the functionsγ1,γ2, and γ3 be continuous on [a,b] and Δ-differentiable on [a,b). If their Δ-derivatives γ1Δ,γ2Δ,andγ3Δ  are bounded and Δ-integrable over [a, b), then the curve γ(t)=(γ1(t),γ2(t),γ3(t)) is rectifiable and its length l(γ) can be evaluated by the formula
(3.3)l(γ)=∫ab[γ1Δ(t)]2+[γ2Δ(t)]2+[γ3Δ(t)]2Δt.Proof.
We show that the curveγ is rectifiable. Let an arbitrary partition of [a,b] be of the form (2.19). Consider l(γ,P) defined by (3.1). Applying to each of the functions γ1,γ2, and γ3 the mean value theorem (see [5, Theorem 4.2]) on [ti-1,ti] for i∈{1,2,…,n}, we get that there exist points ξi,ξi′,ηi,ηi′ and ζi,ζi′ in [ti-1,ti) such that
(3.4)γ1Δ(ξi)(ti-ti-1)≤γ1(ti)-γ1(ti-1)≤γ1Δ(ξi′)(ti-ti-1),(3.5)γ2Δ(ηi)(ti-ti-1)≤γ2(ti)-γ2(ti-1)≤γ2Δ(ηi′)(ti-ti-1),(3.6)γ3Δ(ζi)(ti-ti-1)≤γ3(ti)-γ3(ti-1)≤γ3Δ(ζi′)(ti-ti-1).
From (3.4), (3.5), and (3.6) it follows that
(3.7)|γ1(ti)-γ1(ti-1)|≤Ai(ti-ti-1),|γ2(ti)-γ2(ti-1)|≤Bi(ti-ti-1),|γ3(ti)-γ3(ti-1)|≤Ci(ti-ti-1),where
(3.8)Ai=max⁡{|γ1Δ(ξi)|,|γ1Δ(ξi′)|},Bi=max⁡{|γ2Δ(ηi)|,|γ2Δ(ηi′)|},Ci=max⁡{|γ3Δ(ζi)|,|γ3Δ(ζi′)|}.By the assumption of the theorem, the derivatives γ1Δ,γ2Δ,   and γ3Δ are bounded on [a,b), so that there is a finite positive constant M such that
(3.9)|γjΔ(t)|≤M,j=1,2,3for all t∈[a,b). Thus
(3.10)|γj(ti)-γj(ti-1)|≤M(ti-ti-1),j=1,2,3for all i∈{1,2,…,n}, and we have from (3.1)
(3.11)l(γ,P)≤M3∑i=1n(ti-ti-1)=M3(b-a),so that we get that the curve γ is rectifiable.Now we prove the formula (3.3). Consider the Riemann Δ-sum
(3.12)S=∑i=1n[γ1Δ(ξi)]2+[γ2Δ(ξi)]2+[γ3Δ(ξi)]2(ti-ti-1)of the Δ-integrable function [γ1Δ(t)]2+[γ2Δ(t)]2+[γ3Δ(t)]2, corresponding to the partition P of [a,b] and the choice of intermediate points ξi defined in (3.4). For every δ>0, there exists (see [6, Lemma 5.7]) at least one partition P of [a,b] of the form (2.19) such that for each i∈{1,2,…,n} either ti-ti-1≤δ or ti-ti-1>δ and σ(ti-1)=ti. Let us denote by 𝒫δ([a,b]) the set of all such partitions P of [a,b]. For an arbitrary ε>0, there exists δ>0 such that
(3.13)|l(γ,P)-I|<ε2∀P∈Pδ([a,b]),
where
(3.14)I=∫ba[γ1Δ(t)]2+[γ2Δ(t)]2+[γ3Δ(t)]2Δt.
From (3.4), (3.5), and (3.6) we get
(3.15)0≤γ1(ti)-γ1(ti-1)-γ1Δ(ξi)(ti-ti-1)≤[γ1Δ(ξi′)-γ1Δ(ξi)](ti-ti-1),0≤γ2(ηi)-γ2(ti-1)-γ2Δ(ηi)(ti-ti-1)≤[γ2Δ(ηi′)-γ2Δ(ηi)](ti-ti-1),0≤γ3(ζi)-γ3(ti-1)-γ3Δ(ζi)(ti-ti-1)≤[γ3Δ(ζi′)-γ3Δ(ζi)](ti-ti-1),and, consequently,
(3.16)γ1(ti)-γ1(ti-1)=[γ1Δ(ξi)+αi](ti-ti-1),γ2(ti)-γ2(ti-1)=[γ2Δ(ηi)+βi](ti-ti-1),γ3(ti)-γ3(ti-1)=[γ3Δ(ζi)+ωi](ti-ti-1),where
(3.17)0≤αi≤γ1Δ(ξi′)-γ1Δ(ξi)≤Mi-mi,0≤βi≤γ2Δ(ηi′)-γ2Δ(ηi)≤Ni-ni,0≤ωi≤γ3Δ(ζi′)-γ3Δ(ζi)≤Ri-ri,in which Mi and mi are the supremum and infimum of γ1Δ on [ti-1,ti) and Ni, ni and Ri, ri are corresponding numbers for γ2Δ and γ3Δ, respectively. Using the inequality
(3.18)|x2+y2+z2-x12+y12+z12|≤|x-x1|+|y-y1|+|z-z1|for x,y,z,x1,y1,z1∈ℝ, we obtain
(3.19)|[γ1Δ(ξi)+αi]2+[γ2Δ(ηi)+βi]2+[γ3Δ(ζi)+ωi]2-[γ1Δ(ξi)]2+[γ2Δ(ξi)]2+[γ3Δ(ξi)]2|≤|αi|+|γ2Δ(ηi)+βi-γ2Δ(ξi)|+|γ3Δ(ζi)+ωi-γ3Δ(ξi)|≤|αi|+|βi|+|ωi|+|γ2Δ(ηi)-γ2Δ(ξi)|+|γ3Δ(ζi)-γ3Δ(ξi)|≤Mi-mi+2(Ni-ni)+2(Ri-ri).Therefore,
(3.20)|l(γ,P)-S|=|∑i=1n([γ1Δ(ξ)+αi]2+[γ2Δ(ηi)+βi]2+[γ3Δ(ζi)+ωi]2)(ti-ti-1)|≤∑i=1n[Mi-mi+2(Ni-ni)+2(Ri-ri)](ti-ti-1)=U(γ1Δ,P)-L(γ1Δ,P)+2[U(γ2Δ,P)-L(γ2Δ,P)]+2[U(γ3Δ,P)-L(γ3Δ,P)],where U and L denote the upper and lower Darboux Δ-sums, respectively. Since the functions [γ1Δ]2+[γ2Δ]2+[γ3Δ]2,γ1Δ,γ2Δ and γ3Δ are Δ-integrable over [a,b), for arbitrary ε>0, there exists δ>0 such that
(3.21)|S-I|<ε12,U(γjΔ,P)-L(γjΔ,P)<ε12,j=1,2,3for all P∈𝒫δ([a,b]) (see [6, Theorem 5.9]) and the Riemann definition of Δ-integrability therein), where I is defined by (3.14). Therefore, we get
(3.22)|l(Γ,P)-I|≤|l(Γ,P)-S|+|S-I|<ε12+2ε12+2ε12+ε12=ε2,and so the validity of (3.13) is proved. On the other hand, among the partitions P for which (3.13) is satisfied, there is a partition P such that
(3.23)|l(γ,P)-l(γ)|<ε2.
Indeed, there is a partition P* of [a,b] such that
(3.24)0≤l(γ)-l(γ,P*)<ε2.
Next, we refine the partition P* adding to it new partition points so that we get a partition P that belongs to 𝒫δ([a,b]). Then by l(Γ,P)≥l(Γ,P*), (3.24) yields
(3.25)0≤l(γ)-l(γ,P)<ε2,
so that (3.23) is shown. By (3.13) and (3.23), we get
(3.26)|l(Γ)-I|<ε.Since ε>0 is arbitrary, we have l(Γ)=I, and the proof is complete.Definition 3.3.
The curveγ:[a,b]⊂𝕋→ℝ3 is given in the parametric form γ(t)=(γ1(t),γ2(t),γ3(t)). Let P0:=γ(t0),P0σ:=γ(σ(t0)), for t0∈[a,b]κ2and P:=γ(t) for t∈[a,b]. We denote the angle between the forward tangent lines drawn to the curve at P0 and P by θP and the arc length of the segment PP0 of the curve by |PP0̂|. The forward curvature of γ at P0 is defined as
(3.27)κ=lim⁡P→P0σθP|PP0̂|.Theorem 3.4.
Letγ:[a,b]⊂𝕋→ℝ3 be a natural Δ-parametrized Δ-regular curve, there exists γΔΔ(t0) for t0∈[a,b]κ2. If t0 is a right-dense point, then
(3.28)κ=‖γΔΔ(t0)‖.
If t0 is a right-scattered point, then
(3.29)κ=2μ(t0)arcsin⁡(μ(t0)‖γΔΔ(t0)‖2).Proof.
LetP0:=γ(t0),P0σ:=γ(σ(t0)), for a fix point t0∈[a,b]κ2and P:=γ(t) for t∈[a,b]. Let the unit vectors on the forward tangent lines to the curve at points P and P0 be τ(t) and τ(t0), respectively, and assume that θ(t) is the angle between them. We denote l(t)=|PP0̂|. Then we have
(3.30)l(t)=∫t0t‖γΔ(s)‖Δs=∫t0tΔs=t-t0,(3.31)‖τ(t)-τ(t0)‖=2sinθ(t)2.
Assume that t0 is a right-dense point. Then P→P0=P0σ and θ(t)→0 as t→t0. Since
(3.32)lim⁡t→t0θ(t)/(t-t0)θ(t)/l(t)=lim⁡t→t0l(t)t-t0=1,
we obtain
(3.33)lim⁡t→t0θ(t)t-t0=lim⁡t→t0θ(t)l(t).
By using (3.31) we can write
(3.34)‖τ(t)-τ(t0)t-t0‖=2t-t0sinθ(t)2=sin⁡(θ(t)/2)θ(t)/2θ(t)t-t0.
Thus, we find
(3.35)κ=lim⁡t→t0θ(t)l(t)=‖γΔΔ(t0)‖
as t→t0.
Assume thatt0 is a right-scattered point. Then P→P0σ and θ(t)→θ(σ(t0)) as t→σ(t0). If the equality (3.31) is divided by t-t0, taking the limit as t→σ(t0), we have
(3.36)‖γΔΔ(t0)‖=‖τ(σ(t0))-τ(t0)σ(t0)-t0‖=2σ(t0)-t0sinθ(σ(t0))2=2μ(t0)sinθ(σ(t0))2.
(3.16) On the other hand, we find
(3.37)κ=lim⁡t→σ(t0)θ(t)l(t)=θ(σ(t0))σ(t0)-t0=θ(σ(t0))μ(t0).
It follows from (3.16) and (3.37) that
(3.38)κ=2μ(t0)arcsin(μ(t0)‖γΔΔ(t0)‖2).
This completes the proof.Theorem 3.5.
Letγ:[a,b]⊂𝕋→ℝ3 be a curve with arbitrary parameters that is second Δ-differentiable on [a,b]κ2 and ∥γΔ(t)∥≠0 for all t∈[a,b]κ2. Moreover assume that the function h:[a,b]κ2→ℝ defined by h(t):=〈γΔ(t),γΔ(t)〉 is strictly increasing. The forward curvature of γ at the right-dense point t∈[a,b]κ2 is
(3.39)κ=‖γΔ(t)×γΔΔ(t)‖‖γΔ(t)‖3.Proof.
The functionp:[a,b]→p([a,b])=𝕋̃ defined by s=p(t)=∫at∥γΔ(v)∥Δv is continuous and strictly increasing. Therefore 𝕋̃ will be a time scale (see [1, pages 560-561]). The forward jump operator and the derivative operator on this time scale, will be denoted by σ̃ and Δ̃ respectively. Since the curve γ∘p-1 is a natural Δ-parametrized, to find the curvature to the γ at the point t, it is sufficient to find that the curvature to the γ∘p-1 at the point s.                          (3.40)(p-1)Δ̃(s)=1pΔ(t)=1‖γΔ(t)‖,(p-1)Δ̃(σ̃(s))=1pΔ(p-1(σ̃(s)))=1pΔ(σ(t))=1‖γΔ(σ(t))‖=1‖γΔ(t)‖,(p-1)Δ̃Δ̃(s)=-pΔΔ(t)[pΔ(t)]2pΔ(σ(t))=-‖γΔ(t)‖Δ‖γΔ(t)‖3.
By using (3.40), one can easily find that
(3.41)‖(γ∘p-1)Δ̃Δ̃(s)‖=‖γΔ(t)(p-1)Δ̃Δ̃(s)+γΔΔ(t)(p-1)Δ̃(s)(p-1)Δ̃(σ̃(s))‖=‖‖γΔ(t)‖γΔΔ(t)-‖γΔ(t)‖ΔγΔ(t)‖‖γΔ(t)‖3.
As t∈[a,b]κ2 is a right-dense point and the function h(t):=〈γΔ(t),γΔ(t)〉 is strictly increasing, we have
(3.42)‖γΔ(t)‖Δ=〈γΔ(t),γΔΔ(t)〉‖γΔ(t)‖.
Substituting (3.42) into (3.41), we obtain
(3.43)κ=‖(γΔ(t)×γΔΔ(t))×γΔ(t)‖‖γΔ(t)‖4=‖(γΔ(t)×γΔΔ(t))‖‖γΔ(t)‖sin⁡(π/2)‖γΔ(t)‖4=‖(γΔ(t)×γΔΔ(t))‖‖γΔ(t)‖3.
The proof is complete.Remark 3.6.
It is easy to see that, for the case𝕋=ℝ, the results (3.28) and (3.39) generalize the following formulas stated in classical differential geometry:
(3.44)κ=‖γ′′(t0)‖,κ=‖γ′(t)×γ′′(t)‖‖γ′(t)‖3.Theorem 3.7.
Letγ:[a,b]⊂𝕋→ℝ3 be a curve with arbitrary parameters that is second Δ differentiable on [a,b]κ2and has continuous Δ derivative on [a,b]κ and ∥γΔ(t)∥≠0 for all t∈[a,b]κ. Moreover assume that the function h:[a,b]κ→ℝ defined by h(t):=〈γΔ(t),γΔ(t)〉 is strictly increasing. The forward curvature of γ at the right-scattered point t∈[a,b]κ2 is
(3.45)κ=2μ(t)‖γΔ(t)‖arcsin(μ(t)q12q2),
where
(3.46)q1=‖‖γΔ(t)‖(‖γΔ(t)‖+‖γΔ(σ(t))‖)γΔΔ(t)-〈γΔΔ(t),γΔ(t)+γΔ(σ(t))〉γΔ(t)‖,q2=‖γΔ(t)‖‖γΔ(σ(t))‖(‖γΔ(t)‖+‖γΔ(σ(t))‖).Proof.
Lett∈[a,b]κ2 be a right-scattered point. In this case we have
(3.47)μ̃(s)=σ̃(s)-s=p(σ(t))-p(t)=μ(t)‖γΔ(t)‖,(3.48)‖γΔ(t)‖Δ=〈γΔΔ(t),γΔ(t)+γΔ(σ(t))〉‖γΔ(t)‖+‖γΔ(σ(t))‖.
By using (3.41) and (3.48), we find
(3.49)‖(γ∘p-1)Δ̃Δ̃(s)‖=q1‖γΔ(t)‖2‖γΔ(σ(t))‖(‖γΔ(t)‖+‖γΔ(σ(t))‖).
Substituting (3.49) into (3.29), we obtain
(3.50)κ=2μ(t)‖γΔ(t)‖arcsin(μ(t)q12q2).Example 3.8.
Let𝕋=ℤ and γ:𝕋→ℝ3, γ(t)=β(cost,sint,t).The curve γ satisfies the conditions of Theorem 3.4 for the case β=1/3-2cos1. In this case we have σ(t)=t+1,μ(t)=1, and
(3.51)γΔ(t)=β(cos⁡(t+1)-cos⁡t,sin(t+1)-sint,1),γΔΔ(t)=β(cos⁡(t+2)-2cos⁡(t+1)+cos⁡t,sin(t+2)-2sin(t+1)+sint,0).
Since every point of 𝕋 is right-scattered point, the curvature of γ at any point t is
(3.52)κ=2μ(t)arcsin(μ(t)‖γΔΔ(t)‖2)=2arcsin(2βsin2(12)).
This value is the angle between the line through γ(t), γ(t+1) and the line through γ(t+1), γ(t+2).Example 3.9.
Assume thatγ:𝕋={1,1/2,1/3,…}∪{0}→ℝ3,γ(t)=(t,t2,0), is a non-Δ-natural parametrized curve. The only right-dense point of the time scale 𝕋 is t=0, and the other points of the time scale are right-scattered. The forward jump operator and the graininess function are
(3.53)σ(t)={t1-t,t≠1,1,t=1,μ(t)={t21-t,t≠1,0,t=1.
Furthermore we have
(3.54)γΔ(t)=(1,t2-2tt-1,0),fort≠1γΔΔ(t)=(0,2t-22t-1,0),fort≠1,t≠12,
and, from Theorem 3.5, the forward curvature of the curve γ at the point t=0 is
(3.55)κ(0)=‖γΔ(0)×γΔΔ(0)‖‖γΔ(0)‖3=2.
For the right-scattered point t=1/3, by using Theorem 3.7, we find
(3.56)κ(13)=7261arcsin(22793-67934(313+61)).
This value is the ratio of the angle between the line through the points γ(1/3),  γ(1/2) and the line through the points γ(1/2), γ(1) and the distance between the points γ(1/3) and γ(1/2).



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*Source: 805948-2011-06-07.xml*