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We present the normal and osculating planes of the curves parameterized by a compact subinterval of a time scale.
(ProQuest: ... denotes non-US-ASCII text omitted.)
Recommended by Ferhan M. Atici
Matematik Bölümü, Fen-Edebiyat Fakültesi, Mugla Üniversitesi, Mugla, 48000, Turkey
Received 19 March 2010; Accepted 28 June 2010
1. Introduction
Concept of calculus on time scales (or measure chains) was initiated by Hilger and Aulbach [1, 2] in order to unify discrete and continuous analyses.This theory is appealing because it provides a useful tool for modeling dynamical processes. Since a time-scale is a closed subset of the reals [3], curves may have scattered points in multidimensional time scale spaces. Therefore, Δ -differentiation plays a major role in investigation of curves parameterized by an arbitrary time scale.
The results in this paper were motivated by geometric interpretation of the results presented in [4].
In this paper, we consider planes whose normal is Δ -differentiable vector that is each component of the vector is Δ -differentiable (i.e., normal planes) and which contain first and second order Δ -differentiable vectors (i.e., osculating planes). In this study we present the normal and osculating planes of the curves parameterized by a compact subinterval of a time scale. Since we need vector valued functions to study Δ -differentiable vectors of curves, we first define the concept of vector valued functions on time scales in Section 2. In [5] Guseinov and Özylmaz introduced the tangent line for Δ -regular curves in 3-dimensional time scales; then in [4] Bohner and Guseinov obtained the equation of such tangent line. The tangent line can also be studied in the concept of partial Δ -differentiation. In Section 3, we obtain the equations of tangent vectors of planar curves by using partial Δ -differentiation. Then we derive the equation of the normal plane for a Δ -regular curve. In Section 4, we present the basic theorem to construct osculating plane of a curve and obtain the equation of this plane by using first- and-second order Δ -derivatives.
We refer the reader to resources such as [3, 4, 6, 7] and [8, 9] for more detailed discussions on the calculus of time scales and on the differential geometry of curves, respectively.
2. Vector-Valued Functions on Time Scales
Let n be fixed. Let ...i denote a time scale for each i∈{1,2,...,n} . Let us set [figure omitted; refer to PDF] We call Λn an n -dimensional time scale. Λn is also a complete metric space with [figure omitted; refer to PDF]
Let a time-scale parameter t vary in an interval [a,b] . If to each value t∈[a,b] we assign a vector r(t) , then we say that a vector-valued function r(t) with argument t∈[a,b] is given. Assume that coordinates x1 ,x2 ,...,xn are fixed; then the representation of vector-valued function r(t) is equivalent to the representation of scalar functions x1 (t),x2 (t),...,xn (t) ; that is, r(t)={x1 (t),...,xn (t)} .
Definition 2.1.
A vector r0 is called the limit of the vector-valued function r(t) as t[arrow right]t0 if the length of the vector r(t)-r(t0 ) tends to zero as t[arrow right]t0 . Here we write [figure omitted; refer to PDF] It is clear that the vector-valued function r(t) has a limit if and only if each one of the functions x1 (t),...,xn (t) has a limit as t[arrow right]t0 .
Definition 2.2.
Δ -Derivative of a vector-valued function can be obtained by Δ -differentiating components x1 (t),...,xn (t) of r(t) ; that is, [figure omitted; refer to PDF] Precisely, for the Δ -derivative rΔ (t) of the vector-valued function r(t) , we call the limit [figure omitted; refer to PDF] If this limit exists, then r(t) is called Δ -differentiable.
Proposition 2.3.
Let r1 (t) and r2 (t) be vector-valued functions. Then
(i) (r1 (t)+r2 (t))Δ =r1Δ (t)+r2Δ (t),
(ii) (r1r2 )Δ =r1Δr2 +r1σr2Δ .
The Δ -differentiation of the inner products and vector products of vector-valued functions, is computed by the consecutive differentiation of the cofactors.
Proposition 2.4.
Let r1 (t) and r2 (t) be vector-valued functions, let × be Euclidean vector product, · and let Euclidean inner product. Then
(i) (r1 ·r2)Δ =r1Δ ·r2 +r1σ ·r2Δ ,
(ii) (r1 ×r2)Δ =r1Δ ×r2 +r1σ ×r2Δ =r1Δ ×r2σ +r1 ×r2Δ .
Definition 2.5 (Taylor's expansion for vector-valued functions).
Assume that n times Δ -derivative of the vector-valued function r(t) exist and are rd -continuous, then we can write Taylor's expansions for the components; x1 (t),...,xn (t) as [figure omitted; refer to PDF] where h0 (r,s)≡1 , hk+1 (r,s)=∫srhk (τ,s)Δτ for k∈...0 , and [figure omitted; refer to PDF] for i={1,...,n} .
This system of three equations can be written as [figure omitted; refer to PDF] where o(gn (t,t0 )) denotes a vector whose length is an infinitesimal since lim t[arrow right]t0gn (t,t0 )=0 .
Remark 2.6.
There exists one essential difference between Taylor's expansions of vector-valued function and scalar function. If we consider Taylor's expansion for a scalar function f(t) , then we have [figure omitted; refer to PDF] where ξ is a point between ρn-1 (t) and t0 . For a vector-valued function we cannot write similar formula for the corresponding infinitesimal vector, because in general for different components of the vector o(gn (t,t0 )) the corresponding points ξ are different. However, it is more important to note that the length of the vector o(gn (t,t0 )) is an infinitesimal with respect to gn (t,t0 ) .
3. Tangent Line to a Curve
Let ... be a time scale.
Definition 3.1.
A Δ -regular curve Γ is defined as a mapping [figure omitted; refer to PDF] of the segment [a,b]⊂... , a<b , to the space ...3 , where f1 ,f2 ,f3 are real-valued functions defined on [a,b] and Δ -differentiable on [a,b]κ with rd -continuous Δ -derivatives and [figure omitted; refer to PDF]
Definition 3.2.
A line [Lagrangian (script capital L)]0 passing through the point P0 is called the delta tangent line to the curve Γ at the point P0 if the following held.
(i) [Lagrangian (script capital L)]0 passes also through the point P0σ =(x(σ(t0 )),y(σ(t0 )),z(σ(t0 ))).
(ii) If Po is not an isolated point of the curve Γ , then
[figure omitted; refer to PDF] where P is the moving point of the curve Γ , d(P,[Lagrangian (script capital L)]0 ) is the distance from the point P to the line [Lagrangian (script capital L)]0 , and d(P,P0 ) is the distance from the point P to the point P0σ .
Theorem 3.3.
For any point P0 of the curve Γ there exists the tangent to Γ at P0 and the directing vector of the tangent is Δ -differential of its position vector function rΔ (t0 ) , where r(t0 )=P0 for t0 ∈... .
Proof.
This theorem can be proven as in [5], Theorem 3.3.
Let three functions x:...[arrow right]... , y:...[arrow right]... , and z:...[arrow right]... be given. Let us set x(...):=...1 , y(...)=...2 , and z(...)=...3 . We will assume that ...1 , ...2 , and ...3 are time scales. Denote by σ1 Δ1 , σ2 Δ2 , σ3 Δ3 the forward jump operators and delta operators for ...1 , ...2 , and ...3 , respectively.
Under the above assumptions, let functions [varphi]:...×...×...[arrow right]... and [straight phi]:...×...×...[arrow right]... be given.
Consider a space curve given by two equations. [figure omitted; refer to PDF] If x=x(t) , y=y(t) , z=z(t) is the position vector of the considered curve, then, substituting these three functions into (3.4), we obtain two equalities: [figure omitted; refer to PDF] If the functions [varphi] and [straight phi] are σ1 -completely differentiable, then, Δ -differentiation of these two equalities leads [figure omitted; refer to PDF] If [varphi] and [straight phi] are σ2 -completely differentiable, then Δ -differentiation of (3.5) leads us to obtain the following two equations: [figure omitted; refer to PDF] If [varphi] and [straight phi] are σ3 -completely differentiable, then Δ -differentiation of (3.5) leads us to obtain the following two equations: [figure omitted; refer to PDF] Other combinations of σi -completely differentiability of [varphi] and [straight phi] can be shown similarly. The components {xΔ ,yΔ ,zΔ } of the tangent vector satisfy the system consisting of two equations: (3.6), (3.7), and (3.8).
Assume that [varphi] is σ1 -completely differentiable planar curve given by the equations [varphi](x,y)=0 , z=0 satisfying the condition (∂[varphi]/Δ1 x)2 +(∂[varphi]σ1 /Δ2 y)2 ≠0 ; then the components of the tangent vector rΔ ={xΔ ,yΔ } are the solution of the linear equation [figure omitted; refer to PDF]
Therefore, {xΔ ,yΔ }=μ{-∂[varphi]σ1 /Δ2 y,∂[varphi]/Δ1 x} , and the equation of tangent is [figure omitted; refer to PDF] If planar curve [varphi] is σ2 -completely differentiable, then equation of tangent plane becomes [figure omitted; refer to PDF]
Definition 3.4.
Let Γ be a smooth and completely differentiable space curve. The plane passing through points P0 ∈Γ and orthogonal to the vector tangent to Γ at P0 is called the plane normal to Γ at P0 .
Denote by r... the position vector of the normal plane. Since this plane is orthogonal to the vector rΔ and contains the point with position vector r...-r(t0 ) , the equation of the normal plane is [figure omitted; refer to PDF] The vectors orthogonal to the tangent are called the vectors normal to Γ .
4. Osculating Plane of a Curve
Let P0 be a point of a curve Γ . Take two points Q1 ,Q2 ∈Γ situated right side of P0σ . If the points Q1 and Q2 tend to P0σ , then the limit position of the plane containing P0 ,P0σ ,Q1 ,Q2 is called the osculating plane of Γ at the point P0 .
Theorem 4.1.
Let Γ be a Δ -regular curve represented as r=r(t) . Assume that the vectors rΔ and rΔ2 are not collinear at point P0 . Then there exists the osculating plane of Γ at P0 and it is spanned by the vectors rΔ and rΔ2 .
Proof.
If P0 =P0σ , that is, P0 is right-dense point of Γ , then this theorem can be proven as in differential geometry concept.
Let P0 be a right-scattered point of Γ . Then, the positions vector of [arrow right]P0Q1 and [arrow right]P0Q2 are a1 =r(t0 +τ1 )-r(t0 ) and a2 =r(t0 +τ2 )-r(t0 ) , respectively. That is, these vectors, if linearly independent, span the plane E .
This plane is also spanned by the vectors v(i) =ai /τi for i∈{1,2} or by the vectors [figure omitted; refer to PDF] By the means of Taylor's formula, we have [figure omitted; refer to PDF] Hence, we obtain [figure omitted; refer to PDF] Consequently, if τi [arrow right]0 for i∈{1,2} , then v(1) [arrow right]rΔ (t0 ) and w[arrow right]rΔ2 (t0 ) .
These vectors, if linearly independent, determine the limiting position of the plane E passing through the points P0 ,P0σ ,Q1 ,Q2 .
Corollary 4.2.
If the vectors rΔ (t0 ) and rΔ2 (t0 ) are collinear, then the limit position of considering plane is not determined. For instance, take a straight line [figure omitted; refer to PDF] where a,b are constant vectors and t∈... . Then [figure omitted; refer to PDF] so the osculating plane of the straight line is not determined uniquely. If rΔ (t) and rΔ2 (t) are collinear, then the corresponding point of Γ is called the straightening point of Γ .
Theorem 4.3.
The osculating plane of a planar curve coincides with the plane containing this curve.
Proof.
Let us consider the Taylor expansion of the position vector r(t) at the neighborhood of P0 : [figure omitted; refer to PDF] The curve Γ¯ , determined by the expantion, [figure omitted; refer to PDF] is situated in the osculating plane of Γ at P0 ; the difference between the position vectors of Γ and Γ¯ is a sufficiently small vector [figure omitted; refer to PDF] Hence a sufficiently small neighborhood of P0 on the space curve Γ is near to the planar curve Γ¯ situated in the osculating plane of Γ at P0 .
Now let us write the equation of the osculating plane of Γ at P0 . Let r... be the position vector of the osculating plane. Since rΔ and rΔ2 span the osculating plane, the vector product rΔ ×rΔ2 is orthogonal to the osculating plane. The vector r...-r(t0 ) belongs to the osculating plane; therefore, the inner product of these vectors is equal to zero: [figure omitted; refer to PDF]
With respect to coordinate functions, this equation has the following form: [figure omitted; refer to PDF]
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