Normal and Osculating Planes of [Delta] -Regular Curves

Abstract

We present the normal and osculating planes of the curves parameterized by a compact subinterval of a time scale.

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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} =^r1Δ_r2 +^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)=^∫sr_hk (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]

References

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