A Note on the Infinitesimal Bending of a Rectifying Curve

Abstract

Both notions, of an infinitesimal bending of a curve and of a rectifying curve, play important roles in the theory of curves. In this short note, we begin the study of the infinitesimal bending of a rectifying curve.

1. Introduction

Infinitesimal deformation theory is a very active research field and plays an important role in the development of Differential Geometry. It has interesting applications in physics, medicine, biology, architecture, mechanics, engineering, etc. (see [1,2]). A special place in this theory is devoted to the study of infinitesimal bending. Various authors have investigated the infinitesimal bending of surfaces, curves, and manifolds (see [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]).

Infinitesimal bending is characterized by the stationarity of the arc length with appropriate precision. It can be expressed by an equation of the form [6]:

$$d{s}_{\xi }^2-d{s}^2=o\left(\xi \right),\xi >0,\xi \to 0.$$

Infinitesimal bending of curves in Euclidean 3-dimensional space has been studied in many papers (see, for example, [5,6,7,13]).

The behavior of some geometric magnitudes under second-order infinitesimal bending of a curve is described in [5]. In 2018, in [6], the authors considered curves that lie on the ruled surfaces in ${\mathbb{E}}^3$ and studied their infinitesimal bending. An infinitesimal bending field under whose effect all bent curves remain on the same ruled surface as the initial curve was obtained. Their study continued in [13] with infinitesimal bending of curves lying with a given precision also on ruled surfaces in Euclidean 3-dimensional space. Special cases of ruled surfaces were taken (cylinder, hyperbolic paraboloid, and helicoid) and their infinitesimal bending field was obtained. Using program packet Mathematica, good examples were graphically presented.

In [14], in order to obtain a ruled surface, the authors considered the infinitesimal bending of a curve. Recently, in [17], the authors investigated the total torsion of a spherical curve during infinitesimal bending.

A good generalization of the main statements of the theory of infinitesimal bendings in ${\mathbb{E}}^3$ to dual curves in the dual 3-space ${\mathbb{D}}^3$ is given in [18].

On the other hand, in [19], B.Y. Chen introduced the concept of a rectifying curve as a special curve whose position vector field always lies in its rectifying plane. The author proved simple characterizations and classified all rectifying curves in ${\mathbb{E}}^3$.

Space curves satisfying $\tau /k=as+b$ were studied in [20]. The authors proved that a curve has this property if and only if there exists a point such that all of the rectifying planes to the curve pass through this point. Any curve of this type, using a translation, turns out to be a rectifying curve in the sense of the definition given by Chen in [19].

Some fundamental interesting properties of rectifying curves are presented in [19,20,21,22,23]. In [24], geodesics on an arbitrary cone in Euclidean 3-dimensional space were studied via rectifying curves. The author proved that a curve on a cone in Euclidean 3-dimensional space is a geodesic if and only if it is either a rectifying curve or an open portion of a rulling.

Recently, a good characterization of rectifying curves in terms of involutes and evolutes was presented by the second author of this paper and her co-workers in [25].

All of these articles motivated us to initiate the study of the infinitesimal bending of rectifying curves. More precisely, rectifying curves are considered, and their infinitesimal bendings are investigated for particular cases. Geometrical interpretations are given. A conjecture is formulated and new research directions are suggested.

2. Preliminaries

Let $c\left(s\right)$, $s\in I$, be a unit speed curve in the 3-dimensional Euclidean space ${\mathbb{E}}^3$, parameterized by the arc length s. A family of curves $c_{\xi }$ is an infinitesimal bending of the curve c if it is of the form

$$c_{\xi }\left(s\right)=c\left(s\right)+\xi z\left(s\right),$$

with $\xi =$ parameter $\ge 0$, z the infinitesimal bending field of c, with certain properties that will be defined next.

One can choose z, such that $\left|\right|z^{\prime }\left(s\right)\left|\right|=1$ is defined in the points of curve c, i.e.,

$$z\left(s\right)={\alpha }_1\left(s\right)t\left(s\right)+{\alpha }_2\left(s\right)n\left(s\right)+{\alpha }_3\left(s\right)b\left(s\right),$$

with $\{t,n,b\}$ the Frenet–Serret frame of c.

According to [8], z is an infinitesimal bending field of c if and only if

$${\alpha }_1^{\prime }\left(s\right)=k\left(s\right){\alpha }_2\left(s\right),$$

(1)

where $k\left(s\right)$ is the curvature of c.

If $k_{\xi }\left(s\right),{\tau }_{\xi }\left(s\right)$ is the curvature and $c_{\xi }\left(s\right)$ is the torsion, respectively, the following relations hold [8]:

$$k_{\xi }=k+\xi [k^{\prime }{\alpha }_1+{\alpha }_2^{″}+(k^2-{\tau }^2){\alpha }_2-2\tau {\alpha }_3^{\prime }-{\alpha }_3{\tau }^{\prime }],$$

(2)

$${\tau }_{\xi }=\tau +\xi \{{\tau }^{\prime }{\alpha }_1+k({\alpha }_3^{\prime }+2\tau {\alpha }_2)+{\left[{\displaystyle \frac{1}{k}}(2\tau {\alpha }_2^{\prime }+{\tau }^{\prime }{\alpha }_2+{\alpha }_3^{″}-{\tau }^2{\alpha }_3)\right]}^{\prime }\}.$$

(3)

On the other hand, rectifying curves form a special class of curves, with good geometrical properties. More precisely, a unit speed space curve $c\left(s\right)$ with $k\left(s\right)\ne 0$, for any $s\in I$, is a rectifying curve (i.e., its position vector always lies in its rectifying plane-see [19] for the definition and [19,25] for equivalent definitions and important geometric properties and characterizations) if and only if there exist the constants $a,b\in \mathbb{R},a\ne 0,$ such that

$${\displaystyle \frac{\tau \left(s\right)}{k\left(s\right)}}=as+b,s\in I.$$

(4)

Remark 1.

If s is the arc length of the curve c and $s^{*}$ is the arc length of $c_{\xi }$, then

$$s^{*}={\displaystyle \frac{1}{\sqrt{1+{\xi }^2}}}s.$$

(5)

This relation is easy to prove: one has $c_{\xi }\left(s\right)=c\left(s\right)+\xi z\left(s\right)$, with $\left|\right|c^{\prime }\left(s\right)\left|\right|=1$. One can choose $z\left(s\right)$, with $\left|\right|z^{\prime }\left(s\right)\left|\right|=1$. Using the definition of the bending vector field (see [8]),

$$⟨c^{\prime }\left(s\right),z^{\prime }\left(s\right)⟩=0;$$

then

$$\left|\right|c_{\xi }^{\prime }\left(s\right){\left|\right|}^2=⟨c^{\prime }\left(s\right)+\xi z^{\prime }\left(s\right),c^{\prime }\left(s\right)+\xi z^{\prime }\left(s\right)⟩=1+{\xi }^2,$$

which implies (5).

According to (4), it follows that $c_{\xi }$ is rectifying if and only if

$${\displaystyle \frac{{\tau }_{\xi }\left(s^{*}\right)}{k_{\xi }\left(s^{*}\right)}}={\displaystyle \frac{{\tau }_{\xi }\left(s\right)}{k_{\xi }\left(s\right)}}=A{s}^{*}+B=A·{\displaystyle \frac{1}{\sqrt{1+{\xi }^2}}}s+B,$$

(6)

$A,B\in \mathbb{R},A\ne 0.$

3. The Infinitesimal Bending of a Rectifying Curve

In this paper, the following Problem is considered:

Problem. Under which conditions is the infinitesimal bending $c_{\xi }$ of a rectifying curve c also rectifying?

We study some particular cases. The general problem has not yet been solved.

Equations (2)–(6) are applied to provide solutions to our problem in the following specific cases.

In the next two cases (see Section 3.1 and Section 3.2 below), ${\alpha }_2=0$ is considered in order to simplify relations (2) and (3).

3.1. Constant Curvature

Case 1: $z\left(s\right)={\alpha }_1\left(s\right)t\left(s\right)+{\alpha }_2\left(s\right)n\left(s\right)+{\alpha }_3\left(s\right)b\left(s\right),$ with ${\alpha }_1\left(s\right)={\alpha }_1=\mathrm{constant},$ ${\alpha }_2\left(s\right)=0,$ ${\alpha }_3\left(s\right)={\alpha }_3=\mathrm{constant},$ ${\alpha }_1^2+{\alpha }_3^2\ne 0,$ $k\left(s\right)=k=\mathrm{constant}$.

Then, $\tau \left(s\right)=(as+b)k,$ which resuts in ${\tau }^{\prime }\left(s\right)=ak.$

From (2), we have

$$k_{\xi }=k-\xi {\alpha }_{3}ak.$$

From (3), we obtain

$${\tau }_{\xi }=ak(1-2a\xi {\alpha }_3)s+(c+\xi ak{\alpha }_1-2abk\xi {\alpha }_3).$$

Coefficients A and B are calculated after applying (6).

More precisely, from ${\tau }_{\xi }\left(s\right)=(A{s}^{*}+B)·k_{\xi }\left(s\right)$, the following results

$$ak(1-2a\xi {\alpha }_3)s+(c+\xi ak{\alpha }_1-2abk\xi {\alpha }_3)=(A{\displaystyle \frac{1}{\sqrt{1+{\xi }^2}}}s+B)(-\xi {\alpha }_{3}ak+k).$$

By equalizing the coefficients of s and free terms, one obtains

$$A={\displaystyle \frac{a(1-2a\xi {\alpha }_3)\sqrt{1+{\xi }^2}}{1-a\xi {\alpha }_3}},$$

$$B={\displaystyle \frac{b+\xi {\alpha }_{1}a}{1-a\xi {\alpha }_3}}.$$

Obviously, for ${\alpha }_3=0$, one obtains

$$A=a\sqrt{1+{\xi }^2},B=b+\xi {\alpha }_{1}a.$$

If ${\alpha }_3\ne 0,$ the existence of A and B depends on the parameter $\xi$, and then cannot be assured for any parameter $\xi$.

Then, for ${\alpha }_3=0$, $c_{\xi }$ is a rectifying curve, with ${\displaystyle \frac{{\tau }_{\xi }\left(s^{*}\right)}{k_{\xi }\left(s^{*}\right)}}=A{s}^{*}+B.$

We prove the following

Theorem 1.

Let $c\left(s\right)$ be a unit speed rectifying curve with a constant curvature and $c_{\xi }$ is its infinitesimal bending curve, with z representing the infinitesimal bending vector field along the tangent, i.e., $z\left(s\right)={\alpha }_{1}t\left(s\right),$ ${\alpha }_1=constant\ne 0.$ Then, $c_{\xi }$ is a rectifying curve.

3.2. Constant Torsion

Case 2: $z\left(s\right)={\alpha }_1\left(s\right)t\left(s\right)+{\alpha }_2\left(s\right)n\left(s\right)+{\alpha }_3\left(s\right)b\left(s\right),$ with ${\alpha }_1\left(s\right)=\mathrm{constant},$ ${\alpha }_2\left(s\right)=0,$ ${\alpha }_3\left(s\right)=\mathrm{constant},$ ${\alpha }_1^2+{\alpha }_3^2\ne 0,$ $\tau \left(s\right)=\tau =\mathrm{constant}.$

Because $\tau \left(s\right)=\tau =\mathrm{constant},$ then $k\left(s\right)={\displaystyle \frac{\tau }{as+b}}$ and $k^{\prime }=-{\displaystyle \frac{\tau a}{{(as+b)}^2}}.$

From (2), one has

$$k_{\xi }\left(s\right)=k-{\displaystyle \frac{\xi \tau {\alpha }_{1}a}{{(as+b)}^2}}.$$

From (3), one obtains

$${\tau }_{\xi }\left(s\right)=\tau -\xi \tau {\alpha }_{3}a.$$

Now, we calculate coefficients A and B from (6):

$$\tau -\tau \xi {\alpha }_{3}a=\left[A{\displaystyle \frac{1}{\sqrt{1+{\xi }^2}}}s+B\right]\left({\displaystyle \frac{\tau }{as+b}}-{\displaystyle \frac{\tau \xi {\alpha }_{1}a}{{(as+b)}^2}}\right),$$

$$1-\xi {\alpha }_{3}a=\left[A{\displaystyle \frac{1}{\sqrt{1+{\xi }^2}}}s+B\right]\left({\displaystyle \frac{1}{as+b}}-{\displaystyle \frac{\xi {\alpha }_{1}a}{{(as+b)}^2}}\right),$$

$${(as+b)}^2(1-\xi {\alpha }_{3}a)=\left[A{\displaystyle \frac{1}{\sqrt{1+{\xi }^2}}}s+B\right]\left(as+b-\xi {\alpha }_{1}a\right).$$

By equalizing the coefficients of s, one obtains

$$B=(1-\xi {\alpha }_3)(b+\xi {\alpha }_{1}a).$$

By equalizing the free terms of both sides, we obtain

$$a^2{\xi }^2{\alpha }_1^2=0.$$

But, $a,\xi \ne 0$ and it follows that ${\alpha }_1=0,$ and then ${\alpha }_3\ne 0.$

By equalizing the coefficients of $s^2$, we obtain

$$a^2(1-\xi {\alpha }_{3}a)=A{\displaystyle \frac{1}{\sqrt{1+{\xi }^2}}}a.$$

It follows that $A=a(1-\xi {\alpha }_{3}a)\sqrt{1+{\xi }^2},$ and then $A\ne 0$ for $\xi \ne {\displaystyle \frac{1}{{\alpha }_{3}a}}.$

Then, $z\left(s\right)$ is written as

$$z\left(s\right)={\alpha }_{3}b\left(s\right).$$

We proved the following

Theorem 2.

Let $c\left(s\right)$ be a unit speed rectifying curve of constant torsion τ, with ${\displaystyle \frac{\tau }{k\left(s\right)}}=as+b,a\ne 0$ and $c_{\xi }$ its infinitesimal bending curve, with z the infinitesimal bending vector field along the binormal, i.e., $z\left(s\right)={\alpha }_{3}b\left(s\right),$ ${\alpha }_3=constant\ne 0.$ Then, $c_{\xi }$ is a rectifying curve, for any parameter $\xi \ge 0,$ $\xi \ne {\displaystyle \frac{1}{{\alpha }_{3}a}}.$

Remark 2.

$k\left(s\right)$ and $\tau \left(s\right)$ of a rectifying curve $c\left(s\right)$ cannot be constant at the same time, because ${\displaystyle \frac{\tau \left(s\right)}{k\left(s\right)}}$ is a linear function of s for a rectifying curve.

3.3. Other Cases

Some situations when ${\alpha }_2\left(s\right)={\alpha }_2=\mathrm{constant}\ne 0$ will be discussed next.

Case 3: $z\left(s\right)={\alpha }_1\left(s\right)t\left(s\right)+{\alpha }_2\left(s\right)n\left(s\right)+{\alpha }_3\left(s\right)b\left(s\right),$ with ${\alpha }_1\left(s\right),{\alpha }_3\left(s\right)$—linear functions, ${\alpha }_2\left(s\right)={\alpha }_2=$ constant $\ne 0$ and $k\left(s\right)=k=$ constant.

Then, c rectifying does not imply $c_{\xi }$ rectifying.

Short Proof of Case 3.

One obtains $\tau \left(s\right)=(as+b)k$ and then ${\tau }^{\prime }\left(s\right)=ak$.

Applying these relations in (2), (3), and (6), by equalizing the coefficients of $s^3$ in both sides of relation (6), we obtain the equality

$$A{\displaystyle \frac{1}{\sqrt{1+{\xi }^2}}}(-a^2)k^2\xi {\alpha }_2=0,$$

for any parameter $\xi$.

But $A,a,{\alpha }_2,k\ne 0$ and then this case is not possible, i.e., c rectifying does not imply $c_{\xi }$ rectifying. □

Case 4: $z\left(s\right)={\alpha }_1\left(s\right)t\left(s\right)+{\alpha }_2\left(s\right)n\left(s\right)+{\alpha }_3\left(s\right)b\left(s\right),$ with ${\alpha }_1\left(s\right)$—linear function, ${\alpha }_2\left(s\right)={\alpha }_2=$ constant $\ne 0$, ${\alpha }_3\left(s\right)={\alpha }_3=\mathrm{constant}$, and $k\left(s\right)=k=$ constant.

Then, c rectifying does not imply $c_{\xi }$ rectifying.

The Proof of Case 4 is similar to the previous one and we obtain the same contradiction.

Then, from Case 3 and Case 4, we conclude the following

Theorem 3.

Let $c\left(s\right)$ be a unit speed rectifying curve with a constant curvature and $c_{\xi }$ is its infinitesimal bending curve, with z the infinitesimal bending vector field given by $z\left(s\right)={\alpha }_1\left(s\right)t\left(s\right)+{\alpha }_2\left(s\right)n\left(s\right)+{\alpha }_3\left(s\right)b\left(s\right),$ with ${\alpha }_1\left(s\right)$ linear function, ${\alpha }_2\left(s\right)={\alpha }_2=$ constant $\ne 0,$ ${\alpha }_3\left(s\right)={\alpha }_3=$ constant, or ${\alpha }_3\left(s\right)$ linear function. Then, $c_{\xi }$ is not a rectifying curve.

Remark 3.

This is similar to Case 3 and Case 4, but $\tau \left(s\right)=\tau =constant$ can be considered. Both these cases imply $k\left(s\right)=k=constant,$ which is not possible for rectifying curves, according to Remark 2.

Next, we study a more general case, more precisely ${\alpha }_2\left(s\right)\ne 0$ (not necessarily constant) together with other geometrical restrictions.

The most natural choice for ${\alpha }_2\left(s\right)$ is to be a linear function, which represents the hypothesis of the next two cases.

Case 5: $z\left(s\right)={\alpha }_1\left(s\right)t\left(s\right)+{\alpha }_2\left(s\right)n\left(s\right)+{\alpha }_3\left(s\right)b\left(s\right),$ with ${\alpha }_2\left(s\right)={\beta }_{2}s+{\gamma }_2,{\beta }_2\ne 0,$ ${\alpha }_3\left(s\right)={\alpha }_3=\mathrm{constant}$ and $k\left(s\right)=k=$ constant. Then, c rectifying does not imply $c_{\xi }$ rectifying.

Case 6: $z\left(s\right)={\alpha }_1\left(s\right)t\left(s\right)+{\alpha }_2\left(s\right)n\left(s\right)+{\alpha }_3\left(s\right)b\left(s\right),$ with ${\alpha }_2\left(s\right)={\beta }_{2}s+{\gamma }_2,{\beta }_2\ne 0,$ ${\alpha }_3\left(s\right)={\alpha }_3=\mathrm{constant}$ and $\tau \left(s\right)=\tau =$ constant. Then, c rectifying does not imply $c_{\xi }$ rectifying.

Proof of Case 5.

Because of $\left(1\right)$, it is obvious that ${\alpha }_1\left(s\right)={\displaystyle \frac{k{\beta }_2}{2}}{s}^2+k{\gamma }_{2}s.$

A $c\left(s\right)$ rectifying curve with a constant curvature k implies ${\displaystyle \frac{\tau \left(s\right)}{k}}=as+b$ and then $\tau \left(s\right)=aks+bk.$

From (2), the curvature $k_{\xi }\left(s\right)$, and from (3), the torsion ${\tau }_{\xi }\left(s\right)$, are calculated.

After substituting them into (6) and by equalizing the coefficients of $s^4$, we obtain

$$A{a}^2{k}^2{\beta }_2\xi {\displaystyle \frac{1}{\sqrt{1+{\xi }^2}}}=0,$$

for any parameter $\xi$.

But, $a,{\beta }_2,k\ne 0$ and then we obtain $A=0$, which is impossible.

Then, $c_{\xi }$ is not a rectifying curve. □

From Case 5, one obtains the following

Theorem 4.

Let $c\left(s\right)$ be a unit speed rectifying curve with a constant curvature and $c_{\xi }$ is its infinitesimal bending curve, with z the infinitesimal bending vector field given by $z\left(s\right)={\alpha }_1\left(s\right)t\left(s\right)+{\alpha }_2\left(s\right)n\left(s\right)+{\alpha }_3\left(s\right)b\left(s\right),$ with ${\alpha }_2\left(s\right)={\beta }_{2}s+{\gamma }_2,{\beta }_2\ne 0,$ ${\alpha }_3\left(s\right)={\alpha }_3=\mathit{constant}.$ Then, $c_{\xi }$ is not a rectifying curve.

Proof of Case 6.

A $c\left(s\right)$ rectifying curve owith a constant torsion $\tau$ implies ${\displaystyle \frac{\tau }{k\left(s\right)}}=as+b$ and then $k\left(s\right)={\displaystyle \frac{\tau }{as+b}}.$ Obviously, $\tau \ne 0.$

From (1), one obtains

$${\alpha }_1\left(s\right)=\tau \left[{\displaystyle \frac{{\beta }_2}{a}}s+{\displaystyle \frac{1}{a}}({\gamma }_2-{\displaystyle \frac{{\beta }_{2}b}{a}})ln|as+b|\right].$$

We calculate from (2) the curvature $k_{\xi }\left(s\right)$ and from (3) the torsion ${\tau }_{\xi }\left(s\right)$, and substitute them in (6).

After extensive calculations, we obtained that the only possible cases are ${\gamma }_2=b=0,$ i.e., ${\alpha }_2\left(s\right)={\beta }_{2}s$ and ${\displaystyle \frac{\tau }{k\left(s\right)}}=as.$

Under these circumstances, by equalizing the coefficients of $s^4$ in relation (6), one arrives at

$$A{a}^2{\tau }^2{\beta }_2{\displaystyle \frac{1}{\sqrt{1+{\xi }^2}}}=0,$$

for any parameter $\xi ,$ which results in a contradiction, again. □

Next, we obtained from the proof of Case 6.

Theorem 5.

Let $c\left(s\right)$ be a unit speed rectifying curve of constant torsion and $c_{\xi }$ its infinitesimal bending curve, with z the infinitesimal bending vector field given by $z\left(s\right)={\alpha }_1\left(s\right)t\left(s\right)+{\alpha }_2\left(s\right)n\left(s\right)+{\alpha }_3\left(s\right)b\left(s\right),$ with ${\alpha }_2\left(s\right)={\beta }_{2}s,{\beta }_2\ne 0,$ ${\alpha }_3\left(s\right)={\alpha }_3=\mathit{constant}.$ Then, $c_{\xi }$ is not a rectifying curve.

4. Geometrical Interpretations, Conclusions, and Further Developments

Infinitesimal variations and bendings of submanifolds represent a modern topic in Differential Geometry (see [11,12,13,14,15,16,17,18]); in this article, we considered low-dimensional cases, when the submanifold is a curve. Then, we studied the infinitesimal bendings of special curves, more precisely rectifying curves, for some particular cases. The calculations were straightforward, but long, and we had to omit their details and include only the important steps of the proofs.

This article initiated the study of an infinitesimal bending of a rectifying curve. The full classification has not yet been completed, and more cases must be discussed.

However, Theorems 1 and 2 have the following geometrical interpretations: ${\alpha }_2=0$ means that $z\left(s\right)={\alpha }_1\left(s\right)t\left(s\right)+{\alpha }_3\left(s\right)b\left(s\right),$ i.e., is in the plane spanned by the tangent $t\left(s\right)$ and the binormal $b\left(s\right)$, called the rectifying plane. In the case of a constant curvature, with the constant torsion of a unit speed rectifying curve c, we found that if the infinitesimal bending field was along the tangent, along the binormal, then $c_{\xi }$ was also rectifying, except for one singular value of $\xi$, when z was along the binormal.

One observes that for ${\alpha }_2\ne 0$ an infinitesimal bending of a rectifying curve is not rectifying, in the cases considered in Section 3.3 (see Theorems 3–5). This is the motivation to formulate the following

Conjecture. 

If the infinitesimal bending field of a unit speed rectifying curve does not belong to the rectifying plane of the curve, i.e., ${\alpha }_2\left(s\right)\ne 0,$ then its infinitesimal bending is not rectifying.

Besides this, regarding further developments (in [26]), two of the present authors propose some visualizations of particular cases. After this, additional cases will be considered, for example taking ${\alpha }_1\left(s\right),{\alpha }_2\left(s\right),{\alpha }_3\left(s\right)$ as suitable polynomial functions. Furthermore, some rectifying curves on special surfaces, such as conical surfaces (an explicit example can be found in [25]), and their infinitesimal bendings will be studied.

On the other hand, it is possible to extend the present study to surfaces, and respectively, to hypersurfaces in pseudo-Riemannian geometry, specifically in Lorentz–Minkowski or Minkowski spaces, starting with the recent published papers [27,28,29].

Additionally, we intend to explore applications of our study in other areas, such as road design (see [30]).