A Study on the Behavior of Osculating and Rectifying Curves on Smooth Immersed Surfaces in ${\mathbb{E}}^3$

Abstract

This paper presents a detailed investigation into the isometric properties of osculating and rectifying curves on smooth immersed surfaces in ${\mathbb{E}}^3$. We examine the geometric interactions between these curves, specifically when the osculating curve is associated with one surface and the rectifying curve with another. The main objective of this study is to identify the conditions under which these curves exhibit isometric behavior, preserving their intrinsic geometric properties along their respective Frenet frames. Our findings demonstrate that these curves retain isometric characteristics along the tangent, normal, and binormal directions, offering new insights into their structural invariance. This research makes a significant contribution to the broader field of differential geometry, with potential applications in surface theory.

1. Introduction

The study of curves on surfaces in Euclidean 3-space ${\mathbb{E}}^3$ plays a pivotal role in differential geometry, offering essential insights into the local and global geometry of surfaces. Among the various types of curves, osculating and rectifying curves are of particular significance, as they capture both intrinsic and extrinsic geometric properties along a surface. The osculating curve lies in the plane spanned by the tangent and normal vectors, while the rectifying curve lies in the plane defined by the tangent and binormal vectors. These curves serve as tools for analyzing curvature and torsion behaviors on a surface, making them integral to surface theory and geometric modeling [1,2,3,4]. The study of osculating curves has garnered significant attention, with numerous investigations examining their properties in Euclidean spaces such as ${\mathbb{E}}^3$ and ${\mathbb{E}}^4$. Similarly, rectifying curves have been explored in the context of various smooth surfaces [5,6]. However, despite their importance, the geometric properties of these curves, particularly when defined on different smooth immersed surfaces (SISs) in ${\mathbb{E}}^3$, remain relatively underexplored. While the behavior of these curves on space curves has been extensively studied [7], their interactions when situated on distinct SIS in three-dimensional space have not been adequately addressed in the existing literature.

This work builds upon previous studies on osculating and rectifying curves, including the investigation by Shaikh and Ghosh [8], which examines the geometric properties of these curves on smooth surfaces, and the work of Kulahci et al. [9], who classify normal and osculating curves within the framework of Sasakian space. Furthermore, Montiel and Ros [10] provide foundational concepts in the theory of curves and surfaces. Shaikh and Ghosh’s contributions [11] also specifically address the case of curves whose position vectors lie within the tangent plane of the surface.

This paper fills this gap by investigating the isometric behavior of osculating and rectifying curves on different smooth immersed surfaces (SISs) in ${\mathbb{E}}^3$. The primary motivation behind this study is to determine the conditions under which these curves preserve important geometric quantities, particularly in relation to their respective Frenet frames. We seek to answer the following crucial questions: how do these curves maintain their geometric properties—such as lengths and differential characteristics—when they are mapped across different surfaces, and under what conditions can they exhibit isometric behavior?

Our study focuses on examining how osculating and rectifying curves, when associated with distinct surfaces, preserve their tangent, normal, and binormal directions. This analysis offers new insights into the compatibility of local curve frames across different surfaces. Furthermore, we extend our investigation to normal curves, proposing a framework to explore their interaction with rectifying curves on SISs, which could lead to further generalizations in the study of surface curves.

By addressing these questions, we contribute to a deeper understanding of isometries in differential geometry. The results presented here not only enhance the theoretical framework of surface theory but also offer potential applications in geometric modeling, computer graphics, and the mathematical analysis of motion trajectories on surfaces, where such isometric properties can play a critical role in simulating accurate movements.

This paper is a step toward understanding the geometric relationships between rectifying and osculating curves on SISs and their broader implications in both theoretical and applied mathematics. By rigorously treating the isometric properties of specialized curve pairs, our findings open new avenues for further research and application in fields ranging from surface modeling to physics and engineering.

2. Preliminaries

Let ${\mathbb{C}}_o\left(s\right)$ denote the osculating curve on a smooth immersed surface (SIS) in ${\mathbb{R}}^3$, and let ${\mathbb{C}}_r\left(s\right)$ represent the rectifying curve lying on another SIS in ${\mathbb{R}}^3$. The Serret–Frenet frame associated with the osculating curve ${\mathbb{C}}_o\left(s\right)$ is defined by the following system of differential equations:

$$\left\{\begin{array}{cc}{T}^{\prime }\hfill & =\kappa N,\hfill \\ N^{\prime }\hfill & =-\kappa T+\tau B,\hfill \\ B^{\prime }\hfill & =-\tau N,\hfill \end{array}\right.$$

(1)

where T, N, B, $\kappa$, and $\tau$ are the tangent, principal normal, and binormal vector fields, and the curvature and torsion of the curve ${\mathbb{C}}_o\left(s\right)$, respectively.

Similarly, the Serret–Frenet frame corresponding to the rectifying curve ${\mathbb{C}}_r\left(s\right)$ is given by

$$\left\{\begin{array}{cc}{\overline{T}}^{\prime }\hfill & =\kappa \overline{N},\hfill \\ {\overline{N}}^{\prime }\hfill & =-\kappa \overline{T}+\tau \overline{B},\hfill \\ {\overline{B}}^{\prime }\hfill & =-\tau \overline{N},\hfill \end{array}\right.$$

(2)

where $\overline{T}$, $\overline{N}$, $\overline{B}$, $\kappa$, and $\tau$ denote the tangent, normal, and binormal vectors, and the curvature and torsion of the curve ${\mathbb{C}}_r\left(s\right)$, respectively.

Definition 1.

Let $\varphi :S\to {\mathbb{R}}^3$ be a smooth immersion of a surface S into three-dimensional Euclidean space. We assume that the surface is equipped with a smooth unit normal vector field N, which is defined at every point of S.

Consider a smooth regular curve $C:I\subset \mathbb{R}\to S$, where I is an open interval, and $C^{\prime }\left(t\right)\ne 0$ for all $t\in I$. This curve lies entirely on the surface S.

The composition $\overline{C}=\varphi \circ C:I\to {\mathbb{R}}^3$ represents the image of the curve under the immersion. This image is known as the trace of the curve on the surface.

The pair $(C,\varphi )$ defines a curve on the surface, and the trace $\overline{C}$ is the corresponding space curve in ${\mathbb{R}}^3$.

Definition 2.

(see [12]) Let $S\subset {\mathbb{R}}^3$ be a smooth immersed surface (SIS), and let ${\mathbb{C}}_o:I\subset \mathbb{S}$ be a regular curve defined on an open interval $I\subset \mathbb{R}$. The curve ${\mathbb{C}}_o$ is called an osculating curve on the surface $\mathbb{S}$ if it satisfies the following two conditions at every point along its domain.

(a) 

The tangent vector of ${\mathbb{C}}_o$ lies entirely within the tangent plane of the surface at each point. That is, for all $s\in I$, we have

$$\begin{array}{c}\hfill {\mathbb{C}}_o^{\prime }\left(s\right)\in T_{{\mathbb{C}}_o\left(s\right)}\mathbb{S}\end{array}$$

(b) 

The curvature vector $\kappa \left(s\right)$, which describes how the curve bends in space, also lies within the surface’s tangent plane at the corresponding point:

$$\begin{array}{c}\kappa \left(s\right)\in T_{{\mathbb{C}}_o\left(s\right)}\mathbb{S}\end{array}$$

In essence, an osculating curve on a surface is one whose motion and bending are entirely contained within the geometry of the surface itself. Both its direction and curvature are governed solely by the intrinsic structure of the surface, with no component extending into the normal direction.

Definition 3.

(see [13]) Let $\mathbb{S}\subset {\mathbb{R}}^3$ be a smooth immersed surface (SIS), and let ${\mathbb{C}}_r:I\to \mathbb{S}$ be a regular curve on $\mathbb{S}$, where $I\subset \mathbb{R}$ is an open interval. The curve ${\mathbb{C}}_r$ is referred to as a rectifying curve on the surface if, at each point along the curve, its binormal vector is orthogonal to the surface’s unit normal vector. More precisely, the condition

$$\begin{array}{c}\hfill B\left(s\right)·N_{\mathbb{S}}\left(s\right)=0\end{array}$$

holds for all $s\in I$, where

(a) 

$B\left(s\right)$ is the binormal vector of the space curve ${\overline{\mathbb{C}}}_r=\varphi \circ {\mathbb{C}}_r$ in ${\mathbb{R}}^3$;

(b) 

$N_{\mathbb{S}}\left(s\right)$ is the unit normal vector to the surface $\mathbb{S}$ at the point ${\mathbb{C}}_r\left(s\right)$.

This orthogonality condition implies that the binormal vector lies within the tangent plane of the surface at each point. In other words, the rectifying plane of the curve, spanned by the tangent and binormal vectors, remains tangent to the surface. Such curves reflect a specific geometric compatibility between the curve’s spatial behavior and the surface on which it lies.

Definition 4.

(see [14]) Let $\mathbb{S}$ and $\overline{\mathbb{S}}$ be smooth surfaces in ${\mathbb{R}}^3$, and let

$$\begin{array}{c}\hfill f:W\subset \mathbb{S}\to \overline{\mathbb{S}}\end{array}$$

be a smooth mapping defined on a neighborhood W of a point $q\in \mathbb{S}$. The mapping f is said to be a local isometry at q if there exists a neighborhood $V\subset \overline{\mathbb{S}}$ of the point $f\left(q\right)$ such that the restriction

$$\begin{array}{c}\hfill f:W\to V\end{array}$$

preserves the metric; that is, the differential of f at every point in W preserves the lengths of tangent vectors and the angles between them. In this case, f preserves the first fundamental form locally. If such a local isometry exists at every point $q\in \mathbb{S}$, then the surfaces $\mathbb{S}$ and $\overline{\mathbb{S}}$ are said to be locally isometric. Moreover, if f is a diffeomorphism, meaning it is a smooth, bijective map with a smooth inverse, and it is a local isometry at every point in $\mathbb{S}$, then f is called a global isometry between the surfaces. A global isometry implies the complete preservation of intrinsic geometry between $\mathbb{S}$ and $\overline{\mathbb{S}}$.

3. Osculating and Rectifying Curves on a Smooth Immersed Surface

In this section, we examine the isometries of the osculating and rectifying curves in relation to the tangent, normal, and binormal vectors.

Suppose that $\mathbb{C}\left(s\right)$ is a curve on a smooth surface $\mathbb{S}$. The unit tangent vector ${\mathbb{C}}^{\prime }\left(s\right)$ is perpendicular to the unit normal vector N of the surface. Consequently, the vectors ${\mathbb{C}}^{″}\left(s\right)$, N, and ${\mathbb{C}}^{\prime }\left(s\right)\times N$ are mutually orthogonal.

Since N and T are orthogonal, the acceleration vector ${\mathbb{C}}^{″}\left(s\right)$ is also orthogonal to ${\mathbb{C}}^{\prime }\left(s\right)$. Therefore, we can express ${\mathbb{C}}^{″}\left(s\right)$ as a linear combination of $N\times {\mathbb{C}}^{\prime }\left(s\right)$ and N as follows:

$${\mathbb{C}}^{″}\left(s\right)={\kappa }_{e}N+{\kappa }_f(N\times {\mathbb{C}}^{\prime }\left(s\right)),$$

where ${\kappa }_e$ denotes the normal curvature, and ${\kappa }_f$ represents the geodesic curvature [15]. Their values are given by

$$\left\{\begin{array}{cc}{\kappa }_e\hfill & ={\mathbb{C}}^{″}\left(s\right)·N,\hfill \\ {\kappa }_f\hfill & ={\mathbb{C}}^{″}\left(s\right)·(N\times {\mathbb{C}}^{\prime }\left(s\right)).\hfill \end{array}\right.$$

The osculating curve on a smooth immersed surface $\mathbb{S}$ is defined as

$${\mathbb{C}}_o\left(s\right)={\psi }_1\left(s\right)T\left(s\right)+{\psi }_2\left(s\right)N\left(s\right),$$

where ${\psi }_1\left(s\right)$ and ${\psi }_2\left(s\right)$ are smooth functions ($C^{\infty }$).

Consider a smooth immersed surface $\mathbb{S}$ with a coordinate chart $\zeta (m,n)$, where $\zeta (m,n)$ represents the surface immersion mapping that takes the coordinates m and n to points on the surface. Let $\mathbb{C}\left(s\right)=\mathbb{C}\left(m\right(s),n(s\left)\right)$ be a curve on $\mathbb{S}$ parametrized by s, where $m\left(s\right)$ and $n\left(s\right)$ are smooth functions of s.

By applying the chain rule for the curve $\mathbb{C}\left(s\right)$ on the surface $\mathbb{S}$, we differentiate the curve with respect to s. Since ${\zeta }_m$ and ${\zeta }_n$ represent the tangent vectors to the surface at each point, defined as the partial derivatives of the surface immersion $\varphi$ with respect to the coordinates m and n, respectively, we have

$${\zeta }_m={\displaystyle \frac{\partial \varphi }{\partial m}},{\zeta }_n={\displaystyle \frac{\partial \varphi }{\partial n}},$$

which span the tangent plane to the surface at each point $\varphi (m,n)\in \mathbb{S}$.

Now, by the chain rule, we express the derivative of the curve $\mathbb{C}\left(s\right)$ as

$${\mathbb{C}}_o^{\prime }\left(s\right)=m^{\prime }\left(s\right){\zeta }_m+n^{\prime }\left(s\right){\zeta }_n,$$

Here, $m^{\prime }\left(s\right)$ and $n^{\prime }\left(s\right)$ are the derivatives of the coordinates $m\left(s\right)$ and $n\left(s\right)$ with respect to s, representing the rates of change of the curve along the m- and n-directions, respectively.

Alternatively, we can express this as the tangent vector $\overrightarrow{T}\left(s\right)$ to the curve ${\mathbb{C}}_o\left(s\right)$ along the surface, which can be written as follows:

$$\overrightarrow{T}\left(s\right)=m^{\prime }\left(s\right){\zeta }_m+n^{\prime }\left(s\right){\zeta }_n.$$

(3)

This equation shows that the tangent vector to the curve is a linear combination of the basis vectors ${\zeta }_m$ and ${\zeta }_n$ in the tangent plane of the surface.

Differentiating Equation (3) using the Serret–Frenet equations from Equation (1) gives us the necessary expressions for the derivatives of the tangent, normal, and binormal vectors, leading to the subsequent geometric analysis.

$$\overrightarrow{N}\left(s\right)={\displaystyle \frac{1}{\kappa \left(s\right)}}\left[m^{\prime 2}{\zeta }_{mm}+2m^{\prime }{n}^{\prime }{\zeta }_{mn}+n^{\prime 2}{\zeta }_{nn}+m^{″}{\zeta }_m+n^{″}{\zeta }_n\right].$$

(4)

Differentiating Equation (4) again using Equation (1), we get

$$\overrightarrow{B}\left(s\right)={\displaystyle \frac{1}{\kappa \left(s\right)}}\left[\begin{array}{l}{m}^{\prime 3}{\zeta }_m\times {\zeta }_{mm}+2m^{\prime 2}{n}^{\prime }{\zeta }_m\times {\zeta }_{mn}+m^{\prime }{n}^{\prime 2}{\zeta }_m\times {\zeta }_{nn}\hfill \\ +m^{\prime 2}{n}^{\prime }{\zeta }_n\times {\zeta }_{mm}+2m^{\prime }{n}^{\prime 2}{\zeta }_n\times {\zeta }_{mn}+n^{\prime 3}{\zeta }_n\times {\zeta }_{nn}\hfill \\ +(m^{\prime }{n}^{″}-n^{\prime }{m}^{″})\overrightarrow{N}\hfill \end{array}\right].$$

(5)

where $\overrightarrow{N}={\zeta }_m\times {\zeta }_n$. Substituting the values, we obtain

$${\mathbb{C}}_o\left(s\right)={\psi }_1\left(s\right)\left(m^{\prime }{\zeta }_m+n^{\prime }{\zeta }_n\right)+{\displaystyle \frac{{\psi }_2\left(s\right)}{\kappa \left(s\right)}}\left[\begin{array}{c}{m}^{\prime 2}{\zeta }_{mm}+2m^{\prime }{n}^{\prime }{\zeta }_{mn}+n^{\prime 2}{\zeta }_{nn}\hfill \\ +m^{″}{\zeta }_m+n^{″}{\zeta }_n\hfill \end{array}\right],$$

where ${\psi }_1\left(s\right)$ and ${\psi }_2\left(s\right)$ are smooth functions ($C^{\infty }$).

The rectifying curve is defined on a smooth immersed surface $\overline{\mathbb{S}}$ as

$${\mathbb{C}}_r\left(s\right)={\psi }_1\left(s\right)T\left(s\right)+{\psi }_2\left(s\right)B\left(s\right).$$

Since $B\left(s\right)=T\left(s\right)\times N\left(s\right)$, we obtain the rectifying curve as

$${\mathbb{C}}_r\left(s\right)=({\psi }_1\left(s\right)+{\psi }_2\left(s\right))T\left(s\right)+{\psi }_2\left(s\right)N\left(s\right).$$

The Frenet apparatus of the rectifying curves is given by

$$\overline{T}\left(s\right)=m^{\prime }{\overline{\zeta }}_m+n^{\prime }{\overline{\zeta }}_n.$$

Differentiating the above equation using Equation (2), we obtain

$$\overline{N}\left(s\right)={\displaystyle \frac{1}{\kappa \left(s\right)}}\left[m^{\prime 2}{\overline{\zeta }}_{mm}+2m^{\prime }{n}^{\prime }{\overline{\zeta }}_{mn}+n^{\prime 2}{\overline{\zeta }}_{nn}+m^{″}{\overline{\zeta }}_m+n^{″}{\overline{\zeta }}_n\right].$$

Differentiating the above equation using (2), we obtain the binormal vector as

$$\overline{B}\left(s\right)={\displaystyle \frac{1}{\kappa \left(s\right)}}\left[\begin{array}{c}{m}^{\prime 3}{\overline{\zeta }}_m\times {\overline{\zeta }}_{mm}+2m^{\prime 2}{n}^{\prime }{\overline{\zeta }}_m\times {\overline{\zeta }}_{mn}+m^{\prime }{n}^{\prime 2}{\overline{\zeta }}_m\times {\overline{\zeta }}_{nn}\hfill \\ +m^{\prime 2}{n}^{\prime }{\overline{\zeta }}_n\times {\overline{\zeta }}_{mm}+2m^{\prime }{n}^{\prime 2}{\overline{\zeta }}_n\times {\overline{\zeta }}_{mn}+n^{\prime 3}{\overline{\zeta }}_n\times {\overline{\zeta }}_{nn}\hfill \\ +(m^{\prime }{n}^{″}-n^{\prime }{m}^{″})\overrightarrow{N_1}\hfill \end{array}\right].$$

Since the rectifying curve exhibits an isometry under the smooth immersed surface, we conclude that

$${\overline{\mathbb{C}}}_r=\overline{T}\left(s\right)=m^{\prime }{\zeta }_m+n^{\prime }{\zeta }_n.$$

(6)

The normal vector is

$$\overline{N}\left(s\right)={\displaystyle \frac{1}{\kappa \left(s\right)}}\left[m^{\prime 2}{\zeta }_{mm}+2m^{\prime }{n}^{\prime }{\zeta }_{mn}+n^{\prime 2}{\zeta }_{nn}+m^{″}{\zeta }_m+n^{″}{\zeta }_n\right].$$

(7)

Finally, the binormal vector is

$$\overline{B}\left(s\right)={\displaystyle \frac{1}{\kappa \left(s\right)}}\left[\begin{array}{c}{m}^{\prime 3}{\zeta }_m\times {\zeta }_{mm}+2m^{\prime 2}{n}^{\prime }{\zeta }_m\times {\zeta }_{mn}+m^{\prime }{n}^{\prime 2}{\zeta }_m\times {\zeta }_{nn}\hfill \\ +m^{\prime 2}{n}^{\prime }{\zeta }_n\times {\zeta }_{mm}+2m^{\prime }{n}^{\prime 2}{\zeta }_n\times {\zeta }_{mn}+n^{\prime 3}{\zeta }_n\times {\zeta }_{nn}\hfill \\ +(m^{\prime }{n}^{″}-n^{\prime }{m}^{″})\overrightarrow{N}\hfill \end{array}\right].$$

(8)

As the ${\mathbb{C}}_r\left(S\right)$ curve on an SIS$\left(\mathbb{S}\right)$ shares the same Serret–Frenet frame with the ${\mathbb{C}}_o\left(s\right)$ curve on an SIS($\overline{\mathbb{S}}$), we conclude from Equations (2), (3), and (5)–(8) that

$$\begin{array}{c}\hfill T\left(s\right)=\overline{T}\left(s\right),N\left(s\right)=\overline{N}\left(s\right),\mathrm{and}B\left(s\right)=\overline{B}\left(s\right).\end{array}$$

Theorem 1.

Suppose that $\mathbb{S}$ and $\overline{\mathbb{S}}$ are smooth immersed surfaces in ${\mathbb{E}}^3$, and let f be an isometry between the two surfaces. Let ${\mathbb{C}}_o$ be the osculating curve on $\mathbb{S}$ and ${\mathbb{C}}_r$ be the rectifying curve on $\overline{\mathbb{S}}$. Then, the following results hold:

1. 

The normal curvature is given by

$$\begin{array}{c}\hfill {\kappa }_e=m^{\prime 2}P+2m^{\prime }{n}^{\prime }Q+n^{\prime 2}R,\end{array}$$

where $P,Q,R$ are the coefficients of the second fundamental form of the surface.

2. 

The geodesic curvature, which is preserved under the isometry f, satisfies

$$\begin{array}{c}\hfill {\kappa }_f={\overline{\kappa }}_f,\end{array}$$

where ${\kappa }_f$ and ${\overline{\kappa }}_f$ denote the geodesic curvatures of the osculating and rectifying curves, respectively, on $\mathbb{S}$ and $\overline{\mathbb{S}}$.

Proof. 

By applying the definition of the normal curvature, we obtain

$$\begin{array}{cc}\hfill {\kappa }_e& =\kappa \left(s\right)N\left(s\right)·N,\hfill \\ \hfill {\kappa }_e& =[{\zeta }_{mm}{\left(m^{\prime }\right)}^2+m^{″}{\zeta }_m+n^{″}{\zeta }_n+2m^{\prime }{n}^{\prime }{\zeta }_{mn}+{\zeta }_{nn}{\left(n^{\prime }\right)}^2]·N.\hfill \end{array}$$

This simplifies to

$$\begin{array}{c}\hfill {\kappa }_e=m^{\prime 2}P+2m^{\prime }{n}^{\prime }Q+n^{\prime 2}R,\end{array}$$

where $P,Q,R$ are the coefficients of the second fundamental form of $\mathbb{S}$, representing the normal curvature components.

For the second part of the proof, we compute the geodesic curvature:

$$\begin{array}{cc}\hfill {\kappa }_f& ={\mathbb{C}}^{″}·(N\times {\mathbb{C}}^{\prime }).\hfill \end{array}$$

Since $\mathbb{S}$ and $\overline{\mathbb{S}}$ are isometric surfaces, their first fundamental form coefficients satisfy

$$X=\overline{X},Y=\overline{Y},Z=\overline{Z}.$$

(9)

where $X,Y,Z$ represent the coefficients of the first fundamental form. Differentiating these equations gives

$${\overline{X}}_m={({\overline{\zeta }}_m·{\overline{\zeta }}_m)}_m={({\zeta }_m·{\zeta }_m)}_m=X_m,$$

(10)

Correspondingly, we have

$${\overline{Y}}_m=Y_m,{\overline{Z}}_m=Z_m,{\overline{X}}_n=X_n,{\overline{Y}}_n=Y_n,\mathrm{and}{\overline{Z}}_n=Z_n.$$

(11)

Differentiating Equation (10) with respect to s, we obtain

$$X_m={({\zeta }_m·{\zeta }_m)}_m=2{\zeta }_{mm}·{\zeta }_m.$$

This implies that

$${\zeta }_{mm}·{\zeta }_m={\displaystyle \frac{1}{2}}{X}_m.$$

(12)

Similarly, we derive the following equations:

$${\zeta }_{mn}·{\zeta }_n={\displaystyle \frac{1}{2}}{Z}_m,$$

(13)

$${\zeta }_{mm}·{\zeta }_n=Y_n-{\displaystyle \frac{1}{2}}{X}_n,$$

(14)

$${\zeta }_{mn}·{\zeta }_m={\displaystyle \frac{1}{2}}{X}_n,$$

(15)

$${\zeta }_{nn}·{\zeta }_n={\displaystyle \frac{1}{2}}{Z}_n,$$

(16)

$${\zeta }_{nn}·{\zeta }_m=Y_n-{\displaystyle \frac{1}{2}}{Z}_m.$$

(17)

Hence, the geodesic curvature of the osculating curve on the smooth immersed surface $\mathbb{S}$ is given by

$$\begin{array}{cc}\hfill {\kappa }_f& ={\mathbb{C}}_o^{″}·\left\{({\zeta }_m\times {\zeta }_n)\times (m^{\prime }{\zeta }_m+n^{\prime }{\zeta }_n)\right\},\hfill \\ \hfill {\kappa }_f& =\left\{(m^{\prime 2}{\zeta }_{mm}+2m^{\prime }{n}^{\prime }{\zeta }_{mn}+n^{\prime 2}{\zeta }_{nn})+(m^{″}{\zeta }_m+n^{″}{\zeta }_n)\right\}\hfill \\ \hfill & ·\left\{m^{\prime }(X{\zeta }_n-Y{\zeta }_m)+n^{\prime }(Y{\zeta }_n-Z{\zeta }_m)\right\}.\hfill \end{array}$$

This implies that

$${\kappa }_f=\left\{\begin{array}{c}{m}^{\prime }{m}^{″}(XY-YX)+n^{\prime }{m}^{″}(Y^2-ZX)+m^{\prime }{n}^{″}(XZ-Y^2)\hfill \\ +n^{\prime }{n}^{″}(YZ-ZY)+m^{\prime 3}(X{\zeta }_{mm}·{\zeta }_n-Y{\zeta }_{mm}·{\zeta }_m)\hfill \\ +m^{\prime 2}{n}^{\prime }(Y{\zeta }_{mm}·{\zeta }_n-Z{\zeta }_{mm}·{\zeta }_m)\hfill \\ +2m^{\prime 2}{n}^{\prime }(X{\zeta }_{mn}·{\zeta }_n-Y{\zeta }_{mn}·{\zeta }_m)\hfill \\ +2m^{\prime }{n}^{\prime 2}(Y{\zeta }_{mn}·{\zeta }_n-Z{\zeta }_{mn}·{\zeta }_m)\hfill \\ +m^{\prime }{n}^{\prime 2}(X{\zeta }_{nn}·{\zeta }_n-Y{\zeta }_{nn}·{\zeta }_m)\hfill \\ +n^{\prime 3}(Y{\zeta }_{nn}·{\zeta }_n-Z{\zeta }_{nn}·{\zeta }_m).\hfill \end{array}\right.$$

(18)

By using Equations (12)–(17) in (18), we obtain

$${\kappa }_f=\left\{\begin{array}{c}{n}^{\prime }{m}^{″}(Y^2-ZX)+m^{\prime }{n}^{″}(XZ-Y^2)\hfill \\ +{\displaystyle \frac{1}{2}}{m}^{\prime 3}(2X{Y}_m-X{X}_n-Y{X}_m)\hfill \\ +{\displaystyle \frac{1}{2}}{m}^{\prime 2}{n}^{\prime }(2Y{Y}_m-Y{X}_n-Z{X}_m)\hfill \\ +m^{\prime 2}{n}^{\prime }(Z{X}_m-Y{X}_n)\hfill \\ +m^{\prime }{n}^{\prime 2}(Y{Z}_m-Z{X}_n)\hfill \\ +{\displaystyle \frac{1}{2}}{m}^{\prime }{n}^{\prime 2}(X{Z}_n-2Y{Y}_n+Y{Z}_m)\hfill \\ +{\displaystyle \frac{1}{2}}{n}^{\prime 3}(Y{Z}_n-2Z{Y}_n+Z{Z}_m).\hfill \end{array}\right.$$

(19)

The geodesic curvature of the rectifying curve on an SIS $\overline{\mathbb{S}}$ is

$$\begin{array}{cc}\hfill {\overline{\kappa }}_f& ={\mathbb{C}}_r^{″}·\{({\overline{\zeta }}_m\times {\overline{\zeta }}_n)\times (m^{\prime }{\overline{\zeta }}_m+n^{\prime }{\overline{\zeta }}_n)\},\hfill \\ \hfill {\overline{\kappa }}_f& =\{(m^{\prime 2}{\overline{\zeta }}_{mm}+2m^{\prime }{n}^{\prime }{\overline{\zeta }}_{mn}+n^{\prime 2}{\overline{\zeta }}_{nn})+(m^{″}{\overline{\zeta }}_m+n^{″}{\overline{\zeta }}_n)\}\hfill \\ & ·\{m^{\prime }(\overline{X}{\overline{\zeta }}_n-\overline{Y}{\overline{\zeta }}_m)+n^{\prime }(\overline{Y}{\overline{\zeta }}_n-\overline{Z}{\overline{\zeta }}_m)\}.\hfill \end{array}$$

This implies that

$${\overline{\kappa }}_f=\left\{\begin{array}{c}{m}^{\prime }{m}^{″}(\overline{X}\overline{Y}-\overline{Y}\overline{X})+n^{\prime }{m}^{″}({\overline{Y}}^2-\overline{Z}\overline{X})+m^{\prime }{n}^{″}(\overline{X}\overline{Z}-{\overline{Y}}^2)\hfill \\ +n^{\prime }{n}^{″}(\overline{Y}\overline{Z}-\overline{Z}\overline{Y})+m^{\prime 3}(\overline{X}{\overline{\zeta }}_{mm}·{\overline{\zeta }}_n-\overline{Y}{\overline{\zeta }}_{mm}·{\overline{\zeta }}_m)\hfill \\ +m^{\prime 2}{n}^{\prime }(\overline{Y}{\overline{\zeta }}_{mm}·{\overline{\zeta }}_n-\overline{Z}{\overline{\zeta }}_{mm}·{\overline{\zeta }}_m)+2m^{\prime 2}{n}^{\prime }(\overline{X}{\overline{\zeta }}_{mn}·{\overline{\zeta }}_n-\overline{Y}{\overline{\zeta }}_{mn}·{\overline{\zeta }}_m)\hfill \\ +2m^{\prime }{n}^{\prime 2}(\overline{Y}{\overline{\zeta }}_{mn}·{\overline{\zeta }}_n-\overline{Z}{\overline{\zeta }}_{mn}·{\overline{\zeta }}_m)+m^{\prime }{n}^{\prime 2}(\overline{X}{\overline{\zeta }}_{nn}·{\overline{\zeta }}_n-\overline{Y}{\overline{\zeta }}_{nn}·{\overline{\zeta }}_m)\hfill \\ +n^{\prime 3}(\overline{Y}{\overline{\zeta }}_{nn}·{\overline{\zeta }}_n-\overline{Z}{\overline{\zeta }}_{nn}·{\overline{\zeta }}_m),\hfill \end{array}\right.$$

Using (9)–(17), we obtained

$${\overline{\kappa }}_f=\left\{\begin{array}{c}{n}^{\prime }{m}^{″}(Y^2-ZX)+m^{\prime }{n}^{″}(XZ-Y^2)+{\displaystyle \frac{1}{2}}{m}^{\prime 3}(2X{Y}_m-X{X}_n-Y{X}_m)\hfill \\ +{\displaystyle \frac{1}{2}}{m}^{\prime 2}{n}^{\prime }(2Y{Y}_m-Y{X}_n-Z{X}_m)+m^{\prime 2}{n}^{\prime }(Z{X}_m-Y{X}_n)\hfill \\ +m^{\prime }{n}^{\prime 2}(Y{Z}_m-Z{X}_n)+{\displaystyle \frac{1}{2}}{m}^{\prime }{n}^{\prime 2}(X{Z}_n-2Y{Y}_n+Y{Z}_m)\hfill \\ +{\displaystyle \frac{1}{2}}{n}^{\prime 3}(Y{Z}_n-2Z{Y}_n+Z{Z}_m).\hfill \end{array}\right.$$

(20)

From (19) and (20), we conclude that

$$\begin{array}{c}\hfill {\kappa }_f={\overline{\kappa }}_f.\end{array}$$

Thus, we conclude that the geodesic curvature of the osculating and rectifying curves is preserved under the isometry f, as stated in the theorem. □

3.1. Osculating and Rectifying Curves on Smooth Immersed Surfaces in ${\mathbb{E}}^3$ Along the Tangent Vector

Theorem 2.

Let $f:\mathbb{S}\to \overline{\mathbb{S}}$ be an isometry, let ${\mathbb{C}}_o$ be an osculating curve on $\mathbb{S}$, and let ${\mathbb{C}}_r$ be a rectifying curve on $\overline{\mathbb{S}}$. Then, the tangent components satisfy the following relation:

$${\mathbb{C}}_r·\overline{T}={\mathbb{C}}_o·T,$$

where $T=u{\zeta }_m+v{\zeta }_n$ is the tangent vector to $\mathbb{S}$ for some $u,v\in \mathbb{R}$.

Proof. 

By definition, the osculating curve ${\mathbb{C}}_o$ can be expressed as

$${\mathbb{C}}_o={\psi }_1\left(s\right)T\left(s\right)+{\psi }_2\left(s\right)N\left(s\right).$$

Using the expressions for the tangent and normal vectors from Equation (1), we obtain

$${\mathbb{C}}_o\left(s\right)={\psi }_1\left(s\right)\left(m^{\prime }{\zeta }_m+n^{\prime }{\zeta }_n\right)+{\displaystyle \frac{{\psi }_2\left(s\right)}{\kappa \left(s\right)}}\left[\begin{array}{c}{m}^{\prime 2}{\zeta }_{mm}+2m^{\prime }{n}^{\prime }{\zeta }_{mn}+n^{\prime 2}{\zeta }_{nn}\hfill \\ +m^{″}{\zeta }_m+n^{″}{\zeta }_n\hfill \end{array}\right].$$

(21)

Here, the vectors ${\zeta }_m$ and ${\zeta }_n$ are the tangent vectors of the surface $\mathbb{S}$ in the direction of the coordinates m and n, respectively. More specifically, they are the partial derivatives of the surface immersion $\varphi :\mathbb{S}\to {\mathbb{E}}^3$ with respect to m and n:

$${\zeta }_m={\displaystyle \frac{\partial \varphi }{\partial m}},{\zeta }_n={\displaystyle \frac{\partial \varphi }{\partial n}}.$$

These vectors form the basis of the tangent plane at each point on the surface $\mathbb{S}$. Therefore, the tangent vector T can be expressed as a linear combination of ${\zeta }_m$ and ${\zeta }_n$, i.e., $T=u{\zeta }_m+v{\zeta }_n$, where u and v are scalar coefficients.

Now, we aim to project ${\mathbb{C}}_o\left(s\right)$ onto the tangent plane of the surface $\zeta$ at the point ${\mathbb{C}}_o\left(s\right)$. The tangent plane at ${\mathbb{C}}_o\left(s\right)$ is spanned by the partial derivatives ${\zeta }_m$ and ${\zeta }_n$, so we compute the components of ${\mathbb{C}}_o\left(s\right)$ along these tangent vectors, ${\zeta }_m$ and ${\zeta }_n$.

Thus, the components of ${\mathbb{C}}_o$ along ${\zeta }_m$ and ${\zeta }_n$ are given by

$$\begin{array}{cc}\hfill {\mathbb{C}}_o\left(s\right)·{\zeta }_m=& [{\psi }_1\left(s\right)\left(m^{\prime }{\zeta }_m+n^{\prime }{\zeta }_n\right)+{\displaystyle \frac{{\psi }_2\left(s\right)}{\kappa \left(s\right)}}\{m^{\prime 2}{\zeta }_{mm}+2m^{\prime }{n}^{\prime }{\zeta }_{mn}+n^{\prime 2}{\zeta }_{mn}\hfill \\ \hfill & +m^{″}{\zeta }_m+n^{″}{\zeta }_n\left\}\right]·{\zeta }_m,\hfill \end{array}$$

By applying the results from Equations (9)–(17), we obtain

$$\begin{array}{cc}\hfill {\mathbb{C}}_o\left(s\right)·{\zeta }_m=& {\psi }_1\left(s\right)(m^{\prime }X+n^{\prime }Y)+{\displaystyle \frac{{\psi }_2\left(s\right)}{2\kappa \left(s\right)}}[2m^{″}X+2n^{″}Y+m^{\prime 2}{X}_m+2m^{\prime }{n}^{\prime }{X}_n\hfill \\ \hfill & +2n^{\prime 2}{Y}_n-n^{\prime 2}{Z}_m].\hfill \end{array}$$

(22)

Similarly, we derive

$$\begin{array}{cc}\hfill {\mathbb{C}}_o\left(s\right)·{\zeta }_n=& {\psi }_1\left(s\right)(m^{\prime }Y+n^{\prime }Z)+{\displaystyle \frac{{\psi }_2\left(s\right)}{2\kappa \left(s\right)}}[2m^{″}Y+2n^{″}Z+2m^{\prime 2}{Y}_m-m^{\prime 2}{X}_n\hfill \\ \hfill & +2m^{\prime }{n}^{\prime }{Z}_m+n^{\prime 2}{Z}_n].\hfill \end{array}$$

(23)

To find the value of ${\mathbb{C}}_o·T$, we express it as

$$\begin{array}{c}\hfill {\mathbb{C}}_o·T={\mathbb{C}}_o·(u{\zeta }_m+v{\zeta }_n).\end{array}$$

This implies:

$${\mathbb{C}}_o·(u{\zeta }_m+v{\zeta }_n)=u({\mathbb{C}}_o·{\zeta }_m)+v({\mathbb{C}}_o·{\zeta }_n).$$

By using Equations (22) and (23), we obtain

$$\begin{array}{cc}\hfill {\mathbb{C}}_o·(u{\zeta }_m+v{\zeta }_n)=& u[{\psi }_1\left(s\right)(m^{\prime }X+n^{\prime }Y)+{\displaystyle \frac{{\psi }_2\left(s\right)}{2\kappa \left(s\right)}}(2m^{″}X+2n^{″}Y+m^{\prime 2}{X}_m+\hfill \\ \hfill & 2m^{\prime }{n}^{\prime }{X}_n+2n^{\prime 2}{Y}_n-n^{\prime 2}{Z}_m\left)\right]+v[{\psi }_1\left(s\right)(m^{\prime }Y+n^{\prime }Z)+\hfill \\ \hfill & {\displaystyle \frac{{\psi }_2\left(s\right)}{2\kappa \left(s\right)}}(2m^{″}Y+2n^{″}Z+2m^{\prime 2}{Y}_m-m^{\prime 2}{X}_n+\hfill \\ \hfill & 2m^{\prime }{n}^{\prime }{Z}_m+n^{\prime 2}{Z}_n\left)\right].\hfill \end{array}$$

(24)

The rectifying curve is defined as

$${\mathbb{C}}_r\left(s\right)=({\psi }_1\left(s\right)+{\psi }_2\left(s\right))\overline{T}\left(s\right)+{\psi }_2\left(s\right)\overline{N}\left(s\right).$$

Substituting the values from Equation (2), we obtain

$${\mathbb{C}}_r\left(s\right)=\{{\psi }_1\left(s\right)+{\psi }_2\left(s\right)\}\left(m^{\prime }{\overline{\zeta }}_m+n^{\prime }{\overline{\zeta }}_n\right)+{\displaystyle \frac{{\psi }_2\left(s\right)}{\kappa \left(s\right)}}\left[\begin{array}{c}{m}^{\prime 2}{\overline{\zeta }}_{mm}+2m^{\prime }{n}^{\prime }{\overline{\zeta }}_{mn}+n^{\prime 2}{\overline{\zeta }}_{nn}\hfill \\ +m^{″}{\overline{\zeta }}_m+n^{″}{\overline{\zeta }}_n\hfill \end{array}\right].$$

(25)

Similarly, to find the components of ${\mathbb{C}}_r$ along ${\overline{\zeta }}_m$ and ${\overline{\zeta }}_n$, we have

$$\begin{array}{cc}\hfill {\mathbb{C}}_r\left(s\right)·{\overline{\zeta }}_m=& \{{\psi }_1\left(s\right)+{\psi }_2\left(s\right)\}(m^{\prime }\overline{X}+n^{\prime }\overline{Y})+{\displaystyle \frac{{\psi }_2\left(s\right)}{2\kappa \left(s\right)}}[2m^{″}\overline{X}+2n^{″}\overline{Y}\hfill \\ \hfill & +m^{\prime 2}{\overline{X}}_m+2m^{\prime }{n}^{\prime }{\overline{X}}_n+2n^{\prime 2}{\overline{Y}}_n-n^{\prime 2}{\overline{Z}}_m].\hfill \end{array}$$

(26)

Similarly, we obtain

$$\begin{array}{cc}\hfill {\mathbb{C}}_r\left(s\right)·{\overline{\zeta }}_n=& \{{\psi }_1\left(s\right)+{\psi }_2\left(s\right)\}(m^{\prime }\overline{Y}+n^{\prime }\overline{Z})+{\displaystyle \frac{{\psi }_2\left(s\right)}{2\kappa \left(s\right)}}[2m^{″}\overline{Y}+2n^{″}\overline{Z}\hfill \\ \hfill & +2m^{\prime 2}{\overline{Y}}_m-m^{\prime 2}{\overline{X}}_n+2m^{\prime }{n}^{\prime }{\overline{Z}}_m+n^{\prime 2}{\overline{Z}}_n].\hfill \end{array}$$

(27)

To determine the value of ${\mathbb{C}}_r·\overline{T}$, we express it as

$$\begin{array}{c}\hfill {\mathbb{C}}_r·\overline{T}={\mathbb{C}}_r·(u{\overline{\zeta }}_m+v{\overline{\zeta }}_n).\end{array}$$

This implies

$${\mathbb{C}}_r·(u{\overline{\zeta }}_m+v{\overline{\zeta }}_n)=u({\mathbb{C}}_r·{\overline{\zeta }}_m)+v({\mathbb{C}}_r·{\overline{\zeta }}_n).$$

By using Equations (26) and (27), we obtain

$$\begin{array}{cc}\hfill {\mathbb{C}}_r·(u{\overline{\zeta }}_m+v{\overline{\zeta }}_n)=& u[\{{\psi }_1\left(s\right)+{\psi }_2\left(s\right)\}(m^{\prime }\overline{X}+n^{\prime }\overline{Y})\hfill \\ \hfill & +{\displaystyle \frac{{\psi }_2\left(s\right)}{2\kappa \left(s\right)}}[2m^{″}\overline{X}+2n^{″}\overline{Y}+m^{\prime 2}{\overline{X}}_m+2m^{\prime }{n}^{\prime }{\overline{X}}_n\hfill \\ \hfill & +2n^{\prime 2}{\overline{Y}}_n-n^{\prime 2}{\overline{Z}}_m\left]\right]\hfill \\ \hfill & +v[\{{\psi }_1\left(s\right)+{\psi }_2\left(s\right)\}(m^{\prime }\overline{Y}+n^{\prime }\overline{Z})\hfill \\ \hfill & +{\displaystyle \frac{{\psi }_2\left(s\right)}{2\kappa \left(s\right)}}[2m^{″}\overline{Y}+2n^{″}\overline{Z}+2m^{\prime 2}{\overline{Y}}_m-m^{\prime 2}{\overline{X}}_n+\hfill \\ \hfill & 2m^{\prime }{n}^{\prime }{\overline{Z}}_m+n^{\prime 2}{\overline{Z}}_n\left]\right].\hfill \end{array}$$

(28)

Using the condition ${\psi }_1\left(s\right)+{\psi }_2\left(s\right)={\psi }_1\left(s\right)$ and Equations (9)–(11), (24) and (28), we conclude that

$${\mathbb{C}}_o·(u{\zeta }_m+v{\zeta }_n)={\mathbb{C}}_r·(u{\overline{\zeta }}_m+v{\overline{\zeta }}_n),$$

which proves the statement. □

3.2. Osculating and Rectifying Curves on Smooth Immersed Surfaces in ${\mathbb{E}}^3$ Along the Normal Vector

Theorem 3.

Consider an isometry $f:\mathbb{S}\to \overline{\mathbb{S}}$, let ${\mathbb{C}}_o$ be an osculating curve on $\mathbb{S}$, and let ${\mathbb{C}}_r$ be a rectifying curve on $\overline{\mathbb{S}}$. Then, for the normal components, we have

$${\mathbb{C}}_r·\overline{N}-{\mathbb{C}}_o·N={\displaystyle \frac{{\psi }_2\left(s\right)}{\kappa \left(s\right)}}\left({\overline{\kappa }}_e\left(s\right)-{\kappa }_e\left(s\right)\right).$$

Proof. 

Using Equation (21), the osculating curve ${\mathbb{C}}_o\left(s\right)$ can be expressed as

$${\mathbb{C}}_o\left(s\right)·N={\psi }_1\left(s\right)\left(m^{\prime }{\zeta }_m+n^{\prime }{\zeta }_n\right)+{\displaystyle \frac{{\psi }_2\left(s\right)}{\kappa \left(s\right)}}\left[\begin{array}{c}{m}^{\prime 2}{\zeta }_{mm}+2m^{\prime }{n}^{\prime }{\zeta }_{mn}+n^{\prime 2}{\zeta }_{nn}\hfill \\ +m^{″}{\zeta }_m+n^{″}{\zeta }_n\hfill \end{array}\right]·N.$$

This simplifies to

$${\mathbb{C}}_o\left(s\right)·N={\displaystyle \frac{{\psi }_2\left(s\right)}{\kappa \left(s\right)}}\left[m^{\prime 2}({\zeta }_{mm}·N)+2m^{\prime }{n}^{\prime }({\zeta }_{mn}·N)+n^{\prime 2}({\zeta }_{nn}·N)\right].$$

Rewriting, we obtain

$${\mathbb{C}}_o\left(s\right)·N={\displaystyle \frac{{\psi }_2\left(s\right)}{\kappa \left(s\right)}}\left[P{m}^{\prime 2}+2Q{m}^{\prime }{n}^{\prime }+R{n}^{\prime 2}\right],$$

(29)

where $P,Q,R$ represent the coefficients of the second fundamental form of $\mathbb{S}$.

Similarly, using the isometry condition for ${\mathbb{C}}_r$ along the normal vector and Equation (25), we obtain

$${\mathbb{C}}_r\left(s\right)·\overline{N}=\left\{{\psi }_1\left(s\right)+{\psi }_2\left(s\right)\right\}\left(m^{\prime }{\overline{\zeta }}_m+n^{\prime }{\overline{\zeta }}_n\right)+{\displaystyle \frac{{\psi }_2\left(s\right)}{\kappa \left(s\right)}}\left[\begin{array}{c}{m}^{\prime 2}{\overline{\zeta }}_{mm}+2m^{\prime }{n}^{\prime }{\overline{\zeta }}_{mn}+n^{\prime 2}{\overline{\zeta }}_{nn}\hfill \\ +m^{″}{\overline{\zeta }}_m+n^{″}{\overline{\zeta }}_n\hfill \end{array}\right]·\overline{N}.$$

Using the condition ${\psi }_1\left(s\right)+{\psi }_2\left(s\right)={\psi }_1\left(s\right)$, we simplify this expression to

$${\mathbb{C}}_r\left(s\right)·\overline{N}={\displaystyle \frac{{\psi }_2\left(s\right)}{\kappa \left(s\right)}}\left[m^{\prime 2}({\overline{\zeta }}_{mm}·\overline{N})+2m^{\prime }{n}^{\prime }({\overline{\zeta }}_{mn}·\overline{N})+n^{\prime 2}({\overline{\zeta }}_{nn}·\overline{N})\right].$$

Rewriting, we obtain

$${\mathbb{C}}_r\left(s\right)·\overline{N}={\displaystyle \frac{{\psi }_2\left(s\right)}{\kappa \left(s\right)}}\left[\overline{P}{m}^{\prime 2}+2\overline{Q}{m}^{\prime }{n}^{\prime }+\overline{R}{n}^{\prime 2}\right],$$

(30)

where $\overline{P},\overline{Q},\overline{R}$ represent the coefficients of the second fundamental form of $\overline{\mathbb{S}}$.

Now, combining Equations (29) and (30), we derive

$${\mathbb{C}}_r\left(s\right)·\overline{N}-{\mathbb{C}}_o\left(s\right)·N={\displaystyle \frac{{\psi }_2\left(s\right)}{\kappa \left(s\right)}}\left[\overline{P}{m}^{\prime 2}+2\overline{Q}{m}^{\prime }{n}^{\prime }+\overline{R}{n}^{\prime 2}\right]-{\displaystyle \frac{{\psi }_2\left(s\right)}{\kappa \left(s\right)}}\left[P{m}^{\prime 2}+2Q{m}^{\prime }{n}^{\prime }+R{n}^{\prime 2}\right].$$

Thus, we conclude that

$${\mathbb{C}}_r\left(s\right)·\overline{N}-{\mathbb{C}}_o\left(s\right)·N={\displaystyle \frac{{\psi }_2\left(s\right)}{\kappa \left(s\right)}}\left({\overline{\kappa }}_e\left(s\right)-{\kappa }_e\left(s\right)\right),$$

which completes the proof. □

Corollary 1.

Consider an isometry $f:\mathbb{S}\to \overline{\mathbb{S}}$ and an osculating curve ${\mathbb{C}}_o\left(s\right)$ on $\mathbb{S}$. The normal component of the ${\mathbb{C}}_o\left(s\right)$ curve remains invariant if any of the following conditions hold:

(a) 

The position vector of ${\mathbb{C}}_o\left(s\right)$ points in the direction of its tangent vector.

(b) 

The normal curvature remains constant.

Proof. 

By using Theorem 3, we have

$${\mathbb{C}}_r\left(s\right)·\overline{N}={\mathbb{C}}_o\left(s\right)·N\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}{\displaystyle \frac{{\psi }_2\left(s\right)}{\kappa \left(s\right)}}\left({\overline{\kappa }}_e\left(s\right)-{\kappa }_e\left(s\right)\right)=0,$$

which implies that

$${\psi }_2\left(s\right)=0\mathrm{or}{\overline{\kappa }}_e\left(s\right)={\kappa }_e\left(s\right).$$

According to the definition of an osculating curve, if ${\psi }_2\left(s\right)=0$, then ${\mathbb{C}}_o\left(s\right)={\psi }_1\left(s\right)T\left(s\right)$, meaning that the position vector of ${\mathbb{C}}_o\left(s\right)$ points in the direction of its tangent vector. Otherwise, if ${\psi }_2\left(s\right)\ne 0$, then ${\overline{\kappa }}_e\left(s\right)={\kappa }_e\left(s\right)$, i.e., the normal curvature is invariant. □

Corollary 2.

Let ${\mathbb{C}}_o$ be an osculating curve on $\mathbb{S}$ and let $f:\mathbb{S}\to \overline{\mathbb{S}}$ be an isometry. The curve ${\mathbb{C}}_o\left(s\right)$ is asymptotic if ${\mathbb{C}}_r\left(s\right)$ is asymptotic, the position vector of ${\mathbb{C}}_o\left(s\right)$ is not in the direction of its tangent vector, and the normal component of the osculating curve ${\mathbb{C}}_o\left(s\right)$ is invariant.

Proof. 

Based on Corollary 1, if ${\overline{\kappa }}_e\left(s\right)={\kappa }_e\left(s\right)$, then ${\mathbb{C}}_r\left(s\right)·\overline{N}={\mathbb{C}}_o\left(s\right)·N$, and the position vector of ${\mathbb{C}}_o\left(s\right)$ does not point in the direction of its tangent vector. Therefore, ${\mathbb{C}}_o\left(s\right)$ is asymptotic. If $\kappa \left(s\right)=0$, then $\overline{\kappa }\left(s\right)=0$, indicating that ${\mathbb{C}}_r\left(s\right)$ is asymptotic. □

4. Conclusions

This study provides a comprehensive geometric framework for analyzing the relationship between the osculating curve ${\mathbb{C}}_o\left(s\right)$ and the rectifying curve ${\mathbb{C}}_r\left(s\right)$ on smooth immersed surfaces (SISs) in three-dimensional Euclidean space ${\mathbb{E}}^3$. By leveraging distinct Frenet frames, we establish the geometric correspondence between these two curves and identify the conditions under which they exhibit isometric behavior. This work contributes to the understanding of curve geometry on SISs by offering new insights into their structural invariance under isometries.

The significance of this study lies in its novel exploration of the isometric relationships between osculating and rectifying curves in a new context, expanding upon existing differential geometry identities. Our results show that rectifying curves, in specific cases, correspond to helical motions characterized by consistent twisting along the binormal direction, while osculating curves relate to planar or nearly planar motions, where the curvature vector remains confined to the surface’s tangent plane. This distinction opens up possibilities for further investigation into the classification of such curves and their role in geometric modeling, computer graphics, and other areas where geometric properties are pivotal.

However, there are limitations to this work that warrant further exploration. For instance, while we focus on the isometric properties of the curves within the context of SISs in ${\mathbb{E}}^3$, extending this analysis to higher-dimensional spaces or considering other geometric constraints could yield more comprehensive results. Additionally, the influence of curvature variations along the surface needs to be further examined, especially in the presence of more complex surface topologies.

Future directions for this research could include the application of these results to practical areas, such as surface modeling, where the isometry-preserving properties of curves are essential. Another promising avenue of exploration involves the use of Lie algebraic methods to solve the governing equations for the curves, as demonstrated in previous studies [16], which could provide more efficient computational techniques for analyzing isometries. Further investigation into the connection between these geometric properties and real-world phenomena, such as robotic motion planning or material science, could also offer valuable insights.

Overall, this study lays the groundwork for more advanced research into the interaction of osculating and rectifying curves on SISs, presenting new possibilities for both theoretical advancements and practical applications in various fields.