Modified Sweeping Surfaces in Euclidean 3-Space

Abstract

In this study, we explore the sweeping surfaces in Euclidean 3-space, utilizing the modified orthogonal frames with non-zero curvature and torsion, which allows us to consider the spine curves even if their second differentiations vanish. If the curvature of the spine curve of a sweeping surface has discrete zero points, the Frenet frame might undergo a discontinuous change in orientation. Therefore, the conventional parametrization with the Frenet frame of such a surface cannot be given. Thus, we introduce two types of modified sweeping surfaces by considering two types of spine curves; the first one’s curvature is not identically zero and the second one’s torsion is not identically zero. Then, we determine the criteria for classifying the coordinate curves of these two types of modified sweeping surfaces as geodesic, asymptotic, or curvature lines. Additionally, we delve into determining criteria for the modified sweeping surfaces to be minimal, developable, or Weingarten. Through our analysis, we aim to clarify the characteristics defining these surfaces. We present graphical representations of sample modified sweeping surfaces to enhance understanding and provide concrete examples that showcase their properties.

1. Introduction

Sweeping is a widely used technique in geometric modeling to define the geometry of three-dimensional objects. The geometric and computational properties of this technique have been examined via various approaches where planar curves sweep along the trajectories called the spine curves. For instance, in the process of generating sweeping surfaces, the superiority of using the rotation minimizing frame (RMF) with the Frenet frame was compared in [1,2]. The sweeping surfaces are generated by a plane curve known as a profile curve or generatrix that is continuously moving in the same direction with the normal vector field. Strings, canal surfaces, and tubular (pipe) surfaces are prominent varieties of sweeping surfaces that are the ultimate outcomes of this process. In [3], Xu et al. represented the characterizations of a particular kind of sweeping surface known as a canal surface. Ro and Yoon provided an analysis of another type of sweeping surface called tube that satisfied particular equations based on surface curvatures in [4]. Research findings also suggest that sweeping surfaces generated by the use of RMF can also be categorized as developable surfaces [5]. The investigation of sweeping surfaces along a curve based on the Darboux frame was documented in [6]. Furthermore, the properties of right conoids hypersurfaces in Minkowski 4-space were studied in [7,8]. The theory of singularity has been thoroughly studied in [9] and it is a vital tool in many various fields of inquiry, including differential geometry, the field of optics, robotics, and visual computing. The involutive sweeping surfaces were presented as a novel surface type by Köseoğlu and Bilici, who also explored their singularities [10]. Several scholars have examined the singularities and characteristic features of the surfaces [11,12,13,14,15,16,17].

On the other hand, it is observed that at the points where a space curve’s second derivative is zero, the Frenet frame becomes insufficient. That is, the principal normal and binormal vector fields of such a curve are not continuous at these points. Sasai created the modified orthogonal frame (MOF) and inferred the formulae corresponding to the Frenet formulae in [18] to address this problem. Recently, Bükcü and Karacan described two MOFs with non-zero curvature and non-zero torsion in Minkowski 3-space [19]. A number of investigations [20,21,22,23,24,25] revisited some specific curves as well as surfaces such as ruled surfaces, Hasimoto surfaces, tubular surfaces, and the evolution of curves using MOF.

In numerous above-mentioned investigations, RMF was typically utilized to generate sweeping surfaces to correct the unwanted rotation of the Frenet frame. In fact, RMF is a frame developed to minimize the rotation of the normal and binormal vectors as they move along the curve. On the other hand, to preserve orthogonality without requiring minimal rotation, the normal and binormal vectors are modified according to curvature and torsion in the MOFs with non-zero curvature and non-zero torsion, respectively.

In this study, we modify the parametric representation of a classical sweeping surface defined along a spine curve with non-zero curvatures everywhere. In this modification, the spine curve may have discrete zero curvatures. This can lead to a discontinuity in the directions of the principal normal vector, and thus, the binormal vector, when approaching a zero-curvature point from either side. Thus, the Frenet frame may not be well-defined, particularly because the normal and binormal directions can become ambiguous. Even if a spine curve has discrete zero curvatures, an MOF is well-defined at any point where the curvature is non-zero, and another MOF is also well-defined at any point where the torsion is non-zero. Thus, two types of modified surfaces occur since there are two types of MOFs with non-vanishing curvature and non-vanishing torsion. The novelty of our study is generating sweeping surfaces by using two types of MOFs, which ensure the orthogonality and continuity of the frame vectors of the spine curves. We determine the criteria for the coordinate curves of these types of modified sweeping surfaces to be geodesic, asymptotic, or lines of curvature. Furthermore, we derive criteria for the modified sweeping surfaces to be developable, minimal, or Weingarten. Finally, we provide examples of these sweeping surfaces and illustrate their graphical representations.

2. Preliminaries

Let $\beta$ be a moving space curve with arc-length parameter s in $E^3$. If the tangent, principal normal, and binormal unit vectors of the space curve $\beta$ are noted by t, n, and b, respectively, then the moving Frenet frame of the unit speed curve $\beta$ satisfies

$$\begin{array}{c}{t}_s=\kappa n,\hfill \\ n_s=-\kappa t+\tau b,\hfill \\ b_s=-\tau n,\hfill \end{array}$$

where the curvature and torsion of $\beta$ are $\kappa$ and $\tau$, respectively. The subscript symbol represents differentiation with respect to the variable s. However, if the principal normal or binormal vectors of a space curve are not continuous at the points where the curvature function has discrete zero points, the Frenet frame cannot be defined. Then, the alternative frame of Sasai can be used instead of the Frenet frame [18].

If $\kappa \left(s_0\right)=0$, then the Frenet frame can display a change at the points $s\in \left(s_0-\epsilon ,s_0+\epsilon \right)$ for any $\epsilon >0$. However, a frame can be defined at the points $s\in \left(s_0-\epsilon ,s_0+\epsilon \right)\setminus \left\{s_0\right\}$ where $\kappa \left(s\right)\ne 0$ called MOF with non-vanishing curvature, and another frame can be defined for $\tau \left(s\right)\ne 0$ called MOF with non-vanishing torsion as explained subsequently. Assume that the curvature $\kappa$ of a general analytic curve $\beta$ is not identically zero; then, the elements of MOF with the non-vanishing curvature of a curve are defined as

$$T={\displaystyle \frac{d\beta }{ds}},N={\displaystyle \frac{dT}{ds}},B=T\times N$$

where “×” represents the vector product. At non-zero curvature values of $\kappa$, the frames $\left\{T,N,B\right\}$ and $\left\{t,n,b\right\}$ are related to each other as

$$T\left(s\right)=t\left(s\right),N\left(s\right)=\kappa \left(s\right)n\left(s\right),B\left(s\right)=\kappa \left(s\right)b\left(s\right).$$

Then, the derivative formulae of MOF are written in matrix form as follows:

$${\left[\begin{array}{c}T\hfill \\ N\hfill \\ B\hfill \end{array}\right]}_s=\left[\begin{array}{ccc}0& 1& 0\\ -{\kappa }^2& {\displaystyle \frac{{\kappa }_s}{\kappa }}& \tau \\ 0& -\tau & {\displaystyle \frac{{\kappa }_s}{\kappa }}\end{array}\right]\left[\begin{array}{c}T\\ N\\ B\end{array}\right]$$

(1)

when $\kappa \ne 0$ and

$$\tau ={\displaystyle \frac{det\left({\beta }_s,{\beta }_{ss},{\beta }_{sss}\right)}{{\kappa }^2}}$$

is the torsion of the space curve $\beta$. Moreover, the MOF with non-zero curvature satisfies

$$〈T,T〉=1,〈N,N〉={\kappa }^2=〈B,B〉.$$

On the other hand, the relations between the elements of the other MOF and the Frenet frame at non-zero torsion values of $\tau$ are

$$T\left(s\right)=t\left(s\right),N\left(s\right)=\tau \left(s\right)n\left(s\right),B\left(s\right)=\tau \left(s\right)b\left(s\right).$$

In this case, we obtain the following MOF with non-zero torsion hold:

$${\left[\begin{array}{c}T\hfill \\ N\hfill \\ B\hfill \end{array}\right]}_s=\left[\begin{array}{ccc}0& {\displaystyle \frac{\kappa }{\tau }}& 0\\ -\kappa \tau & {\displaystyle \frac{{\tau }_s}{\tau }}& \tau \\ 0& -\tau & {\displaystyle \frac{{\tau }_s}{\tau }}\end{array}\right]\left[\begin{array}{c}T\\ N\\ B\end{array}\right]$$

(2)

where

$$〈T,T〉=1,〈N,N〉={\tau }^2=〈B,B〉.$$

In the cases of $\kappa =1$ and $\tau =1$, respectively, the derivative Formulas (1) and (2) are coincident with Frenet derivative formulae.

Let ${\phi }_s={\displaystyle \frac{\partial \phi }{\partial s}}$ and ${\phi }_v={\displaystyle \frac{\partial \phi }{\partial v}}$ be tangent vectors of a surface M parametrized by $\phi \left(s,\upsilon \right)$; then, the equation of the normal vector field of the surface is

$$\Delta \left(s,\upsilon \right)={\displaystyle \frac{{\phi }_s\times {\phi }_{\upsilon }}{∥{\phi }_s\times {\phi }_{\upsilon }∥}}.$$

(3)

The Gaussian and mean curvatures of M are

$$K={\displaystyle \frac{km-l^2}{EG-F^2}}\mathrm{and}H={\displaystyle \frac{Em-2El+Gk}{2\left(EG-F^2\right)}},$$

(4)

respectively, where the elements of $\left\{E,F,G\right\}$ and $\left\{k,l,m\right\}$ denote the coefficients of the first and second fundamental forms of the surface with parametrization $\phi \left(s,\upsilon \right)$ as

$$E=〈{\displaystyle \frac{\partial \phi }{\partial s}},{\displaystyle \frac{\partial \phi }{\partial s}}〉,F=〈{\displaystyle \frac{\partial \phi }{\partial s}},{\displaystyle \frac{\partial \phi }{\partial \upsilon }}〉,G=〈{\displaystyle \frac{\partial \phi }{\partial \upsilon }},{\displaystyle \frac{\partial \phi }{\partial \upsilon }}〉$$

(5)

and

$$k=〈{\displaystyle \frac{{\partial }^2\phi }{\partial s^2}},\Delta 〉,l=〈{\displaystyle \frac{{\partial }^2\phi }{\partial s\partial \upsilon }},\Delta 〉,m=〈{\displaystyle \frac{{\partial }^2\phi }{\partial {\upsilon }^2}},\Delta 〉,$$

(6)

respectively. It is common knowledge that a surface M can be described by the following characterizations given in [13]:

• M is a developable surface if and only if $K=0$ everywhere.
• M is a minimal surface if and only if $H=0$ everywhere.
• M is a Weingarten-type surface if and only if $K_s{H}_{\upsilon }-K_{\upsilon }{H}_s=0$ everywhere.

3. Modified Sweeping Surfaces Using the MOFs

In this section, the parametric expression of sweeping surfaces along a spine curve $\beta$ is modified in order to fix the cases in which $\beta$ has discrete zero curvatures. The modified sweeping surfaces are generated with the spine curves even if $\kappa \left(s_0\right)=0$ for some $s_0$. A modified sweeping surface of the MOF with non-vanishing curvature $\kappa$ along at least twice continuously differentiable spine curve $\beta$ is defined by the modified principal normal and modified binormal vectors $N\left(s\right)=\kappa \left(s\right)n\left(s\right)$ and $B\left(s\right)=\kappa \left(s\right)b\left(s\right)$ when $\kappa \left(s\right)\ne 0$ at $s\in \left(s_0-\epsilon ,s_0+\epsilon \right)\setminus \left\{s_0\right\}$ for any $\epsilon >0$. Moreover, a modified sweeping surface of the MOF with non-vanishing torsion $\tau$ along at least three times continuously differentiable spine curve $\beta$ is defined by the modified principal normal and modified binormal vectors $N\left(s\right)=\tau \left(s\right)n\left(s\right)$ and $B\left(s\right)=\tau \left(s\right)b\left(s\right)$ when $\tau \left(s\right)\ne 0$ at $s\in \left(s_0-\epsilon ,s_0+\epsilon \right)\setminus \left\{s_0\right\}$ for any $\epsilon >0$. Now, let us outline a simple method for illustrating the modified sweeping surfaces. The parameter along the curve $\beta$ is chosen as one of the variables, and the position vector $\phi$ is established by connecting a point on the curve $\beta$ to another point on the modified sweeping surface.

The parametric equation of a modified sweeping surface M, formed by the spine curve $\beta$ and the planar profile (cross-section) curve $\delta \left(\upsilon \right)={\left(0,r\left(\upsilon \right),p\left(\upsilon \right)\right)}^{\perp }$, where the symbol “^⊥” represents transpose, is

$$\phi \left(s,\upsilon \right)=\beta \left(s\right)+\delta \left(\upsilon \right)\Gamma \left(s\right)=\beta \left(s\right)+r\left(\upsilon \right)N\left(s\right)+p\left(\upsilon \right)B\left(s\right),$$

(7)

where $\Gamma \left(s\right)=\left(T\left(s\right),N\left(s\right),B\left(s\right)\right)$ denotes the orthogonal matrix with the elements of MOFs with non-vanishing curvature and non-vanishing torsion along the spine curve $\beta \left(s\right)$. Thus, two types of modified sweeping surfaces exist that are investigated in the following subsections.

3.1. Modified Sweeping Surfaces of the MOF with Non-Vanishing Curvature $\kappa$

In this subsection, the modified sweeping surface M from (7) is investigated according to the MOF with non-vanishing curvature $\kappa$. Referring to (1), the first-order partial differentiations of $\phi \left(s,\upsilon \right)$ with respect to s and $\upsilon$ are found as

$${\phi }_s=\left(1-r{\kappa }^2\right)T+\left({\displaystyle \frac{r{\kappa }_s}{\kappa }}-p\tau \right)N+\left(r\tau +{\displaystyle \frac{p{\kappa }_s}{\kappa }}\right)B$$

and

$${\phi }_{\upsilon }=r_{\upsilon }N+p_{\upsilon }B.$$

Thus, by a straightforward computation from the last two equations and Equation (3), the unit normal vector field $\Delta$ of the surface is found as

$$\Delta \left(s,\upsilon \right)={\displaystyle \frac{\left({\lambda }_2{p}_{\upsilon }-{\lambda }_3{r}_{\upsilon }\right)T-{\lambda }_1{p}_{\upsilon }N+{\lambda }_1{r}_{\upsilon }B}{\sqrt{{{\lambda }_1}^2{\kappa }^2\left({p_{\upsilon }}^2+{r_{\upsilon }}^2\right)+{\left({\lambda }_2{p}_{\upsilon }-{\lambda }_3{r}_{\upsilon }\right)}^2}}},$$

(8)

where

$${\lambda }_1\left(s,\upsilon \right)=1-r{\kappa }^2,{\lambda }_2\left(s,\upsilon \right)={\displaystyle \frac{r{\kappa }_s}{\kappa }}-p\tau ,\mathrm{and}{\lambda }_3\left(s,\upsilon \right)={\displaystyle \frac{p{\kappa }_s}{\kappa }}+r\tau .$$

(9)

Theorem 1. 

Let M be a modified sweeping surface formed by the MOF with non-vanishing curvature κ. Then, the Gaussian curvature of M is given with:

$$K\left(s,\upsilon \right)={\displaystyle \frac{{\lambda }_1{\kappa }^2{\eta }_2\left(p_{\upsilon }\left({\lambda }_2{\mu }_1-{\kappa }^2{\lambda }_1{\mu }_2\right)-r_{\upsilon }\left({\lambda }_1{\mu }_1-{\kappa }^2{\lambda }_1{\mu }_3\right)\right)-{\left(\begin{array}{c}{\lambda }_1{\kappa }^2\left(r_{\upsilon }{\lambda }_{3\upsilon }-p_{\upsilon }{\lambda }_{2\upsilon }\right)\hfill \\ +{\lambda }_{1\upsilon }\left({\lambda }_2{p}_{\upsilon }-{\lambda }_3{r}_{\upsilon }\right)\hfill \end{array}\right)}^2}{{∥{\phi }_s\times {\phi }_{\upsilon }∥}^2{\kappa }^2\left({\kappa }^2{\left({\lambda }_2{p}_{\upsilon }-{\lambda }_3{r}_{\upsilon }\right)}^2+{\lambda }_1^2{\eta }_1\right)}}.$$

The mean curvature of M is obtained as

$$H\left(s,\upsilon \right)={\displaystyle \frac{{\eta }_1\left(\begin{array}{c}{\mu }_1\hfill \\ -{\lambda }_1{\kappa }^2\hfill \end{array}\right)\left(\begin{array}{c}{p}_{\upsilon }{\lambda }_2\hfill \\ -r_{\upsilon }{\lambda }_3\hfill \end{array}\right)-{\lambda }_1{\eta }_2\left(\begin{array}{c}{\lambda }_1^2\hfill \\ +{\kappa }^2\left({\lambda }_2^2+{\lambda }_3^2\right)\hfill \end{array}\right)-2\left(\begin{array}{c}{\lambda }_3{p}_{\upsilon }\hfill \\ +{\lambda }_2{r}_{\upsilon }\hfill \end{array}\right)\left(\begin{array}{c}{p}_{\upsilon }\left({\lambda }_2{\lambda }_{1\upsilon }-{\kappa }^2{\lambda }_1{\lambda }_{2\upsilon }\right)\hfill \\ +r_{\upsilon }\left({\kappa }^2{\lambda }_1{\lambda }_{3\upsilon }-{\lambda }_3{\lambda }_{1\upsilon }\right)\hfill \end{array}\right)}{2∥{\phi }_s\times {\phi }_{\upsilon }∥\left({\kappa }^2{\left({\lambda }_2{p}_{\upsilon }-{\lambda }_3{r}_{\upsilon }\right)}^2+{\lambda }_1^2{\eta }_1\right)}}.$$

Here, the following notations are employed for the sake of brevity:

$$\begin{array}{c}∥{\phi }_s\times {\phi }_{\upsilon }∥=\sqrt{{{\lambda }_1}^2{\kappa }^2\left({p_{\upsilon }}^2+{r_{\upsilon }}^2\right)+{\left({\lambda }_2{p}_{\upsilon }-{\lambda }_3{r}_{\upsilon }\right)}^2},\hfill \\ {\mu }_1\left(s,\upsilon \right)={\lambda }_{1s}-{\kappa }^2{\lambda }_2,\hfill \\ {\mu }_2\left(s,\upsilon \right)={\lambda }_1+{\lambda }_{2s}-{\lambda }_3\tau +{\displaystyle \frac{{\lambda }_2{\kappa }_s}{\kappa }},\hfill \\ {\mu }_3\left(s,\upsilon \right)={\lambda }_2\tau +{\lambda }_{3s}+{\displaystyle \frac{{\lambda }_3{\kappa }_s}{\kappa }},\hfill \\ {\eta }_1\left(\upsilon \right)={p_{\upsilon }}^2+{r_{\upsilon }}^2,\hfill \\ {\eta }_2\left(\upsilon \right)=r_{\upsilon }{p}_{\upsilon \upsilon }-p_{\upsilon }{r}_{\upsilon \upsilon }.\hfill \end{array}$$

Proof. 

Let M be a modified sweeping surface generated by the MOF with non-vanishing curvature $\kappa$. From the Equation (5), the coefficients of the first fundamental form of M are

$$\left\{\begin{array}{c}E\left(s,\upsilon \right)={{\lambda }_1}^2+{\kappa }^2\left({{\lambda }_2}^2+{{\lambda }_3}^2\right),\hfill \\ F\left(s,\upsilon \right)={\kappa }^2\left({\lambda }_2{r}_{\upsilon }+{\lambda }_3{p}_{\upsilon }\right),\hfill \\ G\left(s,\upsilon \right)={\kappa }^2\left({p_{\upsilon }}^2+{r_{\upsilon }}^2\right).\hfill \end{array}\right.$$

By considering Equation (1), the second-order partial differentiations of $\phi \left(s,\upsilon \right)$ given in (7) with respect to s and $\upsilon$ are obtained as follows:

$$\begin{array}{c}{\phi }_{ss}={\mu }_{1}T+{\mu }_{2}N+{\mu }_{3}B,\hfill \\ {\phi }_{s\upsilon }={\lambda }_{1\upsilon }T+{\lambda }_{2\upsilon }N+{\lambda }_{3\upsilon }B,\hfill \\ {\phi }_{\upsilon \upsilon }=r_{\upsilon \upsilon }N+p_{\upsilon \upsilon }B.\hfill \end{array}$$

(10)

Note that for the sake of simplicity, we use the following notations:

$$\begin{array}{c}{\mu }_1\left(s,\upsilon \right)={\lambda }_{1s}-{\kappa }^2{\lambda }_2,\hfill \\ {\mu }_2\left(s,\upsilon \right)={\lambda }_1+{\lambda }_{2s}-{\lambda }_3\tau +{\displaystyle \frac{{\lambda }_2{\kappa }_s}{\kappa }},\hfill \\ {\mu }_3\left(s,\upsilon \right)={\lambda }_2\tau +{\lambda }_{3s}+{\displaystyle \frac{{\lambda }_3{\kappa }_s}{\kappa }}.\hfill \end{array}$$

From the Equation (6), the coefficients k, l, and m for M are found as follows:

$$\left\{\begin{array}{c}k\left(s,\upsilon \right)={\displaystyle \frac{\left({\mu }_1\left({\lambda }_2{p}_{\upsilon }-{\lambda }_3{r}_{\upsilon }\right)-{\kappa }^2{\lambda }_1\left({\mu }_2{p}_{\upsilon }-{\mu }_3{r}_{\upsilon }\right)\right)}{∥{\phi }_s\times {\phi }_{\upsilon }∥}},\hfill \\ l\left(s,\upsilon \right)={\displaystyle \frac{\left({\lambda }_{1\upsilon }\left({\lambda }_2{p}_{\upsilon }-{\lambda }_3{r}_{\upsilon }\right)-{\kappa }^2{\lambda }_1\left({\lambda }_{2\upsilon }{p}_{\upsilon }-{\lambda }_{3\upsilon }{r}_{\upsilon }\right)\right)}{∥{\phi }_s\times {\phi }_{\upsilon }∥}},\hfill \\ m\left(s,\upsilon \right)={\displaystyle \frac{{\kappa }^2{\lambda }_1{\eta }_2}{∥{\phi }_s\times {\phi }_{\upsilon }∥}}.\hfill \end{array}\right.$$

If the magnitudes E, F, G, k, l, and m are substituted into Equation (4), the Gaussian and mean curvatures K and H of M are obtained as in the hypothesis. □

Corollary 1. 

Let M be a modified sweeping surface generated by the MOF with non-vanishing curvature κ. Then, M is developable if and only if

$$\begin{array}{c}{\lambda }_1{\kappa }^2{\eta }_2\left({\lambda }_2{\mu }_1{p}_{\upsilon }-{\lambda }_3{\mu }_1{r}_{\upsilon }+{\lambda }_1{\kappa }^2\left(-{\mu }_2{p}_{\upsilon }+{\mu }_3{r}_{\upsilon }\right)\right)={\left({\lambda }_1{\kappa }^2\left(-p_{\upsilon }{\lambda }_{2\upsilon }+r_{\upsilon }{\lambda }_{3\upsilon }\right)+{\lambda }_{1s}\left({\lambda }_2{p}_{\upsilon }-{\lambda }_3{r}_{\upsilon }\right)\right)}^2.\hfill \end{array}$$

M is minimal if and only if

$$\begin{array}{c}{\eta }_1\left({\mu }_1\left({\lambda }_2{p}_{\upsilon }-{\lambda }_3{r}_{\upsilon }\right)+{\lambda }_1{\kappa }^2\left({\mu }_3{r}_{\upsilon }-{\mu }_2{p}_{\upsilon }\right)\right)+{\lambda }_1{\eta }_2\left({\lambda }_1^2+{\kappa }^2\left({\lambda }_2^2+{\lambda }_3^2\right)\right)\hfill \\ -2\left({\lambda }_3{p}_{\upsilon }+{\lambda }_2{r}_{\upsilon }\right)\left({\lambda }_1{\kappa }^2\left(r_{\upsilon }{\lambda }_{3\upsilon }-p_{\upsilon }{\lambda }_{2\upsilon }\right)+\left({\lambda }_2{p}_{\upsilon }-{\lambda }_3{r}_{\upsilon }\right){\lambda }_{1s}\right)=0.\hfill \end{array}$$

Theorem 2. 

Let M be a modified sweeping surface generated by the MOF with non-vanishing curvature κ. Then, M is a Weingarten-type surface if and only if

$$\left(\begin{array}{c}{\kappa }^2{\epsilon }_1\left({\lambda }_1{\eta }_{2\upsilon }+{\eta }_2{\lambda }_{1\upsilon }\right)\hfill \\ +\left({\lambda }_1{\kappa }^2{\eta }_2-2{\epsilon }_1\right){\epsilon }_{1\upsilon }\hfill \end{array}\right)\left(\begin{array}{c}{\eta }_1{\delta }_{1s}-2{\delta }_{3s}\hfill \\ -{\eta }_2\left({\delta }_2{\lambda }_{1s}+{\lambda }_1{\delta }_{2s}\right)\hfill \end{array}\right)=\left(\begin{array}{c}{\delta }_1{\eta }_{1\upsilon }-{\delta }_2\left({\lambda }_1{\eta }_{2\upsilon }+{\eta }_2{\lambda }_{1\upsilon }\right)\hfill \\ +{\eta }_1{\delta }_{1\upsilon }-{\lambda }_1{\eta }_2{\delta }_{2\upsilon }-2{\delta }_{3\upsilon }\hfill \end{array}\right)\left(\begin{array}{c}\kappa {\epsilon }_1{\eta }_2\left(2{\lambda }_1{\kappa }_s+\kappa {\lambda }_{1s}\right)\hfill \\ +{\epsilon }_{1s}\left({\lambda }_1{\kappa }^2{\eta }_2-2{\epsilon }_1\right)\hfill \end{array}\right)$$

where

$$\begin{array}{c}{\epsilon }_1\left(s,\upsilon \right)={\displaystyle \frac{p_{\upsilon }\left({\lambda }_2{\mu }_1-{\kappa }^2{\lambda }_1{\mu }_2\right)-r_{\upsilon }\left({\lambda }_1{\mu }_1-{\kappa }^2{\lambda }_1{\mu }_3\right)}{{∥{\phi }_s\times {\phi }_{\upsilon }∥}^2{\kappa }^2\left({\kappa }^2{\left({\lambda }_2{p}_{\upsilon }-{\lambda }_3{r}_{\upsilon }\right)}^2+{\lambda }_1^2{\eta }_1\right)}},\hfill \\ {\epsilon }_2\left(s,\upsilon \right)={\displaystyle \frac{p_{\upsilon }\left({\lambda }_2{\mu }_1-{\kappa }^2{\lambda }_1{\mu }_2\right)-r_{\upsilon }\left({\lambda }_1{\mu }_1-{\kappa }^2{\lambda }_1{\mu }_3\right)}{∥{\phi }_s\times {\phi }_{\upsilon }∥\kappa \sqrt{\left({\kappa }^2{\left({\lambda }_2{p}_{\upsilon }-{\lambda }_3{r}_{\upsilon }\right)}^2+{\lambda }_1^2{\eta }_1\right)}}},\hfill \\ {\delta }_1\left(s,\upsilon \right)={\displaystyle \frac{\left({\mu }_1-{\lambda }_1{\kappa }^2\right)\left(p_{\upsilon }{\lambda }_2-r_{\upsilon }{\lambda }_3\right)}{2∥{\phi }_s\times {\phi }_{\upsilon }∥\left({\kappa }^2{\left({\lambda }_2{p}_{\upsilon }-{\lambda }_3{r}_{\upsilon }\right)}^2+{\lambda }_1^2{\eta }_1\right)}},\hfill \\ {\delta }_2\left(s,\upsilon \right)={\displaystyle \frac{{\lambda }_1^2+{\kappa }^2\left({\lambda }_2^2+{\lambda }_3^2\right)}{2∥{\phi }_s\times {\phi }_{\upsilon }∥\left({\kappa }^2{\left({\lambda }_2{p}_{\upsilon }-{\lambda }_3{r}_{\upsilon }\right)}^2+{\lambda }_1^2{\eta }_1\right)}},\hfill \\ {\delta }_3\left(s,\upsilon \right)={\displaystyle \frac{\left({\lambda }_3{p}_{\upsilon }+{\lambda }_2{r}_{\upsilon }\right)\left(p_{\upsilon }\left({\lambda }_2{\lambda }_{1\upsilon }-{\kappa }^2{\lambda }_1{\lambda }_{2\upsilon }\right)+r_{\upsilon }\left({\kappa }^2{\lambda }_1{\lambda }_{3\upsilon }-{\lambda }_3{\lambda }_{1\upsilon }\right)\right)}{2∥{\phi }_s\times {\phi }_{\upsilon }∥\left({\kappa }^2{\left({\lambda }_2{p}_{\upsilon }-{\lambda }_3{r}_{\upsilon }\right)}^2+{\lambda }_1^2{\eta }_1\right)}}.\hfill \end{array}$$

Proof. 

Let M be a modified sweeping surface formed by the MOF with non-vanishing curvature $\kappa$. If the Gaussian and mean curvatures K and H of M are differentiated in terms of s and $\upsilon$, we have

$$\left\{\begin{array}{c}{K}_s=\kappa {\epsilon }_1{\eta }_2\left(2{\lambda }_1{\kappa }_s+\kappa {\lambda }_{1s}\right)+\left(-2{\epsilon }_1+{\lambda }_1{\kappa }^2{\eta }_2\right){\epsilon }_{1s},\hfill \\ K_{\upsilon }={\kappa }^2{\epsilon }_1\left({\lambda }_1{\eta }_{2\upsilon }+{\eta }_2{\lambda }_{1\upsilon }\right)+\left(-2{\epsilon }_1+{\lambda }_1{\kappa }^2{\eta }_2\right){\epsilon }_{1\upsilon },\hfill \\ H_s={\eta }_1{\delta }_{1s}-{\eta }_2\left({\delta }_2{\lambda }_{1s}+{\lambda }_1{\delta }_{2s}\right)-2{\delta }_{3s},\hfill \\ H_{\upsilon }={\delta }_1{\eta }_{1\upsilon }-{\delta }_2\left({\lambda }_1{\eta }_{2\upsilon }+{\eta }_2{\lambda }_{1\upsilon }\right)+{\eta }_1{\delta }_{1\upsilon }-{\lambda }_1{\eta }_2{\delta }_{2\upsilon }-2{\delta }_{3\upsilon }.\hfill \end{array}\right.$$

By considering these equalities, the condition specified in the hypothesis is satisfied if and only if $K_s{H}_{\upsilon }-K_{\upsilon }{H}_s=0$, which requires M to be a Weingarten-type surface. □

Theorem 3. 

If M is a modified sweeping surface generated by the MOF with non-vanishing curvature κ, then, the $s-$coordinate curves of M are geodesic if and only if

$${\displaystyle \frac{p_{\upsilon }}{r_{\upsilon }}}={\displaystyle \frac{{\lambda }_1\left({\mu }_1-{\mu }_2\right)+{\lambda }_3\left({\mu }_3-{\mu }_2\right)}{{\lambda }_1\left({\mu }_3-{\mu }_1\right)+{\lambda }_2\left({\mu }_3-{\mu }_2\right)}}.$$

The $s-$coordinate curves of M are asymptotic if and only if

$${\displaystyle \frac{p_{\upsilon }}{r_{\upsilon }}}={\displaystyle \frac{{\mu }_1{\lambda }_3-{\kappa }^2{\mu }_3{\lambda }_1}{{\mu }_1{\lambda }_2-{\kappa }^2{\mu }_2{\lambda }_1}}.$$

Proof. 

To ensure that the coordinate curves meet the criteria for being geodesic curves, the acceleration vector of the coordinate curve must be perpendicular to the surface, and, thereby, parallel to the surface’s normal vector.

From the equation of the normal vector field given in (8) and the first equality of (10), the following equation is obtained:

$$\begin{array}{c}\Delta \times {\phi }_{ss}={\lambda }_1\left({\mu }_3{p}_{\upsilon }+{\mu }_2{r}_{\upsilon }\right)T+\left({\mu }_3{\lambda }_2{p}_{\upsilon }-\left({\mu }_3{\lambda }_3+{\mu }_1{\lambda }_1\right)r_{\upsilon }\right)N\hfill \\ +\left({\mu }_2{\lambda }_3{r}_{\upsilon }-\left({\mu }_2{\lambda }_2+{\mu }_1{\lambda }_1\right)p_{\upsilon }\right)B.\hfill \end{array}$$

By the fact that the vectors T, N, and B are linearly independent, if $\Delta \times {\phi }_{ss}=0$, then the last equation requires

$$\begin{array}{c}{\lambda }_1\left({\mu }_3{p}_{\upsilon }+{\mu }_2{r}_{\upsilon }\right)=0,\hfill \\ {\mu }_3{\lambda }_2{p}_{\upsilon }-\left({\mu }_3{\lambda }_3+{\mu }_1{\lambda }_1\right)r_{\upsilon }=0,\hfill \\ {\mu }_2{\lambda }_3{r}_{\upsilon }-\left({\mu }_2{\lambda }_2+{\mu }_1{\lambda }_1\right)p_{\upsilon }=0.\hfill \end{array}$$

Thus, the criterion in the hypothesis is verified by the common solution of these equalities. It is a fact that the $s-$coordinate curves are geodesics under the criterion stated in the hypothesis.

Moreover, a curve on a surface is classified as asymptotic if the acceleration vector of the curve lies entirely in the tangent plane of the surface, which means that the inner product of the second derivative of the curve and the normal vector field of the surface must be zero. To check the condition for the $s-$coordinate curves of M to be asymptotic, we take the Equation (8) and the first equality in (10) for $\Delta$ and ${\phi }_{ss}$, respectively. Then, we obtain

$$〈\Delta ,{\phi }_{ss}〉=\left({\mu }_1{\lambda }_2-{\kappa }^2{\mu }_2{\lambda }_1\right)p_{\upsilon }+\left({\kappa }^2{\mu }_3{\lambda }_1-{\mu }_1{\lambda }_3\right)r_{\upsilon }.$$

Thus, $\left({\mu }_1{\lambda }_2-{\kappa }^2{\mu }_2{\lambda }_1\right)p_{\upsilon }=\left({\mu }_1{\lambda }_3-{\kappa }^2{\mu }_3{\lambda }_1\right)r_{\upsilon }$ if and only if $〈\Delta ,{\phi }_{ss}〉=0$, and we can state that the $s-$coordinate curves are asymptotic under the criterion stated in the hypothesis. □

Theorem 4. 

Let M be a modified sweeping surface generated by the MOF with non-vanishing curvature κ. Then, the $\upsilon -$coordinate curves of M are geodesic if and only if

$${\lambda }_2{p}_{\upsilon }-{\lambda }_3{r}_{\upsilon }=0,p_{\upsilon \upsilon }+r_{\upsilon \upsilon }=0,\mathrm{and}{\lambda }_1\left(p_{\upsilon }{p}_{\upsilon \upsilon }+r_{\upsilon }{r}_{\upsilon \upsilon }\right)=0.$$

Also, the $\upsilon -$coordinate curves of M are asymptotic if and only if

$$r{\kappa }^2=1\mathrm{or}{r}_{\upsilon }{p}_{\upsilon \upsilon }-p_{\upsilon }{r}_{\upsilon \upsilon }=0.$$

Proof. 

From the equation of the normal vector field given by (8) and the third equality of (10), the following equation is obtained:

$$\Delta \times {\phi }_{\upsilon \upsilon }={\lambda }_1\left(p_{\upsilon }{p}_{\upsilon \upsilon }+r_{\upsilon }{r}_{\upsilon \upsilon }\right)T+p_{\upsilon \upsilon }\left({\lambda }_2{p}_{\upsilon }-{\lambda }_3{r}_{\upsilon }\right)N-r_{\upsilon \upsilon }\left({\lambda }_2{p}_{\upsilon }-{\lambda }_3{r}_{\upsilon }\right)B.$$

Since the vectors T, N, and B are linearly independent, if $\Delta \times {\phi }_{\upsilon \upsilon }=0$, we have

$$\begin{array}{c}\left(p_{\upsilon }{p}_{\upsilon \upsilon }+r_{\upsilon }{r}_{\upsilon \upsilon }\right){\lambda }_1=0,\hfill \\ \left({\lambda }_2{p}_{\upsilon }-{\lambda }_3{r}_{\upsilon }\right)p_{\upsilon \upsilon }=0,\hfill \\ \left({\lambda }_2{p}_{\upsilon }-{\lambda }_3{r}_{\upsilon }\right)r_{\upsilon \upsilon }=0.\hfill \end{array}$$

Thus, the criterion in the hypothesis is verified by these last three equations. Consequently, we can say that the $\upsilon -$coordinate curves of M are geodesics under the criterion stated in the hypothesis.

Now, let us check whether the $\upsilon -$coordinate curves of M are asymptotic. Therefore, we take the Equation (8) and the third equality in (10) and we obtain

$$〈\Delta ,{\phi }_{\upsilon \upsilon }〉={\kappa }^2{\lambda }_1\left(r_{\upsilon }{p}_{\upsilon \upsilon }-p_{\upsilon }{r}_{\upsilon \upsilon }\right).$$

This gives us ${\lambda }_1=0$ or $r_{\upsilon }{p}_{\upsilon \upsilon }-p_{\upsilon }{r}_{\upsilon \upsilon }=0$ if and only if $〈\Delta ,{\phi }_{\upsilon \upsilon }〉=0$. Since ${\lambda }_1=1-r{\kappa }^2$ from (9), it is obvious that the $\upsilon -$coordinate curves of M are asymptotic under the criteria stated in the hypothesis. □

Theorem 5. 

Let M be a modified sweeping surface generated by the MOF with non-vanishing curvature κ. If the s and $\upsilon -$coordinate curves of M are lines of curvature, then

$${\displaystyle \frac{{\lambda }_2}{{\lambda }_3}}={\displaystyle \frac{{\kappa }^2{\lambda }_1{\lambda }_{3\upsilon }-{\lambda }_{1\upsilon }{\lambda }_3}{{\lambda }_{1\upsilon }{\lambda }_2-{\kappa }^2{\lambda }_1{\lambda }_{2\upsilon }}}.$$

Proof. 

It is known that the curves that are always tangent to a principal direction are known as lines of curvature. Each coordinate curve follows one of the principal curvature directions on the surface if and only if $F=l=0$, because $F=0$ ensures the orthogonality of the coordinate curves and $l=0$ ensures that each coordinate direction is independent in terms of curvature. Let us recall the equations

$$F\left(s,\upsilon \right)={\kappa }^2\left({\lambda }_3{p}_{\upsilon }+{\lambda }_2{r}_{\upsilon }\right)$$

and

$$l\left(s,\upsilon \right)={\displaystyle \frac{{\lambda }_{1\upsilon }\left({\lambda }_2{p}_{\upsilon }-{\lambda }_3{r}_{\upsilon }\right)-{\kappa }^2{\lambda }_1\left({\lambda }_{2\upsilon }{p}_{\upsilon }-{\lambda }_{3\upsilon }{r}_{\upsilon }\right)}{∥{\phi }_s\times {\phi }_{\upsilon }∥}}.$$

Then, the common solution of $F=0$ and $l=0$ gives us

$${\displaystyle \frac{{\lambda }_2}{{\lambda }_3}=\frac{{\kappa }^2{\lambda }_1{\lambda }_{3\upsilon }-{\lambda }_{1\upsilon }{\lambda }_3}{{\lambda }_{1\upsilon }{\lambda }_2-{\kappa }^2{\lambda }_1{\lambda }_{2\upsilon }}.}$$

This completes the proof. □

3.2. Modified Sweeping Surfaces of the MOF with Non-Vanishing Torsion $\tau$

In this section, the surface given by the Equation (7) is investigated using the MOF with non-vanishing torsion. The first order partial differentiations of $\phi \left(s,\upsilon \right)$ with respect to s and $\upsilon$, are found as follows:

$${\phi }_s=\left(1-r\kappa \tau \right)T+\left({\displaystyle \frac{r{\tau }_s}{\tau }}-\tau p\right)N+\left({\displaystyle \frac{p{\tau }_s}{\tau }}+r\tau \right)B$$

and

$${\phi }_{\upsilon }=r_{\upsilon }N+p_{\upsilon }B.$$

Thus, by a straightforward computation from the last equations and the Equation (3), the normal vector field $\Delta$ of M is obtained as

$$\Delta \left(s,\upsilon \right)={\displaystyle \frac{\left(f_2{p}_{\upsilon }-f_3{r}_{\upsilon }\right)T-f_1{p}_{\upsilon }N+f_1{r}_{\upsilon }B}{\sqrt{{\tau }^2{f_1}^2\left({p_{\upsilon }}^2+{r_{\upsilon }}^2\right)+{\left(f_2{p}_{\upsilon }-f_3{r}_{\upsilon }\right)}^2}}},$$

(11)

where

$$f_1\left(s,\upsilon \right)=1-r\kappa \tau ,f_2\left(s,\upsilon \right)={\displaystyle \frac{r{\tau }_s}{\tau }}-\tau p,\mathrm{and}{f}_3\left(s,\upsilon \right)={\displaystyle \frac{p{\tau }_s}{\tau }}+r\tau .$$

Theorem 6. 

Let M be a modified sweeping surface constructed by the MOF with non-vanishing torsion. Then, the Gaussian curvature of M is given by

$$K\left(s,\upsilon \right)={\displaystyle \frac{f_1{\eta }_2\left(\left(f_2{g}_1-f_1{g}_2\right)p_{\upsilon }+\left(f_1{g}_3-f_3{g}_1\right)r_{\upsilon }\right)-{\left(\left(f_2{p}_{\upsilon }-f_3{r}_{\upsilon }\right)f_{1\upsilon }-f_1\left(p_{\upsilon }{f}_{2\upsilon }-r_{\upsilon }{f}_{3\upsilon }\right)\right)}^2}{{∥{\phi }_s\times {\phi }_{\upsilon }∥}^2{\tau }^2\left({f_1}^2{\eta }_1+{\tau }^2\left(\left({f_2}^2+{f_3}^2\right){\eta }_1-{\left(f_3{p}_{\upsilon }+f_2{r}_{\upsilon }\right)}^2\right)\right)}}.$$

Under the same condition, the mean curvatures of M is given by

$$H\left(s,\upsilon \right)={\displaystyle \frac{{\tau }^2\left(f_3{r}_{\upsilon }-f_2{p}_{\upsilon }\right)\left(\begin{array}{c}2f_{1\upsilon }\left(f_3{p}_{\upsilon }+f_2{r}_{\upsilon }\right)\hfill \\ -{\eta }_1{g}_1\hfill \end{array}\right)+f_1\left({f_1}^2{\eta }_2+{\tau }^2\left(\begin{array}{c}\left({f_2}^2+{f_3}^2\right){\eta }_2+{\eta }_1\left(+g_3{r}_{\upsilon }-g_2{p}_{\upsilon }\right)\hfill \\ +2\left(f_3{p}_{\upsilon }+f_2{r}_{\upsilon }\right)\left(p_{\upsilon }{f}_{2\upsilon }-r_{\upsilon }{f}_{3\upsilon }\right)\hfill \end{array}\right)\right)}{2∥{\phi }_x\times {\phi }_{\upsilon }∥{\tau }^2\left({f_1}^2{\eta }_1+{\tau }^2\left({\eta }_1\left({f_2}^2+{f_3}^2\right)-{\left(f_3{p}_{\upsilon }+f_2{r}_{\upsilon }\right)}^2\right)\right)}}$$

where

$$\begin{array}{c}∥{\phi }_s\times {\phi }_{\upsilon }∥=\sqrt{{\tau }^2{f_1}^2\left({p_{\upsilon }}^2+{r_{\upsilon }}^2\right)+{\left(f_2{p}_{\upsilon }-f_3{r}_{\upsilon }\right)}^2},\hfill \\ g_1\left(s,\upsilon \right)=f_{1s}-f_2\kappa \tau ,\hfill \\ g_2\left(s,\upsilon \right)=f_{2s}-f_3\tau +{\displaystyle \frac{f_1\kappa }{\tau }}+{\displaystyle \frac{f_2{\tau }_s}{\tau }},\hfill \\ g_3\left(s,\upsilon \right)=f_2\tau +f_{3s}+{\displaystyle \frac{f_3{\tau }_s}{\tau }}.\hfill \end{array}$$

Proof. 

Assume that M is a modified sweeping surface generated by the MOF with non-vanishing torsion. From the Equation (5), the coefficients of the first fundamental form of this sweeping surface are found as follows:

$$\left\{\begin{array}{c}E\left(s,\upsilon \right)={f_1}^2+{\tau }^2\left({f_2}^2+{f_3}^2\right),\hfill \\ F\left(s,\upsilon \right)={\tau }^2\left(f_2{r}_{\upsilon }+f_3{p}_{\upsilon }\right),\hfill \\ G\left(s,\upsilon \right)={\tau }^2\left({p_{\upsilon }}^2+{r_{\upsilon }}^2\right),\hfill \end{array}\right.$$

where $f_1(s,v),f_2(s,v)$, and $f_3(s,v)$ are as mentioned above. The second-order partial differentiations of $\phi \left(s,\upsilon \right)$ with respect to s and $\upsilon$ are as follows:

$$\begin{array}{c}{\phi }_{ss}=g_{1}T+g_{2}N+g_{3}B,\hfill \\ {\phi }_{s\upsilon }=f_{1\upsilon }T+f_{2\upsilon }N+f_{3\upsilon }B,\hfill \\ {\phi }_{\upsilon \upsilon }=r_{\upsilon \upsilon }N+p_{\upsilon \upsilon }B,\hfill \end{array}$$

(12)

where

$$\begin{array}{c}{g}_1\left(s,\upsilon \right)=f_{1s}-f_2\kappa \tau ,\hfill \\ g_2\left(s,\upsilon \right)=f_{2s}-f_3\tau +{\displaystyle \frac{f_1\kappa }{\tau }}+{\displaystyle \frac{f_2{\tau }_s}{\tau }},\hfill \\ g_3\left(s,\upsilon \right)=f_2\tau +f_{3s}+{\displaystyle \frac{f_3{\tau }_s}{\tau }}.\hfill \end{array}$$

By Equations (6) and (12), the second fundamental form coefficients of M are found as follows:

$$\left\{\begin{array}{c}k\left(s,\upsilon \right)={\displaystyle \frac{g_1\left(f_2{p}_{\upsilon }-f_3{r}_{\upsilon }\right)+{\tau }^2{f}_1\left(g_3{r}_{\upsilon }-g_2{p}_{\upsilon }\right)}{∥{\phi }_s\times {\phi }_{\upsilon }∥}},\hfill \\ l\left(s,\upsilon \right)={\displaystyle \frac{f_{1\upsilon }\left(f_2{p}_{\upsilon }-f_3{r}_{\upsilon }\right)+{\tau }^2{f}_1\left(f_{3\upsilon }{r}_{\upsilon }-f_{2\upsilon }{p}_{\upsilon }\right)}{∥{\phi }_s\times {\phi }_{\upsilon }∥}},\hfill \\ m\left(s,\upsilon \right)={\displaystyle \frac{{\tau }^2{f}_1{\eta }_2}{∥{\phi }_s\times {\phi }_{\upsilon }∥}}.\hfill \end{array}\right.$$

If the coefficients of the first and second fundamental forms of this type of modified sweeping surface are substituted into Equation (4), the Gaussian and mean curvatures K and H of M are obtained as in the hypothesis. □

Corollary 2. 

If M is a modified sweeping surface generated by the MOF with non-vanishing torsion, then, M is developable if and only if

$$f_1{\eta }_2\left(\left(f_2{g}_1-f_1{g}_2\right)p_{\upsilon }+\left(f_1{g}_3-f_3{g}_1\right)r_{\upsilon }\right)={\left(\left(f_2{p}_{\upsilon }-f_3{r}_{\upsilon }\right)f_{1\upsilon }-f_1\left(p_{\upsilon }{f}_{2\upsilon }-r_{\upsilon }{f}_{3\upsilon }\right)\right)}^2.$$

The modified sweeping surface generated by the MOF with non-vanishing torsion is the minimal surface if and only if

$${\tau }^2\left(f_3{r}_{\upsilon }-f_2{p}_{\upsilon }\right)\left(2f_{1\upsilon }\left(f_3{p}_{\upsilon }+f_2{r}_{\upsilon }\right)-{\eta }_1{g}_1\right)=f_1\left({f_1}^2{\eta }_2+{\tau }^2\left(\begin{array}{c}{\eta }_2\left({f_2}^2+{f_3}^2\right)+{\eta }_1\left(g_3{r}_{\upsilon }-g_2{p}_{\upsilon }\right)\hfill \\ +2\left(f_3{p}_{\upsilon }+f_2{r}_{\upsilon }\right)\left(p_{\upsilon }{f}_{2\upsilon }-r_{\upsilon }{f}_{3\upsilon }\right)\hfill \end{array}\right)\right).$$

Theorem 7. 

Let M be a modified sweeping surface generated by the MOF with non-vanishing torsion. M is a Weingarten-type surface if and only if

$$\left(\begin{array}{c}{\xi }_1\left(f_1{\eta }_{2\upsilon }+{\eta }_2{f}_{1\upsilon }\right)\hfill \\ +\left(f_1{\eta }_2-2{\xi }_1\right){\xi }_{1\upsilon }\hfill \end{array}\right)\left(\begin{array}{c}{\zeta }_3{f}_{1s}+f_1{\zeta }_{3s}\hfill \\ +\tau \left(\tau {\zeta }_2{\zeta }_{1s}+{\zeta }_1\left(2{\zeta }_2{\tau }_s+\tau {\zeta }_{2s}\right)\right)\hfill \end{array}\right)=\left(\begin{array}{c}{\zeta }_3{f}_{1\upsilon }+f_1{\zeta }_{3\upsilon }\hfill \\ +{\tau }^2\left({\zeta }_2{\zeta }_{1\upsilon }+{\zeta }_1{\zeta }_{2\upsilon }\right)\hfill \end{array}\right)\left(\begin{array}{c}{\eta }_2{\xi }_1{f}_{1s}\hfill \\ +\left(f_1{\eta }_2-2{\xi }_1\right){\xi }_{1s}\hfill \end{array}\right)$$

where

$$\begin{array}{c}{\xi }_1\left(s,\upsilon \right)={\displaystyle \frac{\left(f_2{g}_1-f_1{g}_2\right)p_{\upsilon }+\left(f_1{g}_3-f_3{g}_1\right)r_{\upsilon }}{{∥{\phi }_s\times {\phi }_{\upsilon }∥}^2{\tau }^2\left({f_1}^2{\eta }_1+{\tau }^2\left(\left({f_2}^2+{f_3}^2\right){\eta }_1-{\left(f_3{p}_{\upsilon }+f_2{r}_{\upsilon }\right)}^2\right)\right)}},\hfill \\ {\xi }_2\left(s,\upsilon \right)={\displaystyle \frac{\left(f_2{p}_{\upsilon }-f_3{r}_{\upsilon }\right)f_{1\upsilon }-f_1\left(p_{\upsilon }{f}_{2\upsilon }-r_{\upsilon }{f}_{3\upsilon }\right)}{∥{\phi }_s\times {\phi }_{\upsilon }∥\tau \sqrt{\left({f_1}^2{\eta }_1+{\tau }^2\left(\left({f_2}^2+{f_3}^2\right){\eta }_1-{\left(f_3{p}_{\upsilon }+f_2{r}_{\upsilon }\right)}^2\right)\right)}}},\hfill \\ {\zeta }_1\left(s,\upsilon \right)={\displaystyle \frac{f_3{r}_{\upsilon }-f_2{p}_{\upsilon }}{2∥{\phi }_s\times {\phi }_{\upsilon }∥{\tau }^2\left({f_1}^2{\eta }_1+{\tau }^2\left({\eta }_1\left({f_2}^2+{f_3}^2\right)-{\left(f_3{p}_{\upsilon }+f_2{r}_{\upsilon }\right)}^2\right)\right)}},\hfill \\ {\zeta }_2\left(s,\upsilon \right)={\displaystyle \frac{2f_{1\upsilon }\left(f_3{p}_{\upsilon }+f_2{r}_{\upsilon }\right)-{\eta }_1{g}_1}{2∥{\phi }_s\times {\phi }_{\upsilon }∥{\tau }^2\left({f_1}^2{\eta }_1+{\tau }^2\left({\eta }_1\left({f_2}^2+{f_3}^2\right)-{\left(f_3{p}_{\upsilon }+f_2{r}_{\upsilon }\right)}^2\right)\right)}},\hfill \\ {\zeta }_3\left(s,\upsilon \right)={\displaystyle \frac{{f_1}^2{\eta }_2+{\tau }^2\left(\left({f_2}^2+{f_3}^2\right){\eta }_2+{\eta }_1\left(+g_3{r}_{\upsilon }-g_2{p}_{\upsilon }\right)+2\left(f_3{p}_{\upsilon }+f_2{r}_{\upsilon }\right)\left(p_{\upsilon }{f}_{2\upsilon }-r_{\upsilon }{f}_{3\upsilon }\right)\right)}{2∥{\phi }_s\times {\phi }_{\upsilon }∥{\tau }^2\left({f_1}^2{\eta }_1+{\tau }^2\left({\eta }_1\left({f_2}^2+{f_3}^2\right)-{\left(f_3{p}_{\upsilon }+f_2{r}_{\upsilon }\right)}^2\right)\right)}}.\hfill \end{array}$$

Proof. 

Let M be a modified sweeping surface generated by the MOF with non-vanishing torsion. By differentiating the curvatures K and H of M in terms of s and $\upsilon$, we obtain

$$\left\{\begin{array}{c}{K}_s={\eta }_2{\xi }_1{f}_{1s}+\left(f_1{\eta }_2-2{\xi }_1\right){\xi }_{1s},\hfill \\ K_{\upsilon }={\xi }_1\left(f_1{\eta }_{2\upsilon }+{\eta }_2{f}_{1\upsilon }\right)+\left(f_1{\eta }_2-2{\xi }_1\right){\xi }_{1\upsilon },\hfill \\ H_s={\zeta }_3{f}_{1s}+\tau \left(\tau {\zeta }_2{\zeta }_{1s}+{\zeta }_1\left(2{\zeta }_2{\tau }_s+\tau {\zeta }_{2s}\right)\right)+f_1{\zeta }_{3s},\hfill \\ H_{\upsilon }={\zeta }_3{f}_{1\upsilon }+{\tau }^2\left({\zeta }_2{\zeta }_{1\upsilon }+{\zeta }_1{\zeta }_{2\upsilon }\right)+f_1{\zeta }_{3\upsilon }.\hfill \end{array}\right.$$

From here, the condition specified in the hypothesis is met if and only if $K_s{H}_{\upsilon }-K_{\upsilon }{H}_s=0$. So, one can say that M generated by the MOF with non-vanishing torsion is a Weingarten-type surface under this condition. □

Theorem 8. 

Let M be a modified sweeping surface generated by the MOF with non-vanishing torsion. Then, the $s-$coordinate curves of M are geodesic if and only if

$${\displaystyle \frac{p_{\upsilon }}{r_{\upsilon }}}={\displaystyle \frac{f_3\left(g_3-g_2\right)+f_1{g}_1}{f_2\left(g_3-g_2\right)-f_1{g}_1}}.$$

Also, the $s-$coordinate curves of M are asymptotic if and only if

$${\displaystyle \frac{p_{\upsilon }}{r_{\upsilon }}}={\displaystyle \frac{g_1{f}_3-{\tau }^2{g}_3{f}_1}{g_1{f}_2-{\tau }^2{g}_2{f}_1}}.$$

Proof. 

Let M be a modified sweeping surface generated by the MOF with non-vanishing torsion. From (11) and the first equality in (12), the following equation is obtained:

$$\begin{array}{c}\Delta \times {\phi }_{ss}=f_1\left(g_3{p}_{\upsilon }+g_2{r}_{\upsilon }\right)T+\left(g_3{f}_2{p}_{\upsilon }-\left(g_1{f}_1+g_3{f}_3\right)r_{\upsilon }\right)N\hfill \\ +\left(g_2{f}_3{r}_{\upsilon }-\left(g_1{f}_1+g_2{f}_2\right)p_{\upsilon }\right)B.\hfill \end{array}$$

By the linear independence of the elements of MOF, $\Delta \times {\phi }_{ss}=0$ requires

$$\begin{array}{c}{f}_1\left(g_3{p}_{\upsilon }+g_2{r}_{\upsilon }\right)=0,\hfill \\ g_3{f}_2{p}_{\upsilon }-\left(g_1{f}_1+g_3{f}_3\right)r_{\upsilon }=0,\hfill \\ g_2{f}_3{r}_{\upsilon }-\left(g_1{f}_1+g_2{f}_2\right)p_{\upsilon }=0.\hfill \end{array}$$

Thus, the criterion in the hypothesis is obtained by these last three equations. One can say that the $s-$coordinate curves of the modified sweeping surface are geodesics under the criterion stated in the hypothesis.

From Equations (11) and (12), we have

$$〈\Delta ,{\phi }_{ss}〉=\left(g_1{f}_2-{\tau }^2{g}_2{f}_1\right)p_{\upsilon }+\left({\tau }^2{g}_3{f}_1-g_1{f}_3\right)r_{\upsilon }.$$

Thus, $\left(g_1{f}_2-{\tau }^2{g}_2{f}_1\right)p_{\upsilon }=\left(g_1{f}_3-{\tau }^2{g}_3{f}_1\right)r_{\upsilon }$ if and only if $〈\Delta ,{\phi }_{ss}〉=0$. Consequently, it is proved that the $s-$coordinate curves of M are asymptotic under the criterion stated in the hypothesis. □

Theorem 9. 

Let M be a modified sweeping surface generated by the MOF with non-vanishing torsion. Then, the $\upsilon -$coordinate curves of M are geodesic if and only if

$$f_2{p}_{\upsilon }-f_3{r}_{\upsilon }=0,p_{\upsilon \upsilon }-r_{\upsilon \upsilon }=0\mathit{and}{f}_1\left(p_{\upsilon }{p}_{\upsilon \upsilon }+r_{\upsilon }{r}_{\upsilon \upsilon }\right)=0.$$

Also, the $\upsilon -$coordinate curves of M are asymptotic if and only if

$$r\kappa \tau =1\mathit{or}{r}_{\upsilon }{p}_{\upsilon \upsilon }-p_{\upsilon }{r}_{\upsilon \upsilon }=0.$$

Proof. 

The vector product of the normal vector $\Delta$ given by (11) and ${\phi }_{\upsilon \upsilon }$ given in the third equation of (12) is found as follows:

$$\Delta \times {\phi }_{\upsilon \upsilon }=f_1\left(p_{\upsilon }{p}_{\upsilon \upsilon }+r_{\upsilon }{r}_{\upsilon \upsilon }\right)T+p_{\upsilon \upsilon }\left(f_2{p}_{\upsilon }-f_3{r}_{\upsilon }\right)N+r_{\upsilon \upsilon }\left(f_3{r}_{\upsilon }-f_2{p}_{\upsilon }\right)B.$$

The condition for $\upsilon -$curve to be geodesic is $\Delta \times {\phi }_{\upsilon \upsilon }=0$, which requires

$$\begin{array}{c}{f}_1\left(p_{\upsilon }{p}_{\upsilon \upsilon }+r_{\upsilon }{r}_{\upsilon \upsilon }\right)=0,\hfill \\ p_{\upsilon \upsilon }\left(f_2{p}_{\upsilon }-f_3{r}_{\upsilon }\right)=0,\hfill \\ r_{\upsilon \upsilon }\left(f_3{r}_{\upsilon }-f_2{p}_{\upsilon }\right)=0.\hfill \end{array}$$

Thus, for $\Delta \times {\phi }_{\upsilon \upsilon }=0$, the criterion in the hypothesis must be verified. It is obvious that the $\upsilon -$coordinate curves of M are geodesics under the criterion stated in the hypothesis. From Equation (11) and the third equality in (12), we have

$$〈\Delta ,{\phi }_{\upsilon \upsilon }〉={\tau }^2{f}_1\left(r_{\upsilon }{p}_{\upsilon \upsilon }-p_{\upsilon }{r}_{\upsilon \upsilon }\right).$$

Then, $f_1=0$ or $r_{\upsilon }{p}_{\upsilon \upsilon }-p_{\upsilon }{r}_{\upsilon \upsilon }=0$ if and only if $〈\Delta ,{\phi }_{\upsilon \upsilon }〉=0$. Since $f_1=1-r\kappa \tau$, it is seen that the $\upsilon -$coordinate curves of M are asymptotic under the criteria stated in the hypothesis. □

Theorem 10. 

Let M be a modified sweeping surface generated by the MOF with non-vanishing torsion τ. Then, the s and $\upsilon -$coordinate curves of M are lines of curvature if and only if

$${\displaystyle \frac{p_{\upsilon }}{r_{\upsilon }}}={\displaystyle \frac{{\tau }^2{f}_1{f}_{3\upsilon }-f_{1\upsilon }{f}_3}{f_{1\upsilon }{f}_2-{\tau }^2{f}_1{f}_{2\upsilon }}}.$$

Proof. 

Let M be a modified sweeping surface generated by the MOF with non-vanishing torsion. Then, $F=l=0$ provided that the coordinate curves of M are lines of curvature. Taking into consideration $\tau \ne 0$ and the equations

$$F={\tau }^2\left(f_3{p}_{\upsilon }+f_2{r}_{\upsilon }\right)\mathrm{and}l={\displaystyle \frac{f_{1\upsilon }\left(f_2{p}_{\upsilon }-f_3{r}_{\upsilon }\right)+{\tau }^2{f}_1\left(f_{3\upsilon }{r}_{\upsilon }-f_{2\upsilon }{p}_{\upsilon }\right)}{∥{\phi }_s\times {\phi }_{\upsilon }∥}},$$

the common solution of $F=0$ and $l=0$ gives us

$${\displaystyle \frac{p_{\upsilon }}{r_{\upsilon }}}={\displaystyle \frac{{\tau }^2{f}_1{f}_{3\upsilon }-f_{1\upsilon }{f}_3}{f_{1\upsilon }{f}_2-{\tau }^2{f}_1{f}_{2\upsilon }}}.$$

The last is the condition for the s and $\upsilon -$coordinate curves of M generated by the MOF with non-vanishing torsion to be the lines of curvature. □

Example 1. 

Let us consider a curve β defined by the parametric equation

$$\beta \left(s\right)=\left(sins,sinscoss,s\right).$$

The Frenet apparatuses of β are found as $\left\{t,n,b,\kappa ,\tau \right\}$ where $s\in \left(2k\pi ,\left(2k+1\right)\pi \right)$ and $\left\{t,-n,-b,-\kappa ,\tau \right\}$ where $s\in \left(\left(2k+1\right)\pi ,\left(2k+2\right)\pi \right)$ for $k\in \mathbb{Z}$, such that

$$\begin{array}{c}t=\left({\displaystyle \frac{\sqrt{2}coss}{\sqrt{4+cos2s+cos4s}}},{\displaystyle \frac{\sqrt{2}cos2s}{\sqrt{4+cos2s+cos4s}}},{\displaystyle \frac{\sqrt{2}}{\sqrt{4+cos2s+cos4s}}}\right),\hfill \\ n=\left(\begin{array}{c}{\displaystyle \frac{\left(4cos2s+cos4s-1\right)}{\sqrt{\left(4+cos2s+cos4s\right)\left(27+24cos2s+cos4s\right)}}},\hfill \\ {\displaystyle \frac{-2coss\left(6+cos2s\right)}{\sqrt{\left(4+cos2s+cos4s\right)\left(27+24cos2s+cos4s\right)}}},\hfill \\ {\displaystyle \frac{2coss\left(1+4cos2s\right)}{\sqrt{\left(4+cos2s+cos4s\right)\left(27+24cos2s+cos4s\right)}}}\hfill \end{array}\right),\hfill \\ b=\left(\begin{array}{c}{\displaystyle \frac{4\sqrt{2}coss}{\sqrt{27+24cos2s+cos4s}}},{\displaystyle \frac{-\sqrt{2}}{\sqrt{27+24cos2s+cos4s}}},{\displaystyle \frac{-4-2cos2s}{\sqrt{27+24cos2s+cos4s}}}\hfill \end{array}\right),\hfill \end{array}$$

and

$$\kappa ={\displaystyle \frac{2sins\sqrt{27+24cos2s+cos4s}}{{\left(4+cos2s+cos4s\right)}^{3/2}}},\tau =-{\displaystyle \frac{8sins}{27+24cos2s+cos4s}}.$$

Moreover, at $s=k\pi$ for each $k\in \mathbb{Z}$, the curvature and torsion of β are zero. In addition, the Frenet frame cannot be constituted at these points since normal and binormal vectors are discontinuous, and it is impossible to refer to the Frenet frame in a unique way. So, this problem is solved via the MOF. Here, we express the elements of the MOF with non-zero curvatures for all s as follows:

$$\begin{array}{c}T=\left({\displaystyle \frac{\sqrt{2}coss}{\sqrt{4+cos2s+cos4s}}},{\displaystyle \frac{\sqrt{2}cos2s}{\sqrt{4+cos2s+cos4s}}},{\displaystyle \frac{\sqrt{2}}{\sqrt{4+cos2s+cos4s}}}\right),\hfill \\ N=\left({\displaystyle \frac{2sins\left(4cos2s+cos4s-1\right)}{{\left(4+cos2s+cos4s\right)}^2}},{\displaystyle \frac{-2sin2s\left(6+cos2s\right)}{{\left(4+cos2s+cos4s\right)}^2}},{\displaystyle \frac{2\left(sin2s+2sin4s\right)}{{\left(4+cos2s+cos4s\right)}^2}}\right),\hfill \\ B=\left({\displaystyle \frac{4\sqrt{2}sin2s}{{\left(4+cos2s+cos4s\right)}^{3/2}}},{\displaystyle \frac{-2\sqrt{2}sins}{{\left(4+cos2s+cos4s\right)}^{3/2}}},{\displaystyle \frac{-\sqrt{2}\left(3sins+sin3s\right)}{{\left(4+cos2s+cos4s\right)}^{3/2}}}\right).\hfill \end{array}$$

If we take the planar profile (cross-section) curve $\delta \left(\upsilon \right)=\left(0,cos\upsilon ,sin\upsilon \right)$, the equation of the modified sweeping surface (see Figure 1) generated by the MOF with non-vanishing curvature κ is represented by

$$\phi \left(s,\upsilon \right)=\left(\begin{array}{c}sins+{\displaystyle \frac{2sinscos\upsilon \left(4cos2s+cos4s-1\right)}{{\left(4+cos2s+cos4s\right)}^2}}+{\displaystyle \frac{4\sqrt{2}sin2ssin\upsilon }{{\left(4+cos2s+cos4s\right)}^{3/2}}},\hfill \\ sinscoss+-{\displaystyle \frac{2sin2scos\upsilon \left(6+cos2s\right)}{{\left(4+cos2s+cos4s\right)}^2}}-{\displaystyle \frac{2\sqrt{2}sinssin\upsilon }{{\left(4+cos2s+cos4s\right)}^{3/2}}},\hfill \\ s+{\displaystyle \frac{2cos\upsilon \left(sin2s+2sin4s\right)}{{\left(4+cos2s+cos4s\right)}^2}}-{\displaystyle \frac{\sqrt{2}\left(3sins+sin3s\right)sin\upsilon }{{\left(4+cos2s+cos4s\right)}^{3/2}}}\hfill \end{array}\right).$$

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Example 2. 

Let us consider the Euler (Cornu) spiral parameterized by

$$\beta \left(s\right)=\left({\displaystyle \frac{1}{\sqrt{2}}}\underset{0}{\overset{s}{\int }}cos{\displaystyle \frac{\pi t^2}{2}}dt,{\displaystyle \frac{1}{\sqrt{2}}}\underset{0}{\overset{s}{\int }}sin{\displaystyle \frac{\pi t^2}{2}}dt,{\displaystyle \frac{s}{\sqrt{2}}}\right),$$

where the components $\underset{0}{\overset{s}{\int }}cos{\displaystyle \frac{\pi t^2}{2}}dt$ and $\underset{0}{\overset{s}{\int }}sin{\displaystyle \frac{\pi t^2}{2}}dt$ of this curve are known as Fresnel integrals [21]. The Frenet apparatus of β are found as $\left\{t,n,b,\kappa ,\tau \right\}$ for $s\in {\mathbb{R}}^{+}$ and $\left\{t,-n,-b,-\kappa ,\tau \right\}$ for $s\in {\mathbb{R}}^{-}$ such that

$$\begin{array}{c}t=\left({\displaystyle \frac{1}{\sqrt{2}}}cos{\displaystyle \frac{\pi s^2}{2}},{\displaystyle \frac{1}{\sqrt{2}}}sin{\displaystyle \frac{\pi s^2}{2}},{\displaystyle \frac{1}{\sqrt{2}}}\right),n=\left(-sin{\displaystyle \frac{\pi s^2}{2}},cos{\displaystyle \frac{\pi s^2}{2}},0\right),\hfill \\ b=\left(-{\displaystyle \frac{1}{\sqrt{2}}}cos{\displaystyle \frac{\pi s^2}{2}},-{\displaystyle \frac{1}{\sqrt{2}}}sin{\displaystyle \frac{\pi s^2}{2}},{\displaystyle \frac{1}{\sqrt{2}}}\right),\kappa ={\displaystyle \frac{\pi s}{\sqrt{2}}},\tau ={\displaystyle \frac{\pi s}{\sqrt{2}}}.\hfill \end{array}$$

Here, the Frenet frame does not occur at $s=0$, since the second derivative of $\beta \left(s\right)$ is zero. To fix this problem, we can refer to any MOF. The elements of MOF with non-zero torsion are obtained as

$$\begin{array}{c}T=\left({\displaystyle \frac{1}{\sqrt{2}}}cos{\displaystyle \frac{\pi s^2}{2}},{\displaystyle \frac{1}{\sqrt{2}}}sin{\displaystyle \frac{\pi s^2}{2}},{\displaystyle \frac{1}{\sqrt{2}}}\right),\hfill \\ N=\left(-{\displaystyle \frac{\pi s}{\sqrt{2}}}sin{\displaystyle \frac{\pi s^2}{2}},{\displaystyle \frac{\pi s}{\sqrt{2}}}cos{\displaystyle \frac{\pi s^2}{2}},0\right),\hfill \\ B=\left(-{\displaystyle \frac{\pi s}{2}}cos{\displaystyle \frac{\pi s^2}{2}},-{\displaystyle \frac{\pi s}{2}}sin{\displaystyle \frac{\pi s^2}{2}},{\displaystyle \frac{\pi s}{2}}\right).\hfill \end{array}$$

By taking spine curve β and the planar profile curve $\delta \left(\upsilon \right)=\left(0,cos\upsilon ,sin\upsilon \right)$, a modified sweeping surface (see Figure 2) generated by the MOF with non-vanishing torsion is given with the following equation:

$$\phi \left(s,\upsilon \right)=\left(\begin{array}{c}{\displaystyle \frac{1}{\sqrt{2}}}\underset{0}{\overset{s}{\int }}cos{\displaystyle \frac{\pi t^2}{2}}dt-{\displaystyle \frac{\pi scos\upsilon }{\sqrt{2}}}sin{\displaystyle \frac{\pi s^2}{2}}-{\displaystyle \frac{\pi ssin\upsilon }{2}}cos{\displaystyle \frac{\pi s^2}{2}},\hfill \\ {\displaystyle \frac{1}{\sqrt{2}}}\underset{0}{\overset{s}{\int }}sin{\displaystyle \frac{\pi t^2}{2}}dt+{\displaystyle \frac{\pi scos\upsilon }{\sqrt{2}}}cos{\displaystyle \frac{\pi s^2}{2}}-{\displaystyle \frac{\pi ssin\upsilon }{2}}sin{\displaystyle \frac{\pi s^2}{2}},\hfill \\ {\displaystyle \frac{s}{\sqrt{2}}}+{\displaystyle \frac{\pi ssin\upsilon }{2}}\hfill \end{array}\right).$$

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4. Conclusions

This study explores the application of modified orthogonal frames with non-vanishing curvature and non-vanishing torsion, enabling the generation of two types of modified sweeping surfaces, even when the second derivative of the spine curve is zero. The research involves the following processes:

• Deriving criteria for each type of modified sweeping surface with non-vanishing curvature and non-vanishing torsion to be minimal, developable, or Weingarten surfaces.
• Conducting a comprehensive analysis of the coordinate curves of these modified sweeping surfaces to determine criteria for geodesic, asymptotic, and curvature lines.
• Providing examples of the modified sweeping surfaces along with illustrated graphics.