Developable Ruled Surfaces with Constant Mean Curvature Along a Curve

Abstract

In this work, we study developable ruled surfaces with constant mean curvature along a curve. The mean curvature of developable ruled surfaces generated by indicatrix curves is calculated. The analysis is first carried out in Euclidean three-space and then extended to Lorentz space. For both geometries, we derive the necessary and sufficient conditions under which the developable ruled surfaces exhibit constant mean curvature. In addition, we calculate the mean curvature of the surface using time-like and space-like curves. Later, we give a sufficient condition for the mean curvature of a developable surface to be constant along a striction curve. Finally, we give some examples in Euclidean and Lorentz spaces and present computational examples.

1. Introduction

A developable surface is a surface whose tangent planes are the same along the principal lines of a ruled surface. In other words, a ruled surface $M_{(\alpha ,\gamma )}:I\times P\to E_1^3$ in Lorentz space $E_1^3$ is defined as

$$M_{(\alpha ,\gamma )}(s,u)=\alpha \left(s\right)+u\gamma \left(s\right),$$

where $\alpha :I\to E_1^3$ is called the base curve and $\gamma :I\to E_1^3\setminus \left\{0\right\}$ is referred to as the director curve. The straight lines $u\to \alpha \left(s\right)+u\gamma \left(s\right)$ are the rulings. If the Gaussian curvature of the ruled surface $M_{(\alpha ,\gamma )}$ is zero, then it is called a developable ruled surface, see, e.g., [1]. The ruled surface $M_{(\alpha ,\gamma )}$ is developable if and only if the distribution parameter

$$P_{\gamma }={\displaystyle \frac{det({\alpha }^{\prime },\gamma ,{\gamma }^{\prime })}{⟨{\gamma }^{\prime },{\gamma }^{\prime }⟩}}=0.$$

In [2,3], the authors gave the necessary and sufficient conditions for a ruled surface to be developable in Minkowski space and showed that it is developable if and only if its base curve is a helix. In [4], the mean curvature of the principal normal surface along Bertrand curves was examined. The geometric properties of minimal surfaces and their applications in civil engineering and architecture are discussed in [5]. In [6], the minimality conditions for ruled surfaces with a Legendre base curve were investigated. In [7], constant-mean-curvature surfaces were obtained along a given space-like curve in three-dimensional Lorentz space. In [8], sufficient conditions were presented for determining surfaces in three-dimensional Euclidean space that pass through an arbitrary curve and possess constant mean curvature along that curve. In [9], constant-angle ruled surfaces in Euclidean three-space were studied, and it was shown that they are developable. In this paper, we investigate developable surfaces whose mean curvature is constant along the base curve. First, these surfaces are considered in Euclidean space and then in Lorentz space.

One of the important research areas in differential geometry and physics is Lorentz space, which provides a natural geometric framework for describing the interaction between space and time in the theory of special relativity [10]. In Lorentz space, the relationship between time and space dimensions is based on the principle of the constancy of the speed of light predicted in special relativity. Thus, Lorentz spaces are used to model different physical processes in the theory of special relativity, see, e.g., [11,12,13,14,15].

Various studies on ruled surfaces have also appeared in architectural applications, see, e.g., [16]. An important property of developable surfaces is that the envelope of reflected rays also forms a developable surface. The same phenomenon occurs in the case of fractures. Moreover, reflections and refractions generate focal curves in planar settings [16].

We consider the following developable ruled surface:

$$\Phi (u,s)=\underset{0}{\overset{s}{\int }}\left[f\left(s\right)\gamma \left(s\right)+g\left(s\right){\gamma }^{\prime }\left(s\right)\right]ds+u\gamma \left(s\right),$$

(1)

where $f\left(s\right)$ and $g\left(s\right)$ are differentiable functions, which was discussed in [17]. They proved that ruled surface (1) is developable in three-dimensional Euclidean space when $\alpha$ is a small circle on the unit sphere $S^2.$ We give the condition for ruled surface (1) to have constant mean curvature along a time-like or space-like $\alpha$ curve in Lorentz space. The condition that a surface exhibits constant mean curvature along a curve implies uniform bending behavior in the direction of that curve. If H remains constant along a given curve on a surface, the local geometric behavior of the surface does not change in that direction. This property plays an important role in architectural and structural design, particularly in the context of developable or discretely panelized surfaces. Structures designed with constant-mean-curvature zones often benefit from predictable stress distribution and fabrication efficiency [18]. Therefore, this geometric constraint not only carries mathematical significance but also offers practical advantages in the fields of architectural geometry and computational design.

In this work, we calculate the mean curvature of the surface by using both time-like and space-like curves. When principal normal vector N is time-like, then tangent vector T and binormal vector B are space-like. If N is a space-like vector, then T and B are selected as time-like vectors. We focus on determination of surfaces that have constant mean curvature in Lorentz space. Additionally, we examine some special cases according to the based curves on the ruled surfaces. At the end of this study, some computational examples are given for Euclidean space and Lorentz space.

The rest of this study consists of six sections. Some basic definitions are presented in Section 2. The mean curvature calculations for Euclidean space are given in Section 3. Moreover, we focus on the mean curvature of developable surfaces constructed using indicatrix curves in Euclidean space. In Section 4, the mean-curvature calculations for Lorentz space are presented. In addition, we deal with the mean curvature of developable surfaces constructed using indicatrix curves in Lorentz space. In Section 5, special cases derived from the calculations in Section 4 are discussed. In the last section, computational examples are presented and some final remarks are given.

2. Preliminaries

In this section, we introduce some basic concepts in Euclidean space $E^3$ and Lorentz space $E_1^3$ that will be used throughout this work. For a detailed discussion on the related subjects, we refer to, e.g., [1,19,20].

Let $\alpha =\alpha \left(s\right):I⟶E^3$ be an arbitrary curve in $E^3.$ The curve $\alpha$ is said to be a unit speed if $⟨{\alpha }^{\prime }\left(s\right),{\alpha }^{\prime }\left(s\right)⟩=1$ for any $s\in I.$ Assume that $\left\{T\left(s\right),N\left(s\right),B\left(s\right)\right\}$ is the moving frame of curve $\alpha$, which satisfies the Frenet equations:

$${\displaystyle \frac{d}{ds}}\left(\begin{array}{c}T\left(s\right)\\ N\left(s\right)\\ B\left(s\right)\end{array}\right)=\left(\begin{array}{ccc}0& \kappa \left(s\right)& 0\\ -\kappa \left(s\right)& 0& \tau \left(s\right)\\ 0& -\tau \left(s\right)& 0\end{array}\right)\left(\begin{array}{c}T\left(s\right)\\ N\left(s\right)\\ B\left(s\right)\end{array}\right),$$

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

The Lorentz space is the metric space $E_1^3=\left({\mathbb{R}}^3,⟨,⟩\right)$, where the metric $⟨,⟩$ is given as

$$⟨u,v⟩=u_1{v}_1+u_2{v}_2-u_3{v}_3,u=(u_1,u_2,u_3),v=(v_1,v_2,v_3)$$

and is called Lorentzian metric. Moreover, the vector product of any vectors $u=(u_1,u_2,u_3)$ and $v=(v_1,v_2,v_3)$ in $E_1^3$ is defined by

$$u\times v=\left(\begin{array}{ccc}i& j& -k\\ u_1& u_2& u_3\\ v_1& v_2& v_3\end{array}\right).$$

Let $\beta (s,u)$ be a ruled surface with associated normal vector $N:$

$$N={\displaystyle \frac{{\beta }_u\times {\beta }_s}{\Vert {\beta }_u\times {\beta }_s\Vert }},$$

where ${\beta }_u={\displaystyle \frac{\partial \beta }{\partial u}}$ and ${\beta }_s={\displaystyle \frac{\partial \beta }{\partial s}}.$ Also, expression of the first fundamental form is given below:

$$I=Ed{s}^2+2Fdsdu+Gd{u}^2$$

for the coefficients $E=⟨{\beta }_u,{\beta }_u⟩,F=⟨{\beta }_u,{\beta }_s⟩,G=⟨{\beta }_s,{\beta }_s⟩.$ And the second fundamental form is written as

$$II=ed{s}^2+2fdsdu+gd{u}^2$$

for the coefficients $e=⟨N,{\beta }_{uu}⟩,f=⟨N,{\beta }_{us}⟩,g=⟨N,{\beta }_{ss}⟩.$ Therefore, the mean curvature H and the Gauss curvature K of the surface are defined as follows:

$$\begin{array}{c}\hfill H=ϵ{\displaystyle \frac{1}{2}}{\displaystyle \frac{eG-2fF+gE}{EG-F^2}},K=ϵ{\displaystyle \frac{eg-f^2}{EG-F^2}},\end{array}$$

(2)

where we use the notation $ϵ:=⟨N,N⟩$.

Definition 1. 

Let $E_1^3$ be a Lorentz space, and let $x\in E_1^3$ be a vector. Then:

(i) 

A vector x is said to be space-like if $⟨x,x⟩>0$ or $x=0;$

(ii) 

A vector x is said to be time-like if $⟨x,x⟩<0$;

(iii) 

A vector x is said to be light-like (null) if $⟨x,x⟩=0,$ $x\ne 0$, see, e.g., [10].

If the curve is time-like or space-like, the Frenet formulas are defined as follows:

$$\left(\begin{array}{c}{T}^{\prime }\\ N^{\prime }\\ B^{\prime }\end{array}\right)=\left(\begin{array}{ccc}0& \kappa & 0\\ {\epsilon }_1\kappa & 0& \tau \\ 0& {\epsilon }_2\tau & 0\end{array}\right)\left(\begin{array}{c}T\\ N\\ B\end{array}\right),$$

where $\tau$ is the torsion function, and $\kappa$ is the curvature function. If $\alpha$ is a time-like Frenet curve in $E_1^3$ for ${\epsilon }_1=1$, ${\epsilon }_2=-1$, then Frenet formulas of $\alpha$ are given by:

$$T^{\prime }=\kappa N,N^{\prime }=\kappa T+\tau B,B^{\prime }=-\tau N.$$

For this trihedron, we write

$$T\times N=B,N\times B=-T,B\times T=N.$$

The torsion of the curve $\alpha$ is defined as $\tau =⟨N^{\prime },B⟩$, and the curvature is given by $\kappa =∥T^{\prime }\left(s\right)∥.$

Let $\alpha$ is a space-like Frenet curve in $E_1^3.$ If $T^{\prime }\left(s\right)$ is a space-like vector, then $B\left(s\right)$ is a time-like vector and $N\left(s\right)$ is a space-like vector for ${\epsilon }_1=-1,$ ${\epsilon }_2=1$. Thus, the following Frenet formulas of $\alpha$ are satisfied:

$$T^{\prime }=\kappa N,N^{\prime }=-\kappa T+\tau B,B^{\prime }=\tau N.$$

For this trihedron, we write

$$T\times N=B,N\times B=-T,B\times T=-N.$$

The torsion of the curve $\alpha$ is defined as $\tau =-⟨N^{\prime },B⟩$, and the curvature is given by $\kappa =∥T^{\prime }\left(s\right)∥.$

If $T^{\prime }\left(s\right)$ is a time-like vector, then $B\left(s\right)$ is a space-like vector and $N\left(s\right)$ is a time-like vector for ${\epsilon }_1=1,$ ${\epsilon }_2=1$. Thus, the following Frenet formulas of $\alpha$ are satisfied:

$$T^{\prime }=\kappa N,N^{\prime }=\kappa T+\tau B,B^{\prime }=\tau N.$$

For this trihedron, we have

$$T\times N=B,N\times B=T,B\times T=-N.$$

The torsion of the curve $\alpha$ is defined as $\tau =⟨N^{\prime },B⟩$, and the curvature is given by $\kappa =∥T^{\prime }\left(s\right)∥=\sqrt{-⟨T^{\prime }\left(s\right),T^{\prime }\left(s\right)⟩}.$

Definition 2. 

In [21], the striction curve of the ruled surface $M_{(\alpha ,\gamma )}(s,u)=\alpha \left(s\right)+u\gamma \left(s\right)$ is defined by

$$\tilde{\alpha }\left(s\right)=\alpha \left(s\right)-{\displaystyle \frac{⟨{\alpha }^{\prime }\left(s\right),{\gamma }^{\prime }\left(s\right)⟩}{⟨{\gamma }^{\prime }\left(s\right),{\gamma }^{\prime }\left(s\right)⟩}}\gamma \left(s\right).$$

3. Calculation of Mean Curvature in Euclidean Space

In this section, we investigate the condition under which ruled surface (1) has a constant mean curvature along a general curve in Euclidean space.

Theorem 1. 

Let $\gamma :I\subset \mathbb{R}\to S^2$ be a smooth spherical curve with its Darboux frame

$$\left(\begin{array}{c}{\gamma }^{\prime }\\ T^{\prime }\\ S^{\prime }\end{array}\right)=\left(\begin{array}{ccc}0& m& 0\\ -m& 0& n\\ 0& -n& 0\end{array}\right)\left(\begin{array}{c}\gamma \\ T\\ S\end{array}\right).$$

If $u_0={\displaystyle \frac{n-cmg}{mc}},$ then the mean curvature of the ruled surface $\Phi (u,s)$ is constant along the curve $\Phi (u_0,s)$. Moreover, if m and n are constant, the surface $\Phi (u,s)$ becomes a constant-angle surface, and, along the curve $\Phi (u_0,s)$, the mean curvature H remains constant on this surface.

Proof. 

By taking the derivatives of $\Phi$ with respect to u and s, we get

$$\begin{array}{c}\hfill {\Phi }_u=\gamma \left(s\right),{\Phi }_s=f\left(s\right)\gamma \left(s\right)+g\left(s\right){\gamma }^{\prime }\left(s\right)+u{\gamma }^{\prime }\left(s\right)=f\left(s\right)\gamma \left(s\right)+{\gamma }^{\prime }\left(s\right)(g\left(s\right)+u),\end{array}$$

then the vector product can be written as

$${\Phi }_u\times {\Phi }_s=m(g\left(s\right)+u)S.$$

The coefficients of the first fundamental form can be calculated as follows:

$$\begin{array}{}\mathrm{(3)}& \hfill E& =& ⟨{\Phi }_u,{\Phi }_u⟩=⟨\gamma \left(s\right),\gamma \left(s\right)⟩=1,\hfill \mathrm{(4)}& \hfill F& =& ⟨{\Phi }_u,{\Phi }_s⟩=f\left(s\right),\hfill \mathrm{(5)}& \hfill G& =& ⟨{\Phi }_s,{\Phi }_s⟩=f^2\left(s\right)+{(g\left(s\right)+u)}^2{m}^2.\hfill \end{array}$$

Also, the normal vector N is equal to S. To obtain the coefficients of the second fundamental form, the derivatives of ${\Phi }_u$ with respect to u and s and the derivative of ${\Phi }_s$ with respect to s are calculated as follows:

$$\begin{array}{ccc}\hfill {\Phi }_{uu}& =& 0,{\Phi }_{us}=mT\left(s\right),\hfill \\ \hfill {\Phi }_{ss}& =& (f^{\prime }\left(s\right)-m^{2}g\left(s\right)-m^{2}u)\gamma \left(s\right)+(m^{\prime }g\left(s\right)+m^{\prime }u+mf\left(s\right)+m{g}^{\prime }\left(s\right))T\left(s\right)\hfill \\ & & +\left(mng\right(s)+mnu)S\left(s\right).\hfill \end{array}$$

Therefore, the coefficients of the second fundamental form are obtained below:

$$\begin{array}{}\mathrm{(6)}& \hfill e& =& ⟨N,{\Phi }_{uu}⟩=0,\hfill \mathrm{(7)}& \hfill f& =& ⟨N,{\Phi }_{us}⟩=⟨S\left(s\right),mT\left(s\right)⟩=0,\hfill \mathrm{(8)}& \hfill g& =& ⟨N,{\Phi }_{ss}⟩=mn(g\left(s\right)+u).\hfill \end{array}$$

Using (3)–(5) and (6)–(8) in (2), we have

$$H(u,s)={\displaystyle \frac{n}{2\left(g\right(s)+u)m}}.$$

The value of $u_0={\displaystyle \frac{n-cmg}{mc}}$ is obtained from the equality $H={\displaystyle \frac{c}{2}}$, where c is a constant. Hence along the curve $\Phi (u_0,s)=\underset{0}{\overset{s}{\int }}\left[f\left(s\right)\gamma \left(s\right)+g\left(s\right){\gamma }^{\prime }\left(s\right)\right]ds+\left({\displaystyle \frac{n-cmg}{mc}}\right)\gamma \left(s\right),$ the mean curvature H is constant. □

As a special case for $u=0,$ we obtain the following corollary:

Corollary 1. 

Let $\gamma :I\subset \mathbb{R}\to S^2$ be a smooth curve. If $g={\displaystyle \frac{n}{mc}}$, then the mean curvature H of the ruled surface $\Phi (u,s)$ is constant along the base curve $\underset{0}{\overset{s}{\int }}\left[f\left(s\right)\gamma \left(s\right)+\left({\displaystyle \frac{n}{mc}}\right){\gamma }^{\prime }\left(s\right)\right]ds$. In this case, curve γ is a circle.

On the other hand, we investigate the condition under which ruled surface (1) has constant mean curvature along a striction curve. The striction curve of this surface is defined by

$$\tilde{\gamma }=\underset{0}{\overset{s}{\int }}\left[f\left(s\right)\gamma \left(s\right)+g\left(s\right){\gamma }^{\prime }\left(s\right)\right]ds-{\displaystyle \frac{⟨T\left(s\right),{\gamma }^{\prime }\left(s\right)⟩}{⟨{\gamma }^{\prime }\left(s\right),{\gamma }^{\prime }\left(s\right)⟩}}\gamma \left(s\right),$$

where

$$T\left(s\right)={\displaystyle \frac{f\left(s\right)\gamma \left(s\right)+g\left(s\right){\gamma }^{\prime }\left(s\right)}{\sqrt{f^2+g^2}}}$$

and

$$-{\displaystyle \frac{⟨T\left(s\right),{\gamma }^{\prime }\left(s\right)⟩}{⟨{\gamma }^{\prime }\left(s\right),{\gamma }^{\prime }\left(s\right)⟩}}=-{\displaystyle \frac{⟨f\left(s\right)\gamma \left(s\right)+g\left(s\right){\gamma }^{\prime }\left(s\right),{\gamma }^{\prime }\left(s\right)⟩}{\sqrt{f^2+g^2}⟨{\gamma }^{\prime }\left(s\right),{\gamma }^{\prime }\left(s\right)⟩}}=-{\displaystyle \frac{g}{m\sqrt{f^2+g^2}}}.$$

Then we have the following result:

Corollary 2. 

Let $\gamma :I\subset \mathbb{R}\to S^2$ be a smooth curve. The mean curvature of the ruled surface $\Phi (u,s)$ remains constant for

$$u_0={\displaystyle \frac{n-cmg}{mc}}=-{\displaystyle \frac{g}{m\sqrt{f^2+g^2}}}$$

along the striction curve $\tilde{\gamma }.$ In particular, if $g=f$, then we obtain the following condition under which the surface has constant mean curvature along a striction curve:

$$g={\displaystyle \frac{1}{cm}}\left(n+{\displaystyle \frac{c}{\sqrt{2}}}\right).$$

Mean Curvature for Developable Surfaces Constructed Using Indicatrix Curves

Let $\alpha :I\subset \mathbb{R}\to {\mathbb{R}}^3$ be a smooth curve, and let $\left\{T,N,B\right\}$ be the Frenet frame of $\alpha$. In this section, for the developable surface $\Phi$, we replace the directrix $\gamma$ by the indicatrix curves $T,B$ of the curve $\alpha$. The corresponding ruled surfaces are denoted by ${\beta }_{T,N}$ and ${\beta }_{B,N}$. We then compute the mean curvature of these surfaces, defined by

$${\beta }_{T,N}(u,s)=\int (fT\left(s\right)+gN\left(s\right))ds+uT\left(s\right),$$

$${\beta }_{B,N}(u,s)=\int (fB\left(s\right)+gN\left(s\right))ds+uB\left(s\right).$$

Finally, we determine the conditions under which the mean curvature remains constant.

Theorem 2. 

The mean curvature of the ruled surface ${\beta }_{T,N}(u,s)$ is constant along the curve ${\beta }_{T,N}(u_0,s)$ if $u_0={\displaystyle \frac{\tau -gc}{\kappa c}}$, where κ is the curvature, τ is the torsion, c is a constant, and g is a differentiable function.

Proof. 

By taking the derivatives of ${\beta }_{T,N}$ with respect to u and s, we have

$$\begin{array}{c}\hfill {\beta }_u=T\left(s\right),{\beta }_s=fT\left(s\right)+(g+u\kappa )N\left(s\right),\end{array}$$

then the vector product can be written as

$${\beta }_u\times {\beta }_s=(g+u\kappa )B\left(s\right).$$

The coefficients of the first fundamental form can be calculated as follows:

$$\begin{array}{}\mathrm{(9)}& \hfill E& =& ⟨{\beta }_u,{\beta }_u⟩=⟨T\left(s\right),T\left(s\right)⟩=1,\hfill \mathrm{(10)}& \hfill F& =& ⟨{\beta }_u,{\beta }_s⟩=f,\hfill \mathrm{(11)}& \hfill G& =& ⟨{\beta }_s,{\beta }_s⟩=f^2+{(g+u\kappa )}^2.\hfill \end{array}$$

Also, the normal vector N is equal to $B\left(s\right)$. To obtain the coefficients of the second fundamental form, the derivatives of ${\beta }_u$ with respect to u and s and the derivative of ${\beta }_s$ with respect to s are calculated as follows:

$$\begin{array}{ccc}\hfill {\beta }_{uu}& =& 0,{\beta }_{us}=\kappa N\left(s\right),\hfill \\ \hfill {\beta }_{ss}& =& (f^{\prime }-g\kappa -u{\kappa }^2)T\left(s\right)+(g^{\prime }+f\kappa +u{\kappa }^{\prime })N\left(s\right)+(g\tau +u\kappa \tau )B\left(s\right).\hfill \end{array}$$

Therefore, the coefficients of the second fundamental form are given below:

$$\begin{array}{}\mathrm{(12)}& \hfill e& =& ⟨N,{\beta }_{uu}⟩=0,\hfill \mathrm{(13)}& \hfill f& =& ⟨N,{\beta }_{us}⟩=⟨B\left(s\right),\kappa N\left(s\right)⟩=0,\hfill \mathrm{(14)}& \hfill g& =& ⟨N,{\beta }_{ss}⟩=(g+u\kappa )\tau .\hfill \end{array}$$

Using (9)–(11) and (12)–(14) in (2), we have $H={\displaystyle \frac{\tau }{2(g+u\kappa )}}$. From the equality $H={\displaystyle \frac{c}{2}}$, we obtain $u_0={\displaystyle \frac{\tau -gc}{\kappa c}}$, where c is a constant. Hence, along the curve ${\beta }_{T,N}(u_0,s)=\int (fT\left(s\right)+gN\left(s\right))ds+\left({\displaystyle \frac{\tau -gc}{\kappa c}}\right)T\left(s\right)$, the mean curvature H of the ruled surface ${\beta }_{T,N}$ is constant. If $T\left(s\right)$ is a circle, that is, the curve $\alpha$ is a helix, then the surface becomes a constant-angle surface. □

Corollary 3. 

As a special case for $u=0,$ the mean curvature H of the ruled surface ${\beta }_{T,N}$ is constant along the base curve $\int (fT\left(s\right)+\left({\displaystyle \frac{\tau }{c}}\right)N\left(s\right))ds.$

Theorem 3. 

The mean curvature of the ruled surface ${\beta }_{B,N}(u,s)$ is constant along the curve ${\beta }_{B,N}(u_0,s)$ for $u_0={\displaystyle \frac{cg+\kappa }{\tau c}}.$

Proof. 

Following the same procedure as in the proof of Theorem 2, we compute the partial derivatives of ${\beta }_{B,N}$ and, using the Frenet–Serret formulas, obtain

$${\beta }_u=B\left(s\right),{\beta }_s=fB\left(s\right)+gN\left(s\right)-u\tau N\left(s\right),$$

and hence

$${\beta }_u\times {\beta }_s=(u\tau -g)T\left(s\right).$$

The coefficients of the first fundamental form are given by

$$E=1,F=f,G=f^2+{(g-u\tau )}^2,$$

while the unit normal vector field reduces to $N=T\left(s\right)$. Furthermore, the coefficients of the second fundamental form satisfy

$$e=0,f=0,g=\kappa (u\tau -g).$$

Consequently, the mean curvature takes the form $H=-{\displaystyle \frac{1}{2}}{\displaystyle \frac{\kappa }{g-u\tau }}$. Imposing the condition that H is constant and equal to $c/2$, we obtain $u_0={\displaystyle \frac{cg+\kappa }{\tau c}}$. Therefore, the mean curvature of the ruled surface ${\beta }_{B,N}$ is constant along the curve $\int \left(fB\left(s\right)+gN\left(s\right)\right)ds+\left({\displaystyle \frac{cg+\kappa }{\tau c}}\right)B\left(s\right).$ If $B\left(s\right)$ is a circle, then the curve $\alpha$ is a helix, and the surface is a constant-angle surface. □

Corollary 4. 

For $u=0,$ the mean curvature H of the ruled surface ${\beta }_{B,N}$ is constant along the base curve $\int (fB\left(s\right)+\left({\displaystyle \frac{-\kappa }{c}}\right)N\left(s\right))ds$.

4. Mean Curvature for Developable Surfaces in Lorentz Space

In this section, we derive the conditions under which the mean curvature remains constant for the developable ruled surface $\Phi (u,s)$ in Lorentz space. For this aim, we consider two different cases.

Case 1. Let the vector $\gamma \left(s\right)$ be time-like and the vector ${\gamma }^{\prime }\left(s\right)$ be space-like with the Frenet frame. Thus, the condition for the mean curvature of the ruled surface $\Phi$ to be constant is the same as that obtained in Section 3.

Case 2. Let the vector $\gamma \left(s\right)$ be space-like and the vector ${\gamma }^{\prime }\left(s\right)$ be time-like with the Darboux frame:

$$\left(\begin{array}{c}{\gamma }^{\prime }\\ T^{\prime }\\ S^{\prime }\end{array}\right)=\left(\begin{array}{ccc}0& m& 0\\ m& 0& n\\ 0& n& 0\end{array}\right)\left(\begin{array}{c}\gamma \\ T\\ S\end{array}\right).$$

Then the mean curvature of the ruled surface $\Phi (u,s)$ is constant along the curve $\underset{0}{\overset{s}{\int }}\left[f\left(s\right)\gamma \left(s\right)+g\left(s\right){\gamma }^{\prime }\left(s\right)\right]ds-\left({\displaystyle \frac{n+cmg}{mc}}\right)\gamma \left(s\right)$. If $u=0$, we see that H is constant along the base curve $\underset{0}{\overset{s}{\int }}\left[f\left(s\right)\gamma \left(s\right)+\left({\displaystyle \frac{-n}{mc}}\right){\gamma }^{\prime }\left(s\right)\right]ds$.

Mean Curvature for Developable Surfaces Constructed Using Indicatrix Curves in Lorentz Space

In this section, we calculate the mean curvature for the surfaces ${\beta }_{T,N}$ and ${\beta }_{B,N}$, which are given in Section 3 for Lorentz space.

First, we consider the following different cases for the developable ruled surface ${\beta }_{T,N}$:

Case 1. $\alpha :I\to E_1^3$ is a smooth space-like curve. If the principal normal vector $N\left(s\right)$ is time-like and the tangent vector $T\left(s\right)$ is space-like, then we have the following Frenet–Serret-type formula:

$$\left(\begin{array}{c}{T}^{\prime }\\ N^{\prime }\\ B^{\prime }\end{array}\right)=\left(\begin{array}{ccc}0& \kappa & 0\\ \kappa & 0& \tau \\ 0& \tau & 0\end{array}\right)\left(\begin{array}{c}T\\ N\\ B\end{array}\right).$$

Then the mean curvature of the surface ${\beta }_{T,N}$ is constant along the curve

$$\int \left(fT\left(s\right)+gN\left(s\right)\right)ds-\left({\displaystyle \frac{\tau +gc}{\kappa c}}\right)T\left(s\right).$$

In particular, H is constant along the base curve $\int \left(fT\left(s\right)-{\displaystyle \frac{\tau }{c}}N\left(s\right)\right)ds$ for $u=0$.

Case 2. $\alpha :I\to E_1^3$ is a smooth time-like curve. If the principal normal vector $N\left(s\right)$ is space-like and the tangent vector $T\left(s\right)$ is time-like, then the condition for the mean curvature of the ruled surface $\Phi$ to be constant is the same as that obtained in Section 3.

Next, we examine the following two cases for the developable ruled surface ${\beta }_{B,N}$:

Case 1. $\alpha :I\to E_1^3$ is a smooth space-like curve. If the principal normal vector $N\left(s\right)$ is time-like and the binormal vector $B\left(s\right)$ is space-like, then we have the following Frenet–Serret-type formula:

$$\left(\begin{array}{c}{T}^{\prime }\\ N^{\prime }\\ B^{\prime }\end{array}\right)=\left(\begin{array}{ccc}0& \kappa & 0\\ \kappa & 0& \tau \\ 0& \tau & 0\end{array}\right)\left(\begin{array}{c}T\\ N\\ B\end{array}\right).$$

We find that along the curve $\int (fB\left(s\right)+gN\left(s\right))ds+\left({\displaystyle \frac{\kappa -gc}{\tau c}}\right)B\left(s\right),$ the mean curvature H is constant.

Case 2. Let $\alpha :I\to {\mathbb{R}}_1^3$ be a smooth space-like curve. If the principal normal vector $N\left(s\right)$ is space-like and the binormal vector $B\left(s\right)$ is time-like, then we have the following Frenet–Serret-type formula:

$$\left(\begin{array}{c}{T}^{\prime }\\ N^{\prime }\\ B^{\prime }\end{array}\right)=\left(\begin{array}{ccc}0& \kappa & 0\\ -\kappa & 0& \tau \\ 0& \tau & 0\end{array}\right)\left(\begin{array}{c}T\\ N\\ B\end{array}\right).$$

The coefficients of the first fundamental form are calculated as $E=-1,F=-f,G={\left(g+u\tau \right)}^2-f^2.$ We also determine the coefficients of the second fundamental form $e=0,$ $f=0,$ $g=-\kappa (g+u\tau ).$ Hence along the curve $\int (fB\left(s\right)+gN\left(s\right))ds-\left({\displaystyle \frac{\kappa +gc}{\tau c}}\right)B\left(s\right),$ the mean curvature H is constant.

In particular, if $u=0,$ then H is constant along the base curve $\int (fB\left(s\right)+{\displaystyle \frac{\kappa }{c}}N\left(s\right))ds$ in the first case and along the base curve $\int (fB\left(s\right)-{\displaystyle \frac{\kappa }{c}}N\left(s\right))ds$ in the second case.

5. Special Cases

For the developable ruled surfaces, ${\beta }_{T,N}$ and ${\beta }_{B,N},$ assuming $f=0$ and $g=\lambda ,$ may give rise to various distinct special cases:

Case 1. For the ruled surface

$${\beta }_T=\int \lambda N\left(s\right)ds+uT\left(s\right),$$

if the principal normal vector $N\left(s\right)$ is time-like and the tangent vector $T\left(s\right)$ is space-like, then the mean curvature of the ruled surface ${\beta }_T$ is constant along the curve $\int \lambda N\left(s\right)ds+\left({\displaystyle \frac{-\tau -\lambda c}{\kappa c}}\right)T\left(s\right)$. If $\lambda =\kappa$, the surface becomes a conical surface, and, along the curve $\int \kappa N\left(s\right)ds+\left({\displaystyle \frac{-\tau -\kappa c}{\kappa c}}\right)T\left(s\right)$ on the conical surface, the mean curvature remains constant.

Case 2. For the ruled surface ${\beta }_T$, if the principal normal vector $N\left(s\right)$ is space-like and the tangent vector $T\left(s\right)$ is time-like, then the mean curvature of the ruled surface ${\beta }_T$ is constant along the curve $\int \lambda N\left(s\right)ds+\left({\displaystyle \frac{\tau -\lambda c}{\kappa c}}\right)T\left(s\right)$. We also see that H is constant along the base curve $\int \left({\displaystyle \frac{\tau }{c}}\right)N\left(s\right)ds$ for $u=0.$

Now, let us consider the ruled surface

$${\beta }_B=\int \lambda N\left(s\right)ds+uB\left(s\right).$$

Case 1. If the principal normal vector $N\left(s\right)$ is time-like and the binormal vector $B\left(s\right)$ is space-like, then we conclude that the mean curvature of the ruled surface ${\beta }_B$ remains constant along the curve $\int \lambda N\left(s\right)ds+\left({\displaystyle \frac{\kappa -\lambda c}{\tau c}}\right)B\left(s\right)$.

Case 2. If the principal normal vector $N\left(s\right)$ is space-like and the binormal vector $B\left(s\right)$ is time-like, then we see that along the curve $\int \lambda N\left(s\right)ds+\left({\displaystyle \frac{-\kappa -\lambda c}{\tau c}}\right)B\left(s\right)$ that the mean curvature H of the ruled surface ${\beta }_B$ is constant.

In both cases, when $u=0$, H is constant along the base curve, which is given by $\int \left({\displaystyle \frac{\kappa }{c}}\right)N\left(s\right)ds$ in Case 1 and by $\int \left({\displaystyle \frac{-\kappa }{c}}\right)N\left(s\right)ds$ in Case 2.

6. Some Computational Examples

In this section, we present some computational examples that show the curves along which the mean curvature of the ruled surface are constant. Examples 1–3 will be given in Euclidean space, while Examples 4 and 5 will be presented in Lorentz space.

In the following example, let us construct the curve along which the mean curvature is constant on a constant-angle surface.

Example 1. 

Let $\gamma :I\subset \mathbb{R}\to S^2$ be a smooth curve defined by

$$\gamma \left(s\right)=\left({\displaystyle \frac{1}{\sqrt{2}}}coss,{\displaystyle \frac{1}{\sqrt{2}}}sins,{\displaystyle \frac{1}{\sqrt{2}}}\right).$$

Let us choose $T\left(s\right)$ such that $\gamma \left(s\right).T\left(s\right)=0$, i.e., the position vector is orthogonal to the tangent vector $T\left(s\right)=\left(-sins,coss,0\right)$.

Taking the cross-product of the position vector $\gamma \left(s\right)$ and tangent vector $T\left(s\right)$ provides

$$S\left(s\right)=\gamma \left(s\right)\times T\left(s\right)=\left({\displaystyle \frac{-1}{\sqrt{2}}}coss,{\displaystyle \frac{-1}{\sqrt{2}}}sins,{\displaystyle \frac{1}{\sqrt{2}}}\right).$$

On the other hand, we have the curvature $m={\displaystyle \frac{1}{\sqrt{2}}}$, the torsion $n={\displaystyle \frac{1}{\sqrt{2}}}$, and $u_0=1-g$. If we write $\gamma \left(s\right)$ and $T\left(s\right)$ in $\Phi (u,s)=\int \left(f\left(s\right)\gamma \left(s\right)+g\left(s\right)T\left(s\right)\right)ds+u\gamma \left(s\right)$ and choose $f\left(s\right)=\sqrt{2}coss,g\left(s\right)=sins,$ then we obtain the ruled surface

$$\Phi (u,s)=\int {\displaystyle \frac{2}{\sqrt{2}}}\left(coss-sins,sins+coss,1\right)ds+u\left({\displaystyle \frac{1}{\sqrt{2}}}coss,{\displaystyle \frac{1}{\sqrt{2}}}sins,{\displaystyle \frac{1}{\sqrt{2}}}\right).$$

Moreover, for $u_0=1-sins$, the curve $\beta \left(s\right)$ on the ruled surface $\Phi \left(u,s\right)$ is obtained as follows and is shown in Figure 1.

$$\Phi (u_0,s)=\left(\sqrt{2}sins+{\displaystyle \frac{4}{\sqrt{2}}}coss,-\sqrt{2}coss+{\displaystyle \frac{4}{\sqrt{2}}}sins,\sqrt{2}s+{\displaystyle \frac{2}{\sqrt{2}}}\right).$$

Figure 1. The curve $\Phi (u_0,s)$ in red on the ruled surface $\Phi (u,s)$.

Example 2. 

Let $\alpha :I\subset \mathbb{R}\to {\mathbb{R}}^3$ be a smooth curve defined by

$$\alpha \left(s\right)=\left({\displaystyle \frac{1}{\sqrt{2}}},-{\displaystyle \frac{1}{\sqrt{2}}}coss-{\displaystyle \frac{1}{\sqrt{2}}}sins,-{\displaystyle \frac{1}{\sqrt{2}}}sins+{\displaystyle \frac{1}{\sqrt{2}}}coss\right).$$

If we choose $f\left(s\right)=\sqrt{2}sins$, $g\left(s\right)=\sqrt{2}coss$, then the ruled surface

$${\beta }_{T,N}(u,s)=\int \left(0,1,-1\right)ds+u\left(0,{\displaystyle \frac{1}{\sqrt{2}}}sins-{\displaystyle \frac{1}{\sqrt{2}}}coss,-{\displaystyle \frac{1}{\sqrt{2}}}coss-{\displaystyle \frac{1}{\sqrt{2}}}sins\right),$$

is obtained. Moreover, we obtain the curve

$${\beta }_{T,N}(u_0,s)=\left(0,s+{cos}^{2}s-sinscoss,-s+{cos}^{2}s+cosssins\right)$$

on the surface ${\beta }_{T,N}(u,s)$, see Figure 2.

Figure 2. The curve ${\beta }_{T,N}(u_0,s)$ in red on the ruled surface ${\beta }_{T,N}(u,s)$.

Example 3. 

Let $\alpha :I\subset \mathbb{R}\to {\mathbb{R}}^3$ be a smooth curve defined by

$$\alpha \left(s\right)=\left({\displaystyle \frac{s}{\sqrt{2}}},{\displaystyle \frac{1}{\sqrt{2}}}sins,{\displaystyle \frac{1}{\sqrt{2}}}coss\right).$$

Let us take $f=2\sqrt{2}sins$ and $g=coss$. The ruled surface takes the form

$${\beta }_{B,N}(u,s)=\int \left(-2sins,sinscoss,3{sin}^{2}s-1\right)ds+u\left({\displaystyle \frac{-1}{\sqrt{2}}},{\displaystyle \frac{1}{\sqrt{2}}}coss,{\displaystyle \frac{-1}{\sqrt{2}}}sins\right)$$

Moreover, for $u_0=-\sqrt{2}g-1=-\sqrt{2}coss-1,$ the curve on the ruled surface is derived as follows, see Figure 3:

$${\beta }_{B,N}(u_0,s)=\left(3coss+{\displaystyle \frac{1}{\sqrt{2}}},{\displaystyle \frac{-3}{2}}{cos}^{2}s-{\displaystyle \frac{1}{\sqrt{2}}}coss+{\displaystyle \frac{1}{2}},{\displaystyle \frac{-1}{2}}s-{\displaystyle \frac{1}{2}}sinscoss+{\displaystyle \frac{1}{\sqrt{2}}}sins\right).$$

Figure 3. The curve ${\beta }_{B,N}(u_0,s)$ in red on the ruled surface ${\beta }_{B,N}(u,s)$.

Next, we give some examples of the curves in Lorentz space.

Example 4. 

We consider the developable ruled surface ${\beta }_{T,N}$, where the principal normal vector $N\left(s\right)$ is time-like, and the tangent vector $T\left(s\right)$ is space-like. Let $\alpha :I\subset \mathbb{R}\to {\mathbb{R}}_1^3$ be a smooth curve defined by

$$\alpha \left(s\right)=\left({\displaystyle \frac{s}{\sqrt{2}}},sinh{\displaystyle \frac{s}{\sqrt{2}}},cosh{\displaystyle \frac{s}{\sqrt{2}}}\right).$$

Thus, we have the following ruled surface:

$$\begin{array}{ccc}\hfill {\beta }_{T,N}(u,s)& =& \int \left({\displaystyle \frac{1}{2}}cosh{\displaystyle \frac{s}{\sqrt{2}}},{\displaystyle \frac{1}{2}}{cosh}^2{\displaystyle \frac{s}{\sqrt{2}}}+{sinh}^2{\displaystyle \frac{s}{\sqrt{2}}},{\displaystyle \frac{3}{2}}sinh{\displaystyle \frac{s}{\sqrt{2}}}cosh{\displaystyle \frac{s}{\sqrt{2}}}\right)ds\hfill \\ & & +u{\displaystyle \frac{1}{\sqrt{2}}}\left(1,cosh{\displaystyle \frac{s}{\sqrt{2}}},sinh{\displaystyle \frac{s}{\sqrt{2}}}\right).\hfill \end{array}$$

For $u_0=-1-2{sinh}^2{\displaystyle \frac{s}{\sqrt{2}}}-2g=-2sinh{\displaystyle \frac{s}{\sqrt{2}}}\left(sinh{\displaystyle \frac{s}{\sqrt{2}}}+1\right)-1,$ the curve on the ruled surface ${\beta }_{T,N}(u,s)$ is obtained as follows, see Figure 4:

$${\beta }_{T,N}(u_0,s)=\left(\begin{array}{c}{\displaystyle \frac{1}{\sqrt{2}}}\left(-2{sinh}^2{\displaystyle \frac{s}{\sqrt{2}}}-sinh{\displaystyle \frac{s}{\sqrt{2}}}-1\right),\\ {\displaystyle \frac{1}{\sqrt{2}}}\left[{\displaystyle \frac{3}{4}}sinh\sqrt{2}s-{\displaystyle \frac{s}{2\sqrt{2}}}-cosh{\displaystyle \frac{s}{\sqrt{2}}}\left(2{sinh}^2{\displaystyle \frac{s}{\sqrt{2}}}+2sinh{\displaystyle \frac{s}{\sqrt{2}}}+1\right)\right],\\ {\displaystyle \frac{1}{\sqrt{2}}}\left[{\displaystyle \frac{3}{4}}cosh\sqrt{2}s-sinh{\displaystyle \frac{s}{\sqrt{2}}}\left(2{sinh}^2{\displaystyle \frac{s}{\sqrt{2}}}+2sinh{\displaystyle \frac{s}{\sqrt{2}}}+1\right)\right]\end{array}\right).$$

Figure 4. The curve ${\beta }_{T,N}(u_0,s)$ in red on the ruled surface ${\beta }_{T,N}(u,s)$.

Example 5. 

We consider the developable ruled surface ${\beta }_{B,N}$, where the principal normal vector $N\left(s\right)$ is time-like, and the binormal vector $B\left(s\right)$ is space-like. If we take the curve $\alpha \left(s\right)$ in Example 4 and choose $f={\displaystyle \frac{1}{\sqrt{2}}}cosh{\displaystyle \frac{s}{\sqrt{2}}}$ and $g=-sinh{\displaystyle \frac{s}{\sqrt{2}}}$, then the ruled surface

$$\begin{array}{ccc}\hfill {\beta }_{B,N}(u,s)& =& \int \left({\displaystyle \frac{1}{2}}cosh{\displaystyle \frac{s}{\sqrt{2}}},{\displaystyle \frac{1}{2}}{cosh}^2{\displaystyle \frac{s}{\sqrt{2}}}-{sinh}^2{\displaystyle \frac{s}{\sqrt{2}}},{\displaystyle \frac{-3}{2}}sinh{\displaystyle \frac{s}{\sqrt{2}}}cosh{\displaystyle \frac{s}{\sqrt{2}}}\right)ds+\hfill \\ & & u{\displaystyle \frac{1}{\sqrt{2}}}\left(1,cosh{\displaystyle \frac{s}{\sqrt{2}}},-sinh{\displaystyle \frac{s}{\sqrt{2}}}\right)\hfill \end{array}$$

is obtained. Furthermore, for $u_0=1,$ we obtain the following curve on the surface ${\beta }_{B,N}(u,s)$, see Figure 5:

$${\beta }_{B,N}(u_0,s)=\left(\begin{array}{c}{\displaystyle \frac{\sqrt{2}}{2}}sinh{\displaystyle \frac{s}{\sqrt{2}}}+{\displaystyle \frac{1}{\sqrt{2}}},\\ {\displaystyle \frac{3}{4}}s-{\displaystyle \frac{\sqrt{2}}{4}}sinh\sqrt{2}s+{\displaystyle \frac{1}{\sqrt{2}}}cosh{\displaystyle \frac{s}{\sqrt{2}}},\\ {\displaystyle \frac{-3}{4\sqrt{2}}}cosh\sqrt{2}s-{\displaystyle \frac{1}{\sqrt{2}}}sinh{\displaystyle \frac{s}{\sqrt{2}}}\end{array}\right).$$

Figure 5. The curve ${\beta }_{B,N}(u_0,s)$ in red on the ruled surface ${\beta }_{B,N}(u,s)$.

Example 6. 

We consider the developable ruled surface ${\beta }_{T,N}$, where the principal normal vector $N\left(s\right)$ is space-like, and the tangent vector $T\left(s\right)$ is time-like. Let $\alpha :I\subset \mathbb{R}\to {\mathbb{R}}_1^3$ be a smooth curve defined by $\alpha \left(s\right)={\displaystyle \frac{1}{2}}\left(0,cosh2s,sinh2s\right)$. Setting $f=sinh2s$ and $g=2cosh2s$ yields the ruled surface

$$\begin{array}{c}\hfill {\beta }_{T,N}(u,s)=\int \left(0,{sinh}^{2}2s+2{cosh}^{2}2s,3sinh2scosh2s\right)ds+u\left(0,sinh2s,cosh2s\right).\end{array}$$

Finally, we have the following curve on ${\beta }_{T,N}(u,s)$ for $u_0={\displaystyle \frac{-g}{2}}=-cosh2s,$ see Figure 6:

$${\beta }_{T,N}(u_0,s)=\left(0,{\displaystyle \frac{1}{2}}s-{\displaystyle \frac{1}{4}}sinh2scosh2s,{\displaystyle \frac{3}{4}}{sinh}^{2}2s-{cosh}^{2}2s\right).$$

Figure 6. The curve ${\beta }_{T,N}(u_0,s)$ in red on the ruled surface ${\beta }_{T,N}(u,s)$ in the Lorentz space.

7. Conclusions

In this article, we first examine the conditions under which developable ruled surfaces along the general curves in Euclidean and Lorentz spaces have constant mean curvature. Moreover, the case where the mean curvature is constant along the striction line is considered. Then, some special cases for developable surfaces constructed using indicatrix curves in Lorentz space are examined. The condition that the mean curvature is constant along the base curve is presented. Illustrative examples and graphs of the ruled surfaces obtained from these curves are presented. The studies we conduct here in Lorentz space can also be performed in other non-Euclidean spaces.