Family of ruled surfaces generated by equiform Bishop spherical image in Minkowski 3-space

1.   Introduction

One of several primary goals of classical differential geometry is to examine some surface's classes with unique properties in $E^3$, such as developable and minimal. There are numerous types of surfaces, including cyclic, revolution, helicoid, rotational, canal, and governed surfaces. This type of surface plays a vital role and has a wide range of applications Physics, Computer Aided Geometric Design, and the study of design challenges in spatial mechanisms, among other subjects ^[9,12]. Many studies have been done on the features of these surfaces in Euclidean and Minkowski spaces, as well as certain characterizations ^{[2,3,8,10,13,14,16,18,19]}.

In this paper, we look at a type of ruled surface known as equiform Bishop spherical image ruled surfaces using the equiform Bishop frame in $\mathrm{E}_1^3$. The main results are presented in theorems that concert the necessary and sufficient conditions for those ruled surfaces to be equiform Bishop developable and equiform Bishop minimal.

2.   Preliminaries

The Lorentzian product in Minkowski 3-dimentional space $\mathrm{E}_1^3$ is define by

where $(\varsigma_1, \varsigma_2, \varsigma_3)$ is a rectangular coordinate system of $\mathrm{E}_1^3$. An arbitrary $r\in \mathrm{E}_1^3$ vector is one of the following; spacelike if $\cal{P}(r, r) > 0$ or $r = 0$, timelike if $\cal{P}(r, r) < 0$ and null if $\cal{P}(r, r) = 0$ and $r\neq0$. Similarly, a curve $\phi = \phi(s)$ can be spacelike, timelike or null if its $\phi'(s)$ is spacelike, timelike or null ^[15,17].

Let $\varphi = \varphi(s)$ is a spacelike curve with a timelike binormal. Denoted $\{t, n, b\}$ be the moving Frenet frame of spacelike curve $\phi$, then $\{t, n, b\}$ has the following properties ^[7,15,17]:

where $\Big(\, \cdot = \dfrac{d}{ds}\Big)\,$, $\cal{P}(t, t) = 1$, $\cal{P}(n, n) = 1$, $\cal{P}(b, b) = -1$ and $\cal{P}(t, n) = \cal{P}(t, b) = \cal{P}(n, b) = 0$.

The Bishop frame is an alternative approach to defining a moving frame that is well defined even when the curve has vanishing second derivative ^[4,5].

Let $\{t, n_1, n_2\}$ denote the Bishop frame of the spacelike curve $\phi(s)$ with timelike binormal. Then $\{t, n_1, n_2\}$ is expressed as ^[5,11].

where $\cal{P}(t, t) = 1$, $\cal{P}(n_1, n_1) = 1$, $\cal{P}(n_2, n_2) = -1$ and $\cal{P}(t, n_1) = \cal{P}(t, n_2) = \cal{P}(n_1, n_2) = 0$. We call ${\kappa}_1(s)$ and ${\kappa}_2(s)$ as Bishop curvatures. The relation can be expressed as

where

and

Let $\varphi:I \rightarrow\mathrm{E}_1^3$ be a spacelike curve with a timelike binormal in $\mathrm{E}_1^3$. The equiform parameter of $\varphi$ by $\theta = { }\int {\kappa}_1 ds$. Then $\varrho = \dfrac{ds}{d \theta}$, where $\varrho = \dfrac{1}{ {\kappa}_1}$ the radius of curvature of the curve $\varphi$. We recall $\{T, B_1, B_2, \}$ be the moving equiform Bishop frame where $T(\theta) = \varrho\, t(s)$, $B_1(\theta) = \varrho\, n_1(s)$ and $B_2(\theta) = \varrho\, n_2(s)$ and the equiform Bishop curvatures of the curve $\varphi = \varphi(\vartheta)$ are $k_1(\theta) = \dot{\varrho} = \dfrac{d\varrho}{ds}$ and $k_2(\theta) = \left(\dfrac{ {\kappa}_2}{ {\kappa}_1}\right)$. Then, the frame $\{T, B_1, B_2, \}$ of $\varphi$ is given as ^[1,20,21,22]:

where $\Big(\, ' = \dfrac{d}{d \theta}\Big)\,$, $\cal{P}(T, T) = \varrho^2$, $\cal{P}(B_1, B_1) = \varrho^2$, $\cal{P}(B_2, B_2) = -\varrho^2$, and $\cal{P}(T, B_1) = \cal{P}(T, B_2) =$ $\cal{P}(B_1, B_2) = 0$.

The pseudo-Riemannian sphere of unit radius and with center in the origin in the space $\mathrm{E}_1^3$ is defined by

A ruled surface $\Gamma$ in $\mathrm{E}_1^3$ can be representation as

where $\varphi(s)$ is $\Theta$'s base curve and $X(s)$ be the unit represents a space curve which representing the direction of straight line ^[6].

The $\Theta$'s normal vector field $m$ defined by ^[17]:

The $\Theta$'s components of the first and second fundamental forms are

The Gaussian curvature $K$ and the mean curvature $H$ respectively are given by ^[17]:

A ruled surface is developable if and only if $K = 0$ and minimal if and only if $H = 0$ ^[17].

3.   Some characterizations of equiform Bishop spherical image ruled surfaces

In this section, we introduce the equiform Bishop spherical image ruled surfaces in Mikowski 3-space $\mathrm{E }_ 1 ^ 3$ via the equiform Bishop frame $\{T, B_1, B_2\}$. Also, we study some geometric properties that make these surfaces have $K = 0$ and $H = 0$.

3.1.   $T$-equiform Bishop spherical image ruled surface

Definition 3.1. Let $\varphi = \varphi(\theta)$ be a regular spacelike curve in $\mathrm{E}_1^3$ via equiform Bishop frame (2.5). The $T$-equiform Bishop spherical image ruled surface is defined as

Consider $T$-equiform Bishop spherical image ruled surface (3.1), the natural frame is given by

With the above equation, we can derive the component parts of the first and second fundamental forms of $\Phi$ as follows:

where

and

Using the data described above, the equiform Bishop Gaussian curvature $K_\Phi$ and equiform Bishop mean curvature $H_\Phi$ are calculated as follows:

where

Theorem 3.1. Let $\Phi = \Phi(\theta, \upsilon)$ is $T$-equiform Bishop spherical image ruled surface in $\mathrm{E}_1^3$ given by (3.1). Then, $\Phi$ at the point $(\theta, 0)$ is equiform Bishop flat if and only if

Theorem 3.2. Let $\Phi = \Phi(\theta, \upsilon)$ is $T$-equiform Bishop spherical image ruled surface in $\mathrm{E}_1^3$ given by (3.1). Then, $\Phi$ at the point $(\theta, 0)$ is equiform Bishop minimal if and only if the equiform curvatures satisfy the following differential equation

Case 3.1. At $x_1 = 0$, the $T$-equiform Bishop spherical image ruled surface (3.1) has the following:

Corollary 3.1. At the point $(\theta, 0)$ the $T$-equiform Bishop spherical image ruled surface (3.1) with $x_1 = 0$ is:

${\bf{1).}}$ Equiform Bishop flat surface if the equiform curve $\varphi(\theta)$ is contained in a plane.

${\bf{2).}}$ Equiform Bishop flat surface if only if $k_2 = \left(\dfrac{x_2}{x_3}\right)$.

${\bf{3).}}$ Equiform Bishop minimal surface if the equiform curvatures satisfy the following differential equation

Case 3.2. At $x_2 = 0$, the $T$-equiform Bishop spherical image ruled surface (3.1) has the following:

Corollary 3.2. At the point $(\theta, 0)$ the $T$-equiform Bishop spherical image ruled surface (3.1) with $x_2 = 0$ is:

${\bf{1).}}$ Equiform Bishop flat surface if the equiform curve $\varphi(\theta)$ is contained in a plane.

${\bf{2).}}$ Equiform Bishop minimal surface if the equiform curvatures satisfy the following differential equation

Case 3.3. At $x_3 = 0$, the $T$-equiform Bishop spherical image ruled surface (3.1) has the following:

Corollary 3.3. At the point $(\theta, 0)$, the $T$-equiform Bishop spherical image ruled surface (3.1) with $x_3 = 0$ is:

${\bf{1).}}$ Equiform Bishop flat surface if the equiform curve $\varphi(\theta)$ is contained in a plane.

${\bf{2).}}$ Equiform Bishop flat surface if only if $k_1 = -\dfrac{2x_1^2+x_2^2}{x_1x_2}$.

${\bf{3).}}$ Equiform Bishop minimal surface if the equiform curvatures satisfy the following differential equation

Case 3.4. At $x_1 = x_3 = 0, \, \, x_2 = \varrho$, the $T$-equiform Bishop spherical image ruled surface (3.1) has the following:

Corollary 3.4. At the point $(\theta, 0)$ the $T$-equiform Bishop spherical image ruled surface (3.1) with $x_1 = x_3 = 0, \, \, x_2 = \varrho$ is:

${\bf{ 1).}}$ Equiform Bishop flat surface if the equiform curve $\varphi(\theta)$ is contained in a plane.

${\bf{2).}}$ Equiform Bishop minimal surface if the equiform curvatures satisfy the following differential equation

Case 3.5. At $x_2 = x_3 = 0, \, \, x_1 = \varrho$, the $T$-equiform Bishop spherical image ruled surface (3.1) has the following:

Corollary 3.5. At the point $(\theta, 0)$ the $T$-equiform Bishop spherical image ruled surface (3.1) with $x_2 = x_3 = 0, \, \, x_1 = \varrho$ is:

${\bf{1).}}$ Equiform Bishop flat surface if the equiform curve $\varphi(\theta)$ is contained in a plane.

${\bf{2). }}$ Equiform Bishop minimal surface if the equiform curvatures satisfy the following differential equation

3.2.   $B_1$-equiform Bishop spherical image ruled surface

Definition 3.2. Let $\varphi = \varphi(\theta)$ be a regular spacelike curve in $\mathrm{E}_1^3$ via equiform Bishop frame (2.5). The $B_1$-equiform Bishop spherical image ruled surface is defined as

Now, the natural frame $B_1$-equiform Bishop spherical image ruled surface (3.11)is given by

By the above equation, we can surmise the component parts of the first and second fundamental forms of $\Omega$ as shown in:

Where

and

Using the data described above, the equiform Bishop Gaussian curvature $K_\Omega$ and equiform Bishop mean curvature $H_\Omega$ are calculated as follows:

where

Theorem 3.3. Let $\Omega = \Omega(\theta, \upsilon)$ is $B_1$-equiform Bishop spherical image ruled surface in $\mathrm{E}_1^3$ given by (3.11). Then, $\Omega$ at the point $(\theta, 0)$ is equiform Bishop flat if and only if

Theorem 3.4. Let $\Omega = \Omega(\theta, \upsilon)$ is $B_1$-equiform Bishop spherical image ruled surface in $\mathrm{E}_1^3$ given by (3.11). Then, $\Omega$ at the point $(\theta, 0)$ is equiform Bishop minimal if and only if the equiform curvatures satisfy the following differential equation

Case 3.6. At $x_1 = 0$, the $B_1$-equiform Bishop spherical image ruled surface (3.11) has the following:

Corollary 3.6. At the point $(\theta, 0)$ the $B_1$-equiform Bishop spherical image ruled surface (3.11) with $x_1 = 0$ is:

${\bf{1).}}$ Equiform Bishop flat surface if and only if $x_2 = x_3$.

${\bf{2).}}$ Equiform Bishop minimal surface if the equiform curvatures satisfy the following differential equation

Case 3.7. At $x_2 = 0$, the $B_1$-equiform Bishop spherical image ruled surface (3.11) has the following:

Corollary 3.7. At the point $(\theta, 0)$ the $B_1$-equiform Bishop spherical image ruled surface (3.11) with $x_2 = 0$ is:

${\bf{1).}}$ Equiform Bishop flat surface if and only if $x_1k_1k_2(x_1+x_3)-x_3(x_3k_1-x_1) = 0$.

${\bf{2).}}$ Equiform Bishop minimal surface if the equiform curvatures satisfy the following differential equation

Remark 3.1. On the $B_1$-equiform Bishop spherical image ruled surface (3.11) at the point $(\theta, 0)$ with $x_3 = 0$, $x_1 = x_3 = 0, \, \, x_2 = \varrho$ and $x_2 = x_3 = 0, \, \, x_2 = \varrho$ the equiform Bishop mean curvature $H_\Omega$ is undefined.

3.3.   $B_2$-equiform Bishop spherical image ruled surface

Definition 3.3. Let $\varphi = \varphi(\theta)$ be a regular spacelike curve in $\mathrm{E}_1^3$ via equiform Bishop frame (2.5). The $B_2$-equiform Bishop spherical image ruled surface is defined as

Now, the natural frame $B_2$-equiform Bishop spherical image ruled surface (3.18)is given by

By the above equation, we can surmise the component parts of the first and second fundamental forms of $\Psi$ as shown in:

where

and

Using the data described above, the equiform Bishop Gaussian curvature $K_\Psi$ and equiform Bishop mean curvature $H_\Psi$ are calculated as follows:

where

Theorem 3.5. Let $\Psi = \Psi(\theta, \upsilon)$ is $B_2$-equiform Bishop spherical image ruled surface in $\mathrm{E}_1^3$ given by (3.18). Then, $\Psi$ at the point $(\theta, 0)$ is equiform Bishop flat if and only if

Theorem 3.6. Let $\Psi = \Psi(\theta, \upsilon)$ is $B_2$-equiform Bishop spherical image ruled surface in $\mathrm{E}_1^3$ given by (3.18). Then, $\Psi$ at the point $(\theta, 0)$ is equiform Bishop minimal if and only if the equiform curvatures satisfy the following differential equation

Case 3.8. At $x_1 = 0$, the $B_2$-equiform Bishop spherical image ruled surface (3.18) has the following:

Corollary 3.8. At the point $(\theta, 0)$ the $B_1$-equiform Bishop spherical image ruled surface (3.18) with $x_1 = 0$ is:

${\bf{1). }}$ Equiform Bishop flat surface if the equiform curve $\varphi(\theta)$ is a straight line.

${\bf{2). }}$ Equiform Bishop flat surface if and only if $k_2 = \left(\dfrac{x_2}{x_3}\right)$.

${\bf{3).}}$ Equiform Bishop minimal surface if the equiform curvatures satisfy the following differential equation

Case 3.9. At $x_2 = 0$, the $B_2$-equiform Bishop spherical image ruled surface (3.18) has the following:

Corollary 3.9. At the point $(\theta, 0)$ the $B_2$-equiform Bishop spherical image ruled surface (3.18) with $x_2 = 0$ is:

${\bf{1).}}$ Equiform Bishop flat surface if and only if $k_1 = \left(\dfrac{x_1}{x_3}\right)$.

${\bf{2).}}$ Equiform Bishop flat surface if and only if the equiform curve $\varphi(\theta)$ is a circular helix i.e., $\dfrac{k_2}{k_1} = -\left(\dfrac{x_1}{x_3}\right)$.

${\bf{3).}}$ Equiform Bishop minimal surface if the equiform curvatures satisfy the following differential equation

Case 3.10. At $x_3 = 0$, the $B_2$-equiform Bishop spherical image ruled surface (3.18) has the following:

Corollary 3.10. At the point $(\theta, 0)$, the $B_2$-equiform Bishop spherical image ruled surface (3.18) with $x_3 = 0$ is:

${\bf{1).}}$ Equiform Bishop flat surface if only if $x_2(k_1^2+k_2^2)+x_1k_1 = 0$.

${\bf{2). }}$ Equiform Bishop minimal surface if the equiform curvatures satisfy the following differential equation

Case 3.11. At $x_2 = x_3 = 0, \, \, x_1 = \varrho$, the $B_2$-equiform Bishop spherical image ruled surface (3.18) has the following:

Corollary 3.11. At the point $(\theta, 0)$ the $B_2$-equiform Bishop spherical image ruled surface (3.18) with $x_2 = x_3 = 0, \, \, x_1 = \varrho$ is:

${\bf{1).}}$ Equiform Bishop minimal surface surface if the equiform curve $\varphi(\theta)$ is contained in a plane.

${\bf{2). }}$ Equiform Bishop minimal surface if and only if $\varrho = \pm1$.

Remark 3.2. On the $B_2$-equiform Bishop spherical image ruled surface (3.18) at the point $(\theta, 0)$ with $x_1 = x_3 = 0, \, \, x_2 = \varrho$ the equiform Bishop mean curvature $H_\Psi$ is undefined.

3.4.   Example

We build computational example of equiform-Bishop spherical image ruled surfaces curves in $\mathrm{E}_1^3$ using the moving equiform-Bishop frame $\{T, B_1, B_2\}$ of the spacelike equiform-Bishop curve $\psi(s) = \big(s\, \sin{(\ln{s})}, s\, \cos{(\ln{s})}, s\big)$ with timelike binormal vector (see Figure 1). Then it is simple to demonstrate that

The curvature functions are ${\kappa} = \dfrac{\sqrt{2}}{s}$ and ${\tau} = \dfrac{1}{s}$. Also, $\vartheta(s) = { }\int_0^s\left(\frac{1}{s}\right)ds = \ln{s}$. From (2.4), we get ${\kappa}_1(s) = \left(\frac{\sqrt{2}}{s}\right)\cosh{(\ln s)}$, ${\kappa}_2(s) = \left(\frac{\sqrt{2}}{s}\right)\sinh{(\ln s)}$. Also from (2.3), we get

Now, the equiform-Bishop parameter is $\theta = { }\int_0^s {\kappa}_1\, ds = \sqrt{2}\, \sinh{(\ln s)}$. Then we have $s = \left(\frac{ \theta+\sqrt{ \theta^2+2}}{\sqrt{2}}\right)$ and $\varrho = \left(\frac{ \theta+\sqrt{ \theta^2+2}}{\sqrt{2}\sqrt{ \theta^2+2}}\right)$. Furthermore, the equiform-Bishop curvatures are given by

So the equiform-Bishop curve $\psi = \psi(\theta)$ is define as (see Figure 2)

Additionally, the equiform-Bishop frame are given by

Thus, the $T$-equiform Bishop spherical image ruled surface, $B_1$-equiform Bishop spherical image ruled surface and $B_2$-equiform Bishop spherical image ruled surface are respectively given as (see Figures 3–5)

Acknowledgements

The authors extend their appreciation to the Deanship of Scientific Research, Imam Mohammad Ibn Saud Islamic University (IMSIU), Saudi Arabia, for funding this research work through Grant No. (221412004).

Conflicts of interest

The authors declare no competing interest.