Surface Pencil Couple with Bertrand Couple as Joint Principal Curves in Galilean 3-Space

Abstract

A principal curve on a surface plays a paramount role in reasonable implementations. A curve on a surface is a principal curve if its tangents are principal directions. Using the Serret–Frenet frame, the surface pencil couple can be expressed as linear combinations of the components of the local frames in Galilean 3-space ${\mathbb{G}}_3$. With these parametric representations, a family of surfaces using principal curves (curvature lines) are constructed, and the necessary and sufficient condition for the given Bertrand couple to be the principal curves on these surfaces are derived in our approach. Moreover, the necessary and sufficient condition for the given Bertrand couple to satisfy the principal curves and the geodesic requirements are also analyzed. As implementations of our main consequences, we expound upon some models to confirm the method.

1. Introduction

Principal curve is designated as one of the significant curves on a surface and it plays a primary role in differential geometry [1,2,3,4]. It is an advantageous gadget in surface examination for showing contrast of the principal direction. The harmonic principal curvature and principal curves are considerable and associated with the smooth surfaces. Principal curves can direct the realization of surfaces, quietly utilized in geometric designing, and can constitute consistency, surface polygonization, and surface fulfillment. There exists an enormous literature on the topic, including various monographs, for instance: Martin [5] studied systematic surface patches restricted by principal curves, which are named principal patches. Martin presented that the presence of such patches was conditioned upon confirmed situations corresponding on the patch border curves. Alourdas et al. [6] addressed a mode to confirm a net principal curves on a B-spline surface. Maekawa et al. [7] extended a style to take out the generic characteristics of free-shape parametric surfaces for form inspection. They researched the generic advantage of the umbilics and attitude principal curves that go through an umbilic on a parametric free-shape surface. Che and Paul [8] expanded a manner to resolve and calculate the principal curves and their geometric properties specified on an implicit surface. They also offered a new standard for non-umbilical points and umbilical points on an implicit surface. Zhang et al. [9] proved a planner for calculating and envisaging the principal curves pointed on an implicit surface. Kalogerakis et al. [10] derived a powerful substructure for establishing principal curves by point clouds. Their approach is reasonable for surfaces of random genus, with or without borderlines, and is statistically powerful to use with outliers maintaining surface characteristics. They found the approach to be efficient through an area of synthetic and real-world input data collections with changing amounts of noise and outliers. In practical uses, however, crucial work has focused on the backward exploration or reverse issue: given a 3D curve, how can we locate those surfaces that are faced with this curve as a distinctive curve, if possible, rather than locating and furnishing curves on analytical curved surfaces? Wang et al. [11] was the first to handle the issue of assembling a surface family with a designated locative geodesic curve, through which every surface can be a candidate for mode style. They demonstrated the necessary and sufficient conditions for the coefficients to be satisfied with both the isoparametric and the geodesic demands. This scheme has been used by many scholars (see, for example, [12,13,14,15,16,17,18,19,20,21,22,23]).

Galilean geometry is the simplest pattern of a semi-Euclidean geometry for which the isotropic cone reduces to a plane. It is indicated as a bridge from Euclidean geometry to special relativity. The main spine of Galilean geometry is its private gravity, that is, it allows the scientist to search it in detail without using large amounts of time and energy. In other words, the view of Galilean geometry shapes its development with an unpretentious question, and a gross stretch of a modern geometric community is crucial for its efficient comparison with Euclidean geometry. Also, aggregate evolution is practical to supply the scientist with the psychological emphasis of the uniformity of the inspected structure [24,25]. In the $3\mathbb{D}$ (three-dimensional) Galilean space ${\mathbb{G}}_3$, there are several studies dealing with ${\mathbb{G}}_3$, for example, Dede et al. [26,27] resolved tub surfaces, the descriptions of the parallel surfaces. Yuzbasi et al. [28,29] considered a surface family with a curve to be a joint asymptotic curve and geodesic. Jiang et al. [30] considered surface pencil couple with Bertrand couple as joint asymptotic curves. Almoneef and Abdel-Baky [31] designed a surface family with Bertrand curves to be geodesic curves. AL-Jedani and Abdel-Baky [32] considered the surface family and developable surface family with a joint geodesic curve, respectively.

It is known that on any surface there are three main kinds of curves: principal curves (curvature lines), asymptomatic curves, and geodesic curves. In [30], the study focused on how to construct family surfaces via Bertrand curves that are asymptomatic, whereas our work addresses a very new idea related to a differential geometry field based on construction of a family of surfaces using principal curves (curvature lines). Ref. [31] investigated a similar idea but used the geodesic curves instead of principal ones. In addition, a team of researchers, referred to as Li et al. and cited in [32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49], conducted theoretical studies and advancements on soliton theory, submanifold theory, and other related topics. Further motivation can be found in these papers. Their efforts have significantly contributed to the progression of research in these fields.

However, to our knowledge, no additional work has been done to originate surface pencil couples with curve couples that are principal curves. In order to create the surfaces pencil and, specifically ruled ones, we explore Bertrand couples as principal curves and organize a surface pencil couple with a Bertrand couple as joint principal curves. Furthermore, the accessory to the ruled surface pencil is likewise qualified. In addition, some models are exhibited to create the surface pencil, in general, and ruled ones, in particular, with joint Bertrand principal curves.

2. Basic Concepts

The 3$\mathbb{D}$ (3-dimensional) Galilean space ${\mathbb{G}}_3$ is a Cayley–Klein geometry endued with the projective metric of signature (0, 0, +, +) [48,49]. The absolute figure of ${\mathbb{G}}_3$ is contingent on the organized triple {$\pi$, $\mathfrak{L}$, $\mathfrak{I}$}, where $\pi$ is the (absolute) plane in the real 3$\mathbb{D}$ projective space ${\mathbb{P}}^3$($\mathbb{R}$), $\mathfrak{L}$ is the line (absolute line) in $\pi$, and $\mathfrak{L}$ is the steady elliptic involution of points of $\mathfrak{L}$. Homogeneous coordinates in ${\mathbb{G}}_3$ are developed in such a mode that the absolute plane $\pi$ is pointed by $z_0=0$, the absolute line $\mathfrak{L}$ by $z_0=z_1=0$, and the elliptic involution is pointed by $(0:0:z_2:z_3)\to (0:0:z_3:-z_2)$. A plane is organized Euclidean if it includes $\mathfrak{L}$, otherwise it is organized isotropic, that is, planes $z_0$=const are Euclidean, and so is the plane $\pi$. Other planes are isotropic. Furthermore, an isotropic plane does not involve any isotropic direction. For any $\mathit{\upsilon }=\left({\upsilon }_1,{\upsilon }_2,{\upsilon }_3\right)$, and $\mathit{\nu }=\left({\nu }_1,{\nu }_2,{\nu }_3\right)\in {\mathbb{G}}_3$, their scalar product is

$$<\mathit{\upsilon },\mathit{\nu }>=\left\{\begin{array}{c}{\upsilon }_1{\nu }_1,\mathrm{if}{\upsilon }_1\ne 0\vee {\nu }_1\ne 0,\hfill \\ {\upsilon }_2{\nu }_2+{\upsilon }_3{\nu }_3,\mathrm{if}{\upsilon }_1=0\wedge {\nu }_1=0,\hfill \end{array}\right.$$

(1)

and their vector product is

$$\mathit{\upsilon }\times \mathit{\nu }=\left\{\begin{array}{c}\left|\begin{array}{ccc}{\mathbf{x}}_1\hfill & {\mathbf{x}}_2\hfill & {\mathbf{x}}_3\hfill \\ 0\hfill & {\upsilon }_2\hfill & {\upsilon }_3\hfill \\ 0\hfill & {\nu }_2\hfill & {\nu }_3\hfill \end{array}\right|,\mathrm{if}{\upsilon }_1=0\wedge {\nu }_1=0.\\ \left|\begin{array}{ccc}\mathbf{0}\hfill & {\mathbf{x}}_2\hfill & {\mathbf{x}}_3\hfill \\ {\upsilon }_1\hfill & {\upsilon }_2\hfill & {\upsilon }_3\hfill \\ {\nu }_1\hfill & {\nu }_2\hfill & {\nu }_3\hfill \end{array}\right|,\mathrm{if}{\upsilon }_1\ne 0\vee {\nu }_1\ne 0,\end{array}\right.$$

(2)

where ${\mathbf{x}}_1=(1,0,0)$, ${\mathbf{x}}_2=(0,1,0)$, and ${\mathbf{x}}_3=(0,0,1)$ are the standard basis vectors in ${\mathbb{G}}_3$.

A curve $\mathit{\psi }\left(u\right)=\left({\psi }_1\left(u\right),{\psi }_2\left(u\right),{\psi }_3\left(u\right)\right);u\in I\subseteq \mathbb{R}$ is a denominated allowable curve if it has no inflection points, that is, $\stackrel{.}{\mathit{\psi }}\times \stackrel{..}{\mathit{\psi }}\ne \mathbf{0}$, and no isotropic tangents $\stackrel{.}{{\psi }_1}\ne 0$. An allowable curve is comparable to a regular curve in Euclidean space. For an allowable curve $\mathit{\psi }:$$I\subseteq \mathbb{R}\to {\mathbb{G}}_3$ defined by the Galilean invariant arc-length u, we have:

$$\mathit{\psi }\left(u\right)=\left(u,{\psi }_2\left(u\right),{\psi }_3\left(u\right)\right).$$

(3)

The curvature $\kappa \left(u\right)$ and torsion $\tau \left(u\right)$ of the curve $\psi$$\left(u\right)$ are

$$\begin{array}{ccc}\hfill \kappa \left(u\right)& =& ∥{\mathit{\psi }}^{{}^{\prime \prime }}\left(u\right)∥=\sqrt{{\left({\psi }_2^{{}^{\prime \prime }}\left(u\right)\right)}^2+{\left({\psi }_3^{{}^{\prime \prime }}\left(u\right)\right)}^2},\hfill \\ \hfill \tau \left(u\right)& =& \frac{1}{{\kappa }^2\left(u\right)}det\left({\mathit{\psi }}^{{}^{\prime }},{\mathit{\psi }}^{{}^{{}^{\prime \prime }}},{\mathit{\psi }}^{{}^{{}^{{}^{\prime \prime \prime }}}}\right).\hfill \end{array}$$

(4)

Note that an allowable curve has $\kappa \left(u\right)\ne 0$. The Serret–Frenet vectors are:

$$\begin{array}{ccc}\hfill {\mathbf{g}}_1\left(u\right)& =& {\mathit{\psi }}^{{}^{\prime }}\left(u\right)=\left(1,{\psi }_2^{{}^{\prime }}\left(u\right),{\psi }_3^{{}^{\prime }}\left(u\right)\right),\hfill \\ \hfill {\mathbf{g}}_2\left(u\right)& =& \frac{1}{\kappa \left(u\right)}{\mathit{\psi }}^{{}^{\prime }}\left(u\right)=\frac{1}{\kappa \left(u\right)}\left(0,{\psi }_2^{{}^{\prime \prime }}\left(u\right),{\psi }_3^{{}^{\prime \prime }}\left(u\right)\right),\hfill \\ \hfill {\mathbf{g}}_3\left(u\right)& =& \frac{1}{\tau \left(u\right)}\left(0,{\left(\frac{1}{\kappa \left(u\right)}{\psi }_2^{{}^{\prime \prime }}\left(u\right)\right)}^{{}^{\prime }},{\left(\frac{1}{\kappa \left(u\right)}{\psi }_3^{{}^{\prime \prime }}\left(u\right)\right)}^{{}^{\prime }}\right),\hfill \end{array}$$

(5)

where ${\mathbf{g}}_1\left(u\right)$, ${\mathbf{g}}_2\left(u\right)$, and ${\mathbf{g}}_3\left(u\right)$, respectively, are the tangent, principal normal, and binormal vectors. The Serret–Frenet formulae read:

$$\left(\begin{array}{c}{\mathbf{g}}_1^{{}^{\prime }}\hfill \\ {\mathbf{g}}_2^{{}^{\prime }}\hfill \\ {\mathbf{g}}_3^{{}^{\prime }}\hfill \end{array}\right)=\left(\begin{array}{ccc}0\hfill & \kappa \left(u\right)\hfill & 0\hfill \\ 0\hfill & 0\hfill & \tau \left(u\right)\hfill \\ 0\hfill & -\tau \left(u\right)\hfill & 0\hfill \end{array}\right)\left(\begin{array}{c}{\mathbf{g}}_1\hfill \\ {\mathbf{g}}_2\hfill \\ {\mathbf{g}}_3\hfill \end{array}\right).$$

(6)

Definition 1

([31]). Let $\mathit{\psi }\left(u\right)$ and $\widehat{\mathit{\psi }}\left(u\right)$ be curves with Galilean invariant arc-length u, ${\mathbf{g}}_2\left(u\right)$ and ${\widehat{\mathbf{g}}}_2\left(u\right)$ are their principal normals, respectively; the set {$\widehat{\mathit{\psi }}\left(u\right)$, $\mathit{\psi }\left(u\right)$} is an organized Bertrand couple if ${\mathbf{g}}_2\left(u\right)$ and ${\widehat{\mathbf{g}}}_2\left(u\right)$ are linearly related at the conformable points, $\mathit{\psi }\left(u\right)$ is organized as the Bertrand mate of $\widehat{\mathit{\psi }}\left(u\right)$, and

$$\widehat{\mathit{\psi }}\left(u\right)=\mathit{\psi }\left(u\right)+f{\mathbf{g}}_2\left(s\right),$$

(7)

where f is a stationary.

We designate a surface S by

$$S:\mathbf{p}(u,t)=\left(p_1\left(u,t\right),p_2\left(u,t\right),p_3\left(u,t\right)\right),(u,t)\in \mathbb{D}\subseteq {\mathbb{R}}^2.$$

(8)

If ${\mathbf{p}}_j(u,t)=\frac{\partial \mathbf{p}}{\partial j}$, the isotropic surface normal is

$$\mathbf{u}(u,t)={\mathbf{p}}_u\wedge {\mathbf{p}}_t,\mathrm{with}<\mathbf{u},{\mathbf{p}}_u>=<\mathbf{u},{\mathbf{p}}_t>=0.$$

A curve on a surface S can be principal curve by the status specified by the well-known theorem below [1,2].

Theorem 1.

(Monge’s Theorem). A curve on a surface is a principal curve if and only if the surface normals along that curve create a developable surface [1,2].

3. Main Results

This section presents a new aspect for originating a surface pencil couple with a Bertrand couple as joint principal curves in ${\mathbb{G}}_3$. For this aim, let $\widehat{\mathit{\psi }}\left(u\right)$ be an allowable curve, $\mathit{\psi }\left(u\right)$ be its Bertrand mate, and $\{\widehat{\kappa }\left(u\right),\widehat{\tau }\left(u\right),{\widehat{\mathbf{g}}}_1\left(u\right),{\widehat{\mathbf{g}}}_2\left(u\right)$, ${\widehat{\mathbf{g}}}_3\left(u\right)\}$ is the Serret–Frenet instruments of $\widehat{\mathit{\psi }}\left(u\right)$, as in Equation (1). The surface pencil with $\mathit{\psi }\left(u\right)$ can be specified by

$$S:\mathbf{p}(u,t)=\mathit{\psi }\left(u\right)+\mathfrak{a}(u,t){\mathbf{g}}_1\left(u\right)+\mathfrak{b}(u,t){\mathbf{g}}_2\left(u\right)+\mathfrak{c}(u,t){\mathbf{g}}_3\left(u\right),$$

(9)

and the surface pencil with $\widehat{\mathit{\psi }}\left(u\right)$ is

$$\widehat{S}:\widehat{\mathbf{p}}(u,t)=\widehat{\mathit{\psi }}\left(u\right)+\mathfrak{a}(u,t){\widehat{\mathbf{g}}}_1\left(u\right)+\mathfrak{b}(u,t){\widehat{\mathbf{g}}}_2\left(u\right)+\mathfrak{c}(u,t){\widehat{\mathbf{g}}}_3\left(u\right),$$

(10)

where $\mathfrak{a}(u,t),$$\mathfrak{b}(u,t)$, $\mathfrak{c}(u,t)$ are all $C^1$ functions, and $0\le t\le T,0\le u\le L$. If the variable t is defined as the time, the functions $\mathfrak{a}(u,t),$$\mathfrak{b}(u,t)$, and $\mathfrak{c}(u,t)$ can then be explicated as oriented marching distances of a point at the time t in the directions ${\widehat{\mathbf{g}}}_1$, ${\widehat{\mathbf{g}}}_2$, and ${\widehat{\mathbf{g}}}_3$, respectively, and the vector $\widehat{\mathit{\psi }}\left(u\right)$ is the initialization of this point.

Our provocation is to confer necessary and sufficient situations for $\psi$$\left(u\right)$ as an isoparametric principal curve on S. First, let us define a unit vector $\mathbf{g}\left(u\right)$ orthogonal to the curve $\psi$$\left(u\right)$, that is,

$$\mathbf{g}\left(u\right)=cos\varphi {\mathbf{g}}_2\left(u\right)+sin\varphi {\mathbf{g}}_3\left(u\right),\mathrm{with}\varphi =\varphi \left(u\right).$$

(11)

Suppose that the ruled surface

$$\mathbf{y}(u,t)=\mathit{\psi }\left(u\right)+t\mathbf{g}\left(u\right);t\in \mathbb{R},$$

(12)

is a developable one, that is,

$$det({\mathit{\psi }}^{{}^{\prime }},\mathbf{g}\left(u\right),{\mathbf{g}}^{{}^{\prime }}\left(u\right))=\left|\begin{array}{ccc}1& 0& 0\\ 0& cos\varphi & cos\varphi \\ 0& -{\varphi }^{{}^{\prime }}sin\varphi -\tau sin\varphi & {\varphi }^{{}^{\prime }}cos\varphi +\tau cos\varphi \end{array}\right|=0,$$

From which we find

$${\varphi }^{{}^{\prime }}\left(u\right)+\tau \left(u\right)=0\Rightarrow \varphi \left(u\right)={\varphi }_0-\underset{u_0}{\overset{u}{\int }}\tau \left(u\right)du,$$

(13)

where ${\varphi }_0=\varphi \left(u_0\right)$ and $u_0$ is the initial value of arc length.

Second, since $\mathit{\psi }\left(u\right)$ is an isoparametric curve on S, there exists a value $t=t_0$ such that $\mathit{\psi }\left(u\right)=\mathbf{p}(u,t_0)$. Then, we have

$$\begin{array}{ccc}\hfill \mathfrak{a}(u,t_0)& =& \mathfrak{b}(u,t_0)=\mathfrak{c}(u,t_0)=0,\hfill \\ \hfill \frac{\partial \mathfrak{a}(u,t_0)}{\partial u}& =& \frac{\partial \mathfrak{b}(u,t_0)}{\partial u}=\frac{\partial \mathfrak{c}(u,t_0)}{\partial u}=0.\hfill \end{array}$$

Thus, the isotropic surface normal is

$$\mathbf{u}(u,t_0):=\frac{\partial \mathbf{p}(u,t_0)}{\partial u}\times \frac{\partial \mathbf{p}(u,t_0)}{\partial t}=-\frac{\partial \mathfrak{c}(u,t_0)}{\partial t}{\mathbf{g}}_2\left(u\right)+\frac{\partial \mathfrak{b}(u,t_0)}{\partial t}{\mathbf{g}}_3\left(u\right).$$

(14)

Moreover, via Monge’s Theorem, $\mathit{\psi }\left(u\right)$ is a principal curve on S if and only if $\mathbf{u}\left(u\right)$ is parallel to $\mathbf{g}(u,t_0)$. Therefore, from Equations (11) and (14), there exists a function $\chi \left(u\right)\ne 0$ such that

$$-\frac{\partial \mathfrak{c}(u,t_0)}{\partial t}=\chi \left(u\right)cos\varphi ,\frac{\partial \mathfrak{b}(u,t_0)}{\partial t}=\chi \left(u\right)sin\varphi ,$$

(15)

where $\varphi \left(u\right)$ is designated by Equation (13). The functions $\chi \left(u\right)$ and $\varphi \left(u\right)$ are controlling functions.

Hence, we give the following theorem.

Theorem 2.

The expression $\mathit{\psi }\left(u\right)$ is a principal curve on S if and only if

$$\left.\begin{array}{c}\mathfrak{a}(u,t_0)=\mathfrak{b}(u,t_0)=\mathfrak{c}(u,t_0)=0,0\le t_0\le T,0\le u\le L,\hfill \\ -\frac{\partial \mathfrak{c}(u,t_0)}{\partial t}=\chi \left(u\right)cos\varphi ,\frac{\partial \mathfrak{b}(u,t_0)}{\partial t}=\chi \left(u\right)sin\varphi ,\chi \left(u\right)\ne 0,\hfill \\ \varphi \left(u\right)={\varphi }_0-{\int }_{u_0}^u\tau \left(u\right)du,{\varphi }_0=\varphi \left(u_0\right),\hfill \end{array}\right\}$$

(16)

where $u_0$ is the starting value of the arc length.

Any surface $S:\mathbf{p}(u,t)$ recognized by Equation (9) and identified by Theorem 2 is an element of the surface pencil with $\mathit{\psi }\left(u\right)$ as joint principal curve. As reported in [8], for the purpose of resolution and experiment, we also examine the case when $\mathfrak{a}(u,t)$, $\mathfrak{b}(u,t)$, and $\mathfrak{c}(u,t)$ can be realized by

$$\mathfrak{a}(u,t)=\mathfrak{l}\left(u\right)\mathfrak{a}\left(t\right),\mathfrak{b}(u,t)=\mathfrak{m}\left(u\right)\mathfrak{b}\left(t\right),\mathfrak{c}(u,t)=\mathfrak{n}\left(u\right)\mathfrak{c}\left(t\right).$$

(17)

Here $\mathfrak{l}\left(u\right)$, $\mathfrak{m}\left(u\right),$$\mathfrak{n}\left(u\right)$, $\mathfrak{a}$$\left(t\right)$, $\mathfrak{b}$$\left(t\right)$, and $\mathfrak{c}$$\left(t\right)$ are ${\mathfrak{c}}^1$ functions that do not identically vanish. Then, from Theorem 2, we gain:

Corollary 1.

The expression$\mathit{\psi }\left(u\right)$ is a principal curve on S if and only if

$$\left.\begin{array}{c}\mathfrak{a}\left(t_0\right)=\mathfrak{b}\left(t_0\right)=\mathfrak{c}\left(t_0\right)=0,0\le t_0\le T,0\le u\le L,\hfill \\ -\mathfrak{n}\left(u\right)\frac{d\mathfrak{c}\left(t_0\right)}{dt}=\chi \left(u\right)cos\varphi ,\mathfrak{m}\left(u\right)\frac{d\mathfrak{b}\left(t_0\right)}{dt}=\chi \left(u\right)sin\varphi .\hfill \\ \varphi \left(u\right)={\varphi }_0-{\int }_{u_0}^u\tau \left(u\right)du,{\varphi }_0=\varphi \left(u_0\right),\hfill \end{array}\right\}$$

(18)

where $u_0$ is the starting value of the arc length.

However, we can assume that $\mathfrak{a}(u,t)$, $\mathfrak{b}(u,t)$, and $\mathfrak{c}(u,t)$ are based only on the variable t; that is, $\mathfrak{l}\left(u\right)=m\left(u\right)=n\left(u\right)=1$. Then, we treat Equation (18) via the dissimilar terms of $\varphi \left(u\right)$ as follows:

(i) If $\tau \left(u\right)\ne 0$, then $\varphi \left(u\right)$ is a non-stationary function of variable u and Equation (18) can be distinguished by

$$\left.\begin{array}{c}\mathfrak{a}\left(t_0\right)=\mathfrak{b}\left(t_0\right)=\mathfrak{c}\left(t_0\right)=0,\\ -\frac{d\mathfrak{c}\left(t_0\right)}{dt}=\chi \left(u\right)cos\varphi ,\frac{d\mathfrak{b}\left(t_0\right)}{dt}=\chi \left(u\right)sin\varphi .\end{array}\right\}$$

(19)

(ii) If $\tau \left(u\right)=0$, that is, the curve is a planar curve, then $\varphi \left(u\right)={\varphi }_0$ is a stationary and we have:

(a) If ${\varphi }_0\ne 0$, Equation (18) can be distinguished by

$$\left.\begin{array}{c}\mathfrak{a}\left(t_0\right)=\mathfrak{b}\left(t_0\right)=\mathfrak{c}\left(t_0\right)=0,\\ -\frac{d\mathfrak{c}\left(t_0\right)}{dt}=\chi \left(u\right)cos{\varphi }_0,\frac{d\mathfrak{b}\left(t_0\right)}{dt}=\chi \left(u\right)sin{\varphi }_0.\end{array}\right\}$$

(20)

(b) If ${\varphi }_0=0$, Equation (18) can be distinguished by

$$\left.\begin{array}{c}\mathfrak{a}\left(t_0\right)=\mathfrak{b}\left(t_0\right)=\mathfrak{c}\left(t_0\right)=0,\\ -\frac{d\mathfrak{c}\left(t_0\right)}{dt}=\chi \left(u\right),\frac{d\mathfrak{b}\left(t_0\right)}{dt}=0,\end{array}\right\}$$

(21)

and from Eqsuations (14) and (15), we have $\mathbf{u}(u,t_0)\Vert$g${}_2$. In this situation, the curve $\psi$ = $\psi$$\left(u\right)$ is not only a principal curve but also a geodesic. We also let {$\widehat{S}$, S} to indicate the surface pencil couple with a Bertrand couple {$\widehat{\mathit{\psi }}\left(u\right)$, $\psi$$\left(u\right)$} as joint principal curves.

Example 1.

Let $\mathit{\psi }\left(u\right)$ be an admissible helix specified by

$$\mathit{\psi }\left(u\right)=(u,sinu,cosu),0\le u\le 2\pi .$$

Then,

$${\mathit{\psi }}^{{}^{\prime }}\left(u\right)=(1,cosu,-sinu),{\mathit{\psi }}^{{}^{\prime \prime }}\left(u\right)=(0,-sinu,-cosu),{\mathit{\psi }}^{{}^{{}^{\prime \prime \prime }}}\left(u\right)=(0,-cosu,sinu).$$

In view of Equations (3)–(5), we gain $\kappa \left(u\right)=-\tau \left(u\right)=1$, and

$${\mathbf{g}}_1\left(u\right)=(1,cosu,-sinu),{\mathbf{g}}_2\left(u\right)=(0,-sinu,-cosu),{\mathbf{g}}_3\left(u\right)=(0,cosu,-sinu).$$

Then $\varphi \left(u\right)=-u+{\varphi }_0$. If ${\varphi }_0=0$, we gain $\varphi \left(u\right)=-u$. For

$$\begin{array}{ccc}\hfill \mathfrak{l}\left(u\right)& =& \mathfrak{m}\left(u\right)=\mathfrak{n}\left(u\right)=1,\hfill \\ \hfill \mathfrak{a}\left(t\right)& =& t,\mathfrak{b}\left(t\right)=-t\chi \left(u\right)sinu,\mathfrak{c}\left(t\right)=-t\chi \left(u\right)cosu,\chi \left(u\right)\ne 0.\hfill \end{array}$$

The surface pencil S with $\mathit{\psi }\left(u\right)$ is

$$S:\mathbf{p}(u,t)=(u,sinu,cosu)+t(1,-\chi sinu,-\chi cosu)\times \left(\begin{array}{ccc}1& cosu& -sinu\\ 0& -sinu& -cosu\\ 0& cosu& -sinu\end{array}\right).$$

The surface pencil $\widehat{S}$ with $\widehat{\mathit{\psi }}\left(u\right)$ as joint principal curve is as follows: Let $f=2$ in Equation (7), we derive $\widehat{\mathit{\psi }}\left(u\right)=(u,-sinu,-cosu)$. The Serret–Frenet vectors of $\widehat{\mathit{\psi }}\left(u\right)$ are

$${\widehat{\mathbf{g}}}_1\left(u\right)=(1,-cosu,sinu),{\widehat{\mathbf{g}}}_2\left(u\right)=(0,sinu,cosu),{\widehat{\mathbf{g}}}_3\left(u\right)=(0,-cosu,sinu).$$

Then,

$$\widehat{S}:\widehat{\mathbf{p}}(u,t)=(u,-sinu,-cosu)+t(1,-\chi sinu,-\chi cosu)\times \left(\begin{array}{ccc}1& -cosu& sinu\\ 0& sinu& cosu\\ 0& -cosu& sinu\end{array}\right).$$

Therefore, for $\chi \left(u\right)=1$, $-2\le$$t\le 2$, $0\le$$u\le 2\pi$, then {$\widehat{S}$, S} is exhibited in Figure 1, where the blue curve demonstrates$\psi$$\left(u\right)$ and the green curve is $\widehat{\mathit{\psi }}\left(u\right)$.

Example 2.

Suppose we are given an admissible curve by

$$\mathit{\psi }\left(u\right)=(u,1+sinu,sinu),0\le u\le 2\pi .$$

Then,

$${\mathbf{g}}_1\left(u\right)=(1,cosu,cosu),{\mathbf{g}}_2\left(u\right)=(0,-\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}),{\mathbf{g}}_3\left(u\right)=(0,\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}),$$

with $\kappa \left(u\right)=\sqrt{2}sinu$ and $\tau \left(u\right)=0$, which shows that $\varphi \left(u\right)={\varphi }_0$ is a stationary. For

$$\begin{array}{ccc}\hfill \mathfrak{l}\left(u\right)& =& \mathfrak{m}\left(u\right)=\mathfrak{n}\left(u\right)=1,\hfill \\ \hfill \mathfrak{a}\left(t\right)& =& t,\mathfrak{b}\left(t\right)=t\chi \left(u\right)sin{\varphi }_0,-\mathfrak{c}\left(t\right)=t\chi \left(u\right)cos{\varphi }_0,\chi \left(u\right)\ne 0.\hfill \end{array}$$

The surface pencil S over $\mathit{\psi }\left(u\right)$ is

$$S:\mathbf{p}(u,t)=(u,1+sinu,sinu)+t(1,-\chi \left(u\right)sin{\theta }_0,\chi cos{\theta }_0)\left(\begin{array}{ccc}1& cosu& cosu\\ 0& -\frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\\ 0& \frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\end{array}\right).$$

Similarly, let $f=\sqrt{2}$ in Equation (7), we acquire $\widehat{\mathit{\psi }}\left(u\right)=(u,sinu,sinu-1)$,and

$${\widehat{\mathbf{g}}}_1\left(u\right)=(1,cosu,cosu),{\widehat{\mathbf{g}}}_2\left(u\right)=(0,-\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}),{\widehat{\mathbf{g}}}_3\left(u\right)=(0,\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}),$$

Comparably, we have

$$\widehat{S}:\widehat{\mathbf{p}}(u,t)=(u,sinu,sinu-1)+t(1,-\chi \left(u\right)sin{\varphi }_0,\chi \left(u\right)cos{\varphi }_0)\times \left(\begin{array}{ccc}1& cosu& cosu\\ 0& -\frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\\ 0& \frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\end{array}\right).$$

For $\chi \left(u\right)=1$, ${\varphi }_0=0$, $-2.5\le$$t\le 2.5$, $0\le$$u\le 2\pi$, then {$\widehat{S}$, S} is exhibited in Figure 2, where the blue curve displays $\mathit{\psi }$$\left(u\right)$, and the green curve displays $\widehat{\psi }\left(u\right)$. Figure 3 specifies the {$\widehat{S}$, S} for ${\varphi }_0=\pi /2,$$-2.5\le$$t\le 2.5$, and $0\le$$u\le 2\pi$.

Ruled Surface Pencil Couple with Bertrand Couple as Joint Principal Curves

Suppose $S_i:{\mathbf{p}}_i(u,t)$ is a ruled surface with the directrix $\psi$${}_i\left(u\right)$ and $\psi$${}_i\left(u\right)$ is also an isoparametric curve of ${\mathbf{p}}_i(u,t)$; then there exists $t_0$ such that ${\mathbf{p}}_i(u,t_0)=$$\psi$${}_i\left(u\right)$. It follows that

$$S_i:{\mathbf{p}}_i(u,t)-{\mathbf{p}}_i(u,t_0)=(t-t_0){\mathbf{e}}_i\left(u\right),0\le u\le L,\mathrm{with}t,t_0\in [0,T],$$

(22)

where ${\mathbf{e}}_i\left(u\right)$($i=1,2,3)$ explains the orientation of the rulings. In view of Equation (9), we gain

$$(t-t_0){\mathbf{e}}_i\left(u\right)=\mathfrak{a}(u,t){\mathbf{g}}_{1i}\left(u\right)+\mathfrak{b}(u,t){\mathbf{g}}_{2i}\left(u\right)+\mathfrak{c}(u,t){\mathbf{g}}_{3i}\left(u\right),$$

(23)

where $0\le u\le L,$ with $t,t_0\in [0,T]$. In fact, Equation (22) is a regulation of equations with the three unknowns $\mathfrak{a}(u,t),$$\mathfrak{b}(u,t)$, and $\mathfrak{c}(u,t)$. The resolutions can be deduced as

$$\begin{array}{c}\mathfrak{a}(u,t)=(t-t_0)<{\mathbf{e}}_i\left(u\right),{\mathbf{g}}_{1i}\left(u\right)>,\\ \mathfrak{b}(u,t)=(t-t_0)<{\mathbf{e}}_i\left(u\right),{\mathbf{g}}_{2i}\left(u\right)>,\\ \mathfrak{c}(u,t)=(t-t_0)<{\mathbf{e}}_i\left(u\right),{\mathbf{g}}_{3i}\left(u\right)>.\end{array}$$

(24)

In view of Equation (15), if $\psi$$\left(u\right)$ is a principal curve on the surface ${\mathbf{p}}_i(u,t)$, we get

$$\begin{array}{c}\mathfrak{a}(u,t)=0,\\ \chi \left(u\right)sin\varphi =<{\mathbf{e}}_i\left(u\right),{\mathbf{g}}_{2i}\left(u\right)>,\\ -\chi \left(u\right)cos\varphi =<{\mathbf{e}}_i\left(u\right),{\mathbf{g}}_{3i}\left(u\right)>.\end{array}$$

(25)

The above regulations are clearly the necessary and sufficient situations for which ${\mathbf{p}}_i(u,t)$ is a ruled surface with a directrix $\psi$${}_i\left(u\right)$; $i=1,2,3$.

In Galilean 3-space ${\mathbb{G}}_3$, there exist only three types of ruled surfaces, specified as follows [22,23]:

Type I. Non-conoidal or conoidal ruled surfaces for which the wrist (striction) curve does not lie in an Euclidean plane;

Type II. Ruled surfaces for which the wrist curve is in an Euclidean plane;

Type III. Conoidal ruled surfaces for an absolute line as the oriented line in infinity.

We immediately research if the curve $\psi$${}_i\left(u\right)$ is also a principal curve on these three types:

Type I: $\psi$${}_1\left(u\right)=(u,{\psi }_2\left(u\right),$${\psi }_3\left(u\right))$ does not lie in an Euclidean plane and ${\mathbf{e}}_1\left(u\right)=$$(1,e_2\left(u\right),e_3\left(u\right))$ is non-isotropic. Then,

$$\begin{array}{ccc}\hfill {\mathbf{g}}_{11}\left(u\right)& =& \left(1,{\psi }_2^{{}^{\prime }}\left(u\right),{\psi }_3^{{}^{\prime }}\left(u\right)\right),\hfill \\ \hfill {\mathbf{g}}_{21}\left(u\right)& =& \frac{1}{\kappa \left(u\right)}\left(0,{\psi }_2^{{}^{\prime \prime }}\left(u\right),{\psi }_3^{{}^{\prime \prime }}\left(u\right)\right),\hfill \\ \hfill {\mathbf{g}}_{31}\left(u\right)& =& \frac{1}{\kappa \left(u\right)}\left(0,-{\psi }_3^{{}^{\prime \prime }}\left(u\right),{\psi }_2^{{}^{\prime \prime }}\left(u\right)\right),\hfill \end{array}$$

(26)

where $\kappa \left(u\right)=\sqrt{{\left({\psi }_2^{{}^{\prime \prime }}\left(u\right)\right)}^2+{\left({\psi }_3^{{}^{\prime \prime }}\left(u\right)\right)}^2}$. From Equations (1), (24), and (26), we have:

$$\mathfrak{a}(u,t)=(t-t_0),\mathfrak{b}(u,t)=\mathfrak{c}(u,t)=0,$$

(27)

which does not satisfy Theorem 2.

Type II: $\psi$${}_2\left(u\right)=(0,{\psi }_2\left(u\right)),z\left(u\right))$ lie in an Euclidean plane and ${\mathbf{e}}_2\left(u\right)=(1,e_2\left(u\right),e_3\left(u\right))$ is non-isotropic. Then,

$$\begin{array}{ccc}\hfill {\mathbf{g}}_{12}\left(u\right)& =& \left(0,{\psi }_2^{{}^{\prime }}\left(u\right),{\psi }_3^{{}^{\prime }}\left(u\right)\right),\hfill \\ \hfill {\mathbf{g}}_{22}\left(u\right)& =& \frac{1}{\kappa \left(u\right)}\left(0,{\psi }_2^{{}^{\prime \prime }}\left(u\right),{\psi }_3^{{}^{\prime \prime }}\left(u\right)\right),\hfill \\ \hfill {\mathbf{g}}_{32}\left(u\right)& =& \frac{1}{\kappa \left(u\right)}\left(0,0,0\right),\hfill \end{array}$$

(28)

where $\kappa \left(u\right)=\sqrt{{\left({\psi }_2^{{}^{\prime \prime }}\left(u\right)\right)}^2+{\left({\psi }_3^{{}^{\prime \prime }}\left(u\right)\right)}^2}$. From Equations (1), (24), and (28), we gain:

$$\mathfrak{a}(u,t)=\mathfrak{b}(u,t)=\mathfrak{c}(u,t)=0,$$

(29)

which does not satisfy Theorem 2.

Corollary 2.

In ${\mathbb{G}}_3$, there are no ruled surface pencil couples of Type I and Type II with Bertrand couples as joint principal curves.

Type III: $\psi$${}_3\left(u\right)=(u,{\psi }_2\left(u\right),0)$ does not lie in an Euclidean plane and ${\mathbf{e}}_3\left(u\right)=(0,e_2\left(u\right),e_3\left(u\right))$ is non-isotropic. Then,

$$\begin{array}{ccc}\hfill {\mathbf{g}}_{13}\left(u\right)& =& \left(1,{\psi }_2^{{}^{\prime }}\left(u\right),0\right),\hfill \\ \hfill {\mathbf{g}}_{23}\left(u\right)& =& \frac{1}{\kappa \left(u\right)}\left(0,{\psi }_2^{{}^{\prime \prime }}\left(u\right)0\right),\hfill \\ \hfill {\mathbf{g}}_{33}\left(u\right)& =& \frac{1}{\kappa \left(u\right)}\left(0,0,{\psi }_2^{{}^{\prime \prime }}\left(u\right)\right),\hfill \end{array}$$

(30)

where $\kappa \left(u\right)=\sqrt{{\left({\psi }_2^{{}^{\prime \prime }}\left(u\right)\right)}^2}$. From Equations (1), (24), and (29), we have:

$$\left.\begin{array}{c}\mathfrak{a}(u,t)=0,\mathfrak{b}(u,t)=ϵ(t-t_0)e_2\left(u\right),\hfill \\ \mathfrak{c}(u,t)=ϵ(t-t_0)e_3\left(u\right),\hfill \\ e_2\left(u\right)\ne 0,e_3\left(u\right)\ne 0,t_0\ne 0,\hfill \end{array}\right\}$$

(31)

where

$$ϵ=\left\{\begin{array}{c}1,\mathrm{if}{\psi }_2^{{}^{\prime \prime }}\left(u\right)>0.\\ -1,\mathrm{if}{\psi }_2^{{}^{\prime \prime }}\left(u\right)<0.\end{array}\right.$$

(32)

Equation (31) satisfies Theorem 2. Suppose at all points on $\psi$${}_3\left(u\right)$, the ruling ${\mathbf{e}}_3\left(u\right)\in up\{{\mathbf{g}}_{13}\left(u\right),{\mathbf{g}}_{23}\left(u\right)$, ${\mathbf{g}}_{33}\left(u\right)\}$, then

$${\mathbf{e}}_3\left(u\right)=\zeta \left(u\right){\mathbf{g}}_{13}\left(u\right)+\sigma \left(u\right){\mathbf{g}}_{23}\left(u\right)+\mu \left(u\right){\mathbf{g}}_{33}\left(u\right),$$

(33)

for some functions $\lambda \left(u\right)$, $\sigma \left(u\right)$, and $\mu \left(u\right)$. Replacing it into Equation (25), we get

$$\sigma \left(u\right)=\chi \left(u\right)sin\varphi ,\mu \left(u\right)=-\chi \left(u\right)cos\varphi .$$

(34)

Then,

$${\mathbf{e}}_3\left(u\right)=\lambda \left(u\right){\mathbf{g}}_{13}\left(u\right)+\chi \left(u\right)sin\varphi {\mathbf{g}}_{23}\left(u\right)-\chi \left(u\right)cos\varphi {\mathbf{g}}_{33}\left(u\right).$$

(35)

Choosing $\mathfrak{a}(u,t)=t\lambda \left(u\right)$, $\mathfrak{b}(u,t)=t\chi \left(u\right)sin\varphi$, and $\mathfrak{c}(u,t)=-t\chi \left(u\right)cos\varphi$, the ruled surface pencil $S_3$ with $\psi$${}_3\left(u\right)$ can be shown by

$${\mathbf{p}}_3(u,t)={\mathit{\psi }}_3\left(u\right)+t\zeta \left(u\right){\mathbf{g}}_{13}\left(u\right)+t\lambda \left(u\right)(sin\varphi {\mathbf{g}}_{23}\left(u\right)-cos\varphi {\mathbf{g}}_{33}\left(u\right)),0\le u\le L,0\le t\le T.$$

(36)

Then, the ruled surface pencil ${\widehat{S}}_3$ is

$${\widehat{\mathbf{p}}}_3(u,t)={\widehat{\mathit{\psi }}}_3\left(\widehat{u}\right)+t\lambda \left(u\right){\widehat{\mathbf{g}}}_{13}\left(u\right)+t\chi \left(u\right)(sin\varphi {\widehat{\mathbf{g}}}_{23}\left(u\right)-cos\varphi {\widehat{\mathbf{g}}}_{33}\left(u\right)),0\le u\le L,0\le t\le T.$$

(37)

The functions $\lambda \left(u\right)$ and $\chi \left(u\right)$ can control the shape of the surfaces family $S_3$ and ${\widehat{S}}_3$.

Example 3.

Via Example 1, we have:

By taking $\lambda \left(u\right)=\chi \left(u\right)=u$, then {${\widehat{S}}_3$, $S_3$} with {${\widehat{\mathit{\psi }}}_3\left(u\right)$,$\psi$${}_3\left(u\right)$} as joint Bertrand principal curves is (Figure 4):

$$\left\{\begin{array}{c}{\mathbf{p}}_3(u,t)=(u(1+t),sinu+tu(cos-cos2u),cosu+tu(-sinu+sin2u)),\\ {\widehat{\mathbf{p}}}_3(u,t)=(u(1+t),-sinu+tu(-cos+cos2u),-cosu+tu(sinu-sin2u)),\end{array}\right.$$

where the blue curve demonstrates $\psi$${}_3\left(u\right)$, the green curve is ${\widehat{\mathit{\psi }}}_3\left(u\right)$, $-0.7\le t\le 0.7$, and $0\le u\le 2\pi$.

By taking $\lambda \left(u\right)=\chi \left(u\right)=\sqrt{u}$, then {${\widehat{S}}_3$, $S_3$} with {${\widehat{\mathit{\psi }}}_3\left(u\right)$,$\psi$${}_3\left(u\right)$} as joint Bertrand principal curves is (Figure 5):

$$\left\{\begin{array}{c}{\mathbf{p}}_3(u,t)=(\sqrt{u}(\sqrt{u}+t),sinu+t\sqrt{u}(cos-cos2u),cosu+t\sqrt{u}(-sinu+sin2u)),\\ {\widehat{\mathbf{p}}}_3(u,t)=(\sqrt{u}(\sqrt{u}+t),-sinu+t\sqrt{u}(-cos+cos2u),-cos\sqrt{u}+tu(sinu-sin2u)),\end{array}\right.$$

where the blue curve demonstrates $\psi$${}_3\left(u\right)$, the green curve is ${\widehat{\mathit{\psi }}}_3\left(u\right)$, $-0.7\le t\le 0.7$, and $0\le u\le 2\pi$.

Example 4.

Via Example 2, we have:

(1) By taking ${\varphi }_0=0$, and $\lambda \left(u\right)=\chi \left(u\right)=u$, then {${\widehat{S}}_3$, $S_3$} with {${\widehat{\mathit{\psi }}}_3\left(u\right)$,$\psi$${}_3\left(u\right)$} as joint Bertrand principal curves is (Figure 6):

$$\left\{\begin{array}{c}{\mathbf{p}}_3(u,t)=(u(1+t),1+sinu+tu(cos-\frac{1}{\sqrt{2}}),sinu+tu(cosu-\frac{1}{\sqrt{2}})),\\ {\widehat{\mathbf{p}}}_3(u,t)=u(1+t),sinu+tu(cos-\frac{1}{\sqrt{2}}),sinu-1+tu(cosu-\frac{1}{\sqrt{2}})),\end{array}\right.$$

where the blue curve demonstrates $\psi$${}_3\left(u\right)$, the green curve is ${\widehat{\mathit{\psi }}}_3\left(u\right)$, $-0.2\le t\le 0.2$, and $0\le u\le 2\pi$

(2) By taking ${\varphi }_0=\pi /2$, and $\zeta \left(u\right)=\chi \left(u\right)=u$, then {${\widehat{S}}_3$, $S_3$} with {${\widehat{\mathit{\psi }}}_3\left(u\right)$,$\psi$${}_3\left(u\right)$} as joint Bertrand principal curves is (Figure 7):

$$\left\{\begin{array}{c}{\mathbf{y}}_3(u,t)=(u(1+t),1+sinu+tu(cos+\frac{1}{\sqrt{2}}),sinu+tu(cosu+\frac{1}{\sqrt{2}})),\\ {\widehat{\mathbf{y}}}_3(u,t)=u(1+t),sinu+tu(cos+\frac{1}{\sqrt{2}}),sinu-1+tu(cosu+\frac{1}{\sqrt{2}})),\end{array}\right.$$

where the blue curve demonstrates $\psi$${}_3\left(u\right)$, the green curve is ${\widehat{\mathit{\psi }}}_3\left(u\right)$, $-0.2\le t\le 0.2$, and $0\le u\le 2\pi .$

4. Conclusions

This paper studied principal curves and their associated surfaces in Galilean 3-space. Given a 3D curve, we seek surfaces that are faced with this curve as a distinctive curve. The work viewed the Bertrand couples as principal curves and procures a surface pencil couple with a Bertrand couple as joint principal curves. Then, necessary and suffcient conditions for an admissible curve to be an isoparametric principal curve on surface are newly derived in this case. Examples of the surface pencil with principal curves are shown in the figures with a Bertrand couple. Our results in this paper contribute to the work by Jiang et al. [30]. We hope that these outcomes will open modern perceptions for researchers working on geometrical modeling and production evolution procedure in the manufacturing industry. The judgment of stratifying the mechanisms applied here to various spaces such as Lorentz-space, pseudo-Galilean space, and Heisenberg space is already an investigation topic. We will discuss this problem in the future.