Tubular surface generated by a curve lying on a regular surface and its characterizations

1.   Introduction

The envelope of a moving sphere with variable radius is characterized as a canal surface, which is frequently used in computer-aided design (CAD) and computer-aided geometric design (CAGD) for solid and surface modeling. A canal surface is an envelope of a one-parameter set of spheres centered at the center curve $c(s)$ with radius $r(s)$. The spheres that are specified by the radius function $r(s)$ and the center curve $c(s)$ are combined to form a canal surface, which is obtained by the spine curve $c(s)$. These surfaces have a wide range of uses, including form reconstruction, robot movement planning, the creation of blending surfaces, and the easy sight of long and thin objects like pipes, ropes, poles, and live intestines. The term "tubular surface" refers to these canal surfaces if the radius function $r(s)$ is constant (for more details, see ^{[1,2,3,4,5,6,7,8]}).

Tubular surfaces are one of the enormous vital subjects of surface theory. In $\mathbb{R}^{3}$, a tubular surface is a fundamental and well-known device that is used for geometric construction. Due to this place of tubular surfaces, numerous geometers and designers have explored and acquired numerous properties of tubular surfaces, see for instance ^[9,10,11,12].

In this article, we investigate the geometric conditions for the tubular surfaces to have generic singularities as a front (i.e., cuspidal lips, cuspidal beaks, and Swallowtails). Moreover, we study the tubular Weingarten surfaces which fulfill nontrivial connection between components of the set $\{K, K_{II}, H, H_{II}\}$, where $(K, H)$ and $(K_{II}, H_{II})$ are Gaussian curvatures.

The paper can be organized as follows: We provide a brief review of the geometry of surfaces, particularly Frenet and Darboux frames related to our study of tubular surfaces in Section 2. In Section 3, we investigate the singularities of tubular surfaces with a Darboux frame and provide some findings from these surfaces. Section 4 provides tubular Weingarten and linear Weingarten surfaces (W-and LW-surfaces) in accordance with a nontrivial functional relation between their curvatures. To enhance our findings and provide a practical demonstration, we include some computational examples in Section 5. These examples not only serve to illustrate our primary results but also feature graphical representations for clarity.

2.   Preliminaries

In this part, we show a few ideas, equations, and summaries of curves and surfaces in $\mathbb{R}^{3}$ which can be tracked down in the course readings on differential geometry, see ^[1,2,3]. A curve is regular if it admits a tangent line at each point of the curve. In the following, all curves are assumed to be regular. Let $\mathbf{\alpha } (s):I\subseteq \mathbb{\mathbb{R}}\rightarrow \; \mathbb{R}^{3}$ be a unit speed curve in $\mathbb{R}^{3}$; by $\kappa (s)$ and $\tau (s)$ we denote the natural curvature and torsion of $\mathbf{\alpha }$, respectively. The Frenet equations are:

The Darboux frame is an alternative approach to defining a new moving frame constructed on a surface. One can exist on a surface in Euclidean or non-Euclidean spaces ^[13]. The Darboux frame of $\mathbf{\alpha } = \mathbf{\alpha }(s)$ is expressed as follows:

and the relation matrix between Serret-Frenet and Darboux frames is given by

where $\kappa _{g}$ is the geodesic curvature, $\kappa _{n}$ is the normal curvature, and $\tau _{g}$ is the geodesic torsion of $\alpha (s)$.

They are defined as:

In addition, $\kappa _{g}$ and $\tau _{g}$ can be calculated as follows:

Let ${\boldsymbol{n}}(u, v)$ be the standard unit normal vector field on a surface $\mathbf{\Upsilon } = \mathbf{\Upsilon }(u, v)$ defined by $\mathbf{n = }\frac{ \mathbf{\Upsilon }_{u}\times \mathbf{\ X}_{v}}{\left\Vert \mathbf{\Upsilon } _{u}\times \mathbf{\Upsilon }_{v}\right\Vert }$, where $\mathbf{\Upsilon } _{i} = \frac{\partial \mathbf{\Upsilon }}{\partial u_{i}}$ and $i = 1, 2$. Therefore, the metric $(I)$ of $\mathbf{\Upsilon }$ is

where $g_{11} = \langle\mathbf{\Upsilon }_{u}, \mathbf{\Upsilon }_{u}\rangle$, $g_{12} = \langle \mathbf{\Upsilon }_{u}, \mathbf{\Upsilon }_{v}\rangle,$ and $g_{22} = \langle\mathbf{\Upsilon } _{v}, \mathbf{\Upsilon }_{v}\rangle$. Also, the $2^{nd}$ fundamental form $(II)$ of $\mathbf{\Upsilon }$ is

where $h_{11} = \langle\mathbf{\Upsilon }_{uu}, {\boldsymbol{n}}\rangle$, $h_{12} = \langle\mathbf{ \Upsilon }_{uv}, {\boldsymbol{n}}\rangle$, and $h_{22} = \langle\mathbf{\Upsilon }_{vv}, {\boldsymbol{n}}\rangle$. The Gaussian curvature $K$, and mean curvature $H$ are respectively, expressed as:

From Brioschi's formula ^[14,15], the second Gaussian curvature $K_{II}$ is expressed as

where $h = \det (h_{ij}), \; h_{ij, \alpha } = \frac{\partial h_{ij}}{\partial u^{\alpha }},$ and $h_{ij, \alpha \beta } = \frac{\partial ^{2}h_{ij}}{\partial u^{\alpha }\partial u^{\beta }}$. Furthermore the second mean curvature $H_{II}$ is

where

2.1.   Criteria for singularities of fronts

In this subsection, we will utilize a similar strategy on the peculiarity hypothesis for groups of double smooth capabilities. Nitty gritty depictions are viewed as in the books ^[16,17]. Let $U\subset \mathbb{R}^{2}$ be an open set and $f:(U, p)\rightarrow (\mathbb{R}^{3}, \mathbf{0})$ a map germ. Two map germs $f_{i}$: $(\mathbb{R}^{2}$, $\mathbf{0})\rightarrow (\mathbb{R}^{3}, \mathbf{0})$ (i = 1, 2), are $\mathcal{A}$-equivalent if there exist diffeomorphism germs $g_{1}:$($\mathbb{R}^{2}$, $\mathbf{0})\rightarrow (\mathbb{R}^{2}, \mathbf{0})$, and $g_{2}:$($\mathbb{R}^{3}$, $\mathbf{0})\rightarrow (\mathbb{R}^{3}, \mathbf{0})$ such that $f_{2}\circ g_{1} = g_{2}\circ f_{1}$ holds. A map germ $f:U\subseteq \mathbb{R}^{2}\rightarrow \mathbb{R}^{3}$ is called a (wave) front if there exists a unit vector field $\nu$ of $\mathbb{ \ R}^{3}$ along $f$ such that $\mathfrak{L} = \langle f\mathbf{, }\ \nu \rangle$ is a Legendrian immersion. Since $\mathfrak{L} = \langle f, \nu \rangle$ is Legendrian,

hold. For a front $f$, we define a function $\lambda :U\subset \mathbb{R} ^{2}\rightarrow \mathbb{R}$ by $\lambda (u, v) = \det (f_{u}, f_{v}, \nu \mathbf{) }$. The function $\lambda$ is called a discriminant function of $f$.

We call $p\in U$ a singular point of $f$ if rank$(df_{p})\leq 1$. The set of singular points $S(f)$ of $f$ is the zero set of $\lambda$. A singular point $p\in U$ of $f$ is said to be non-degenerate if $d\lambda (p)\neq 0$. Let $p$ be a non-degenerate singular point of a front $f$. Then $S(f)$ is parameterized by a regular curve $\gamma (t):(-\varepsilon, \varepsilon)\rightarrow U$ near $p$. Moreover, there exists a non-vanishing vector field $\eta$ along $\gamma$ such that $df\left(\eta (t)\right) = 0$. This vector field $\eta$ is called a null vector. For further details, see ^[11,12]. Under these notations, we present the criterion for the cuspidal edges, Swallowtails, and cuspidal butterfly as follows:

Proposition 2.1. Let $f:U\subseteq\mathbb{R} ^{2}\rightarrow \mathbb{R}^{3}$ be a front and $p$ a non-degenerate singular point of $f$. Then we have:

(1) $f$ is $\mathcal{A}$-equivalent to cuspidal edge $CE$ at $p$ if and only if $\eta \lambda (p)\neq 0$, where $\eta \lambda$ means the directional derivative $D_{\eta }\lambda;$

(2) $f$ is $\mathcal{A}$-equivalent to Swallowtail $SW$ at $p$ if and only if $\eta \lambda (p) = 0$, and $\eta ^{2}\lambda (p)\neq 0,$

(3) $f$ is $\mathcal{A}$-equivalent to cuspidal butterfly$\ CBF$ at $p$ if and only if $\eta \lambda (p) = \eta ^{2}\lambda (p) = 0$, and $\eta ^{3}\lambda (p)\neq 0$.

Here,

These surfaces are shown in the following figures (see Figures 1, 2a and 2b, respectively).

Now, we turn to degenerate singularities. Let $p$ be a degenerate singular point of the front $f$. If rank$(df_{p}) = 1$, then there exists $\eta$ near $p$; if $q\in S(f)$, then $df_{q}\left(\eta (t)\right) = 0$. Criteria for degenerate singularities are as follows:

Proposition 2.2. Let $f:U\subseteq \mathbb{R}^{2}\rightarrow \mathbb{R}^{3}$ be a front and $p$ a degenerate singular point of $f$. Then we have the following:

(1) $f$ is $\mathcal{A}$-equivalent to cuspidal lips $CLP$ if and only if rank$(df_{p}) = 1$, and the $\det \left(\mathcal{H}\lambda (p)\right) > 0$, where $\det \left(\mathcal{H}\lambda (p)\right)$ denotes the determinant of the Hessian matrix of $\lambda$ at $p$ (see Figure 3b);

(2) $f$ is $\mathcal{A}$-equivalent to cuspidal beaks $CBK$ if and only if rank$(df_{p}) = 1$, $\det \left(\mathcal{H}\lambda (p)\right) < 0$, and $\eta ^{2}\lambda (p)\neq 0$ (see Figure 3a).

Here, for a function $\lambda :(U, u, v)\rightarrow \mathbb{R}$, $\mathcal{H} \lambda$ is the matrix defined by

and

3.   Singularities of tubular surfaces

Here, we study the singularity of a tubular surface, and give the conditions for this surface to be CE, SW, CBF, CLP, and CBK singularities in terms of $\kappa _{g}(s), \kappa _{n}(s)$, and $\tau _{g}(s)$. By utilizing the Darboux frame, the tubular surfaces of radius $r > 0$ about the $\beta (s) = \int {\boldsymbol{T}}\left(s\right) ds$ is the surface with parametrization

3.1.   Local singularities

Singularties are essential for understanding the properties of tubular surfaces. So, after simple calculations, we have

From Eq (3.2), we can show that $\mathbf{\Upsilon }$ has a singularity at $\mathbf{Q}\left(s, \varphi \right)$ if and only if

This is equivalent to

Therefore, the Hessian matrix of $\lambda (s, \varphi)$ at a singular point is given by

and

Now it is easy to prove the following lemma.

Lemma 3.1. The tubular surface $\mathbf{\Upsilon }$ parameterized by Eq (3.1) is a front.

Lemma 3.2. Let $\mathbf{\Upsilon }$ be the tubular surface parameterized by Eq (3.1). Then rank$(dQ_{p})\leq 1$ at $p$ if and only if

is satisfied.

Proof. We suppose that a tubular surface $\mathbf{\Upsilon }$ parameterized by Eq (3.1) has singularity at $p$. Then it satisfies the Eq (3.3). Conversely, if Eq (3.3) holds, in view of Eq (3.2), then

which means that rank$(dQ_{p})\leq 1$. This completes the proof.

Proposition 3.1. Let $\mathbf{\Upsilon }$ be a tubular surface parameterized by Eq (3.1). If $p$ is a non-degenerate singular point for $\mathbf{\Upsilon }$, then $d\lambda (p)\neq 0$ if and only if

or

is satisfied.

Proof. Suppose that $\mathbf{\Upsilon }$ be the tubular surface parameterized by Eq (3.1). Then, it has singularity at $p$ if and only if Eq (3.3) holds. Conversely, if Eq (3.3) holds, it is clear that:

which means that $p$ is a non-degenerate singular point $d\lambda (p)\neq 0$ if and only if $\kappa _{g}^{^{\prime }}\cos \varphi -\kappa _{n}^{^{\prime }}\sin \varphi \neq 0$ or $\kappa _{g}\sin \varphi +\kappa _{n}\cos \varphi \neq 0$. Hence, the proof is completed.

So, we find the following result.

Corollary 3.1. Let $p$ be a degenerate singular point for $\mathbf{\Upsilon }$. Then we have $d\lambda (p) = 0$ if and only if $\kappa _{g}^{^{\prime }}/\kappa _{n}^{^{\prime }} = \tan \varphi,$ and $\kappa _{g}/\kappa _{n} = -\cot \varphi$.

Now we are ready to state our main theorems:

Theorem 3.1. Let $p$ be a non-degenerate singular point for $\mathbf{\Upsilon }$. Then

(1) $\mathbf{\Upsilon }$ is $\mathcal{A}$-equivalent to the $CE$ at $p$ if and only if $\eta \lambda (p)\neq 0$, that is,

(2) $\mathbf{\Upsilon }$ is $\mathcal{A}$-equivalent to the $SW$ at $p$ if and only if $\eta \lambda (p) = 0$, and $\eta ^{2}\lambda (p)\neq 0$, that is,

and

(3) $\mathbf{\Upsilon }$ is $\mathcal{A}$-equivalent to the $CBF$ at $p$ if and only if $\eta \lambda (p) = \eta ^{2}\lambda (p) = 0$, and $\eta ^{3}\lambda (p)\neq 0$, that is,

and

Proof. (1) Since $p$ is a singular point of $\mathbf{\Upsilon }$, we have

Because $p$ is a non-degenerate singular point, the null vector filed $\eta$ is defined as

Therefore,

So, we get: $\eta \lambda (p)\neq 0$ if and only if

(2) Similarly, we have:

By using Case (1), we have: $\eta \lambda (p) = 0$, and $\eta ^{2}\lambda (p)\neq 0$ if and only if

and

(3) By a similar procedure as in Case (1) and Case (2), we have $\eta \lambda (p) = \eta ^{2}\lambda (p) = 0$, and $\eta ^{3}\lambda (p)\neq 0$, if and only if

and

Therefore, using Proposition 1, the proof is complete.

Theorem 3.2. Let $\mathbf{\Upsilon }$ be a tubular surface parameterized by Eq (3.1), and $p$ is a degenerate singular point. Then, one has the followings:

(1) $\mathbf{\Upsilon }$ is $\mathcal{A}$-equivalent to $CLP$ if and only if rank$(df_{p}) = 1$, and

(2) $\mathbf{\Upsilon }$ is $\mathcal{A}$-equivalent to $CBK$ if and only if rank$(df_{p}) = 1$,

and

Proof. Let $p$ be a degenerate singular point of $\mathbf{\Upsilon }$, then Eqs (3.7) and (3.8) are hold. Therefore, using Proposition 2, the proof is complete.

3.2.   The relation among the curvature functions

Presently, we concentrate on tubular surfaces fulfilling a few conditions concerning their curvatures as follows:

According to Eqs $(3.2)$, we find

The normal vector of $\mathbf{Q}$ is

By a straightforward calculation, we get

This prompts the components of $(II)$; $h_{11}$, $h_{12}$, and $h_{22}$ as follows:

Therefore, we get

where $\mu = \left(1-r\ \kappa _{g}\cos \varphi -r\ \kappa _{n}\sin \varphi \right) ^{2}.$ And from Eq (3.16), we get

From Eqs (2.6), (3.16), and (3.18), we find

where

Also, we have:

From Eqs (3.17), (3.19), and (3.20), we obtain the result:

Corollary 3.2. The Gaussian curvatures of $\mathbf{\Upsilon }$ are

and

4.   Tubular LW-surfaces

Now, for a tubular LW-surfaces $\mathbf{\Upsilon }$, an extension of Eq (3.1) for a nontrivial functional relation between a pair $\{A, B\}, \; A\neq B,$ of the curvatures $K, \ K_{II}, \ H$, and $H_{II}$ are studied. Thus, by using Eqs (3.17), (3.19) and (3.20), one can get the differentiation of $K, \ K_{II}$, $H$, and $H_{II}$ concerning $s$ and $\varphi$. In any case, the upsides of these estimations are long to such an extent that we can overlook them. In this manner, we have the accompanying cases:

$\left(i\right) f(K, H) = \left(K\right) _{s}\left(H\right) _{\varphi }-\left(K\right) _{\varphi }\left(H\right) _{s}\ = 0,$

$\left(ii\right) f(K, K_{II}) = \left(K\right) _{s}\left(K_{II}\right) _{\varphi }-\left(K\right) _{\varphi }\ \left(K_{II}\right) _{s} = 0,$

$\left(iii\right) f(H, K_{II}) = \left(H\right) _{s}\left(K_{II}\right) _{\varphi }-\left(H\right) _{\varphi }\ \left(K_{II}\right) _{s} = 0,$

$\left(iv\right) f(H\, H_{II}) = \left(H\right) _{s}\left(H_{II}\right) _{\varphi }-\left(H\right) _{\varphi }\ \left(H_{II}\right) _{s} = 0,$

$\left(v\right) f(K_{II}\, H_{II}) = \left(K_{II}\right) _{s}\left(H_{II}\right) _{\varphi }-\left(K_{II}\right) _{\varphi }\left(H_{II}\right) _{s} = 0.$

From the primary case, one can see that it has evaporated indistinguishably. Hence we have

Corollary 4.1. The tubular surface $\mathbf{\Upsilon }$ is a W-surface.

From the second and third cases, one can get the two Jacobian equations, and we conclude that $\kappa_{g}^{\prime } = \kappa_{n}^{\prime } = 0,$ which leads to $\kappa_{g} = \kappa_{n} = constant$. Consequently, we obtain the following result.

Theorem 4.1. The tubular surface $\mathbf{\Upsilon }$ is a W-surface generated by a circle.

Similarly, from the fourth and fifth cases, the two Jacobian equations are split to sixteen conditions and satisfied when $\kappa_{g} = \tau_{g} = constant$. Subsequently, we find the following theorem.

Theorem 4.2. The tubular surface M is a W-surface generated by a circular helix $\mathbf{ \ \alpha }$ with non-zero constant curvatures (see Figure 4).

Finally, one can see that the following linear relations hold:

Theorem 4.3. For a tubular surface, the following hold:

$\left(i\right) \; aK+bH = c,$ where $a+c\lambda ^{2}\neq 0$ and $\kappa_{g} = \kappa_{n} = 0,$

$\left(ii\right) \; aK+bK_{II} = c \;,$ where $b = c = 0$ and $\kappa_{g} = \kappa_{n} = 0,$

$\left(iii\right) \; aH+bK_{II} = c,$ where $b = 0$, and $\kappa = 0,$

$\left(iv\right) \; aH+bH_{II} = c,$ where $a+b+c\lambda \neq 0, \; \tau_{g} \neq 0$, and $\kappa_{g} = \kappa_{n} = 0,$

$\left(v\right) \; aH_{II}+bK_{II} = c,$ where $a+b+c\lambda \neq 0, \; \tau_{g} \neq 0$, and $\kappa_{g} = \kappa_{n} = 0.$

Here, $a, b$, and $c$ are non-zero arbitrary constants.

As a result, we give the following corollary:

Corollary 4.2. The tubular surface M is an open part of a circular cylinder.

5.   Applications

Now, we will introduce two computational examples for constructing tubular surfaces to support our main results.

Example 5.1. Consider the regular surface parameterized by

Darboux frame vectors of the curve $\mathbf{\alpha }(s) = \left(s, \cos \left(s\right), \sin \left(s\right) \right)$, which lies on the regular surface $S_{1}$, are

Also, we have

Thus, according to Eq (3.1), the constructed tubular surface $\mathbf{ \Upsilon }_{1}(s, v)$ associated with the Darboux frame of radius $r > 0$ along $\mathbf{\alpha }(s)$ is parametrized by (see Figure 4b):

For $\mathbf{\Upsilon } _{1}$, we obtain

Moreover, we obtain

Example 5.2. Let us consider the regular surface parameterized by

Darboux frame vectors of the curve $\mathbf{\beta }(s) = \left(1+\cos \left(s\right), \sin \left(s\right), 2\sin \left(\frac{s}{2}\right) \right)$, which lies on the regular surface $S_{2}$ are:

Also, we have

Thus, according to Eq (3.1), the tubular surface $\mathbf{\Upsilon } _{2}(s, v)$ associated with the Darboux frame of radius $r > 0$ along $\mathbf{\beta } (s)$ is the surface with the parametrization (see Figure 5b):

Remark. It should be noted that the calculations of the tubular surface $\mathbf{\Upsilon }_{2}$ can be calculated using Mathematica.

Example 5.3. Let $\gamma = \gamma (u)$ be a space curve which lies on a regular surface and has a cusp at $u_{0} = 0$ (see Figure 6a),

Darboux frame vectors of $\gamma$ are calculated as follows:

Therefore, the tubular surface associated with these Darboux vectors along $\gamma (u)$ is given by

then

From Eq (5.4), we get

which means $\mathbf{\Upsilon }_{3}$ has a singularity if and only if

Consequently, $\mathbf{\Upsilon }_{3}$ represents a front surface, and among its singular points are those denoted as $(0, n\pi), \ \ i.e., \ \ n = 0, 1, 2, ....$ (see Figure 6b).

6.   Conclusions

In this work, we studied the geometric properties and singularities of tubular surfaces with a Darboux frame in $\mathbb{R}^{3}$. Also, the local singularities of tubular Weingarten surfaces and relations among their curvature functions were studied. This study was intended to clear away to conduct the geometric analysis of tubular surfaces through the geometric conditions for these surfaces to have generic singularities as a front (i.e., cuspidal lips, cuspidal beaks, and Swallowtails).

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors gratefully acknowledge the approval and support of this research study by the grant No. SCAR-2023-12-2124 from the Deanship of Scientific Research at Northern Border University, Arar, {KSA}.

Conflict of interest

The authors declare that there are no conflicts of interest.