Abstract
This study begins with the construction of type- Smarandache ruled surfaces, whose base curves are Smarandache curves derived by rotation-minimizing Darboux frame vectors of the curve in . The direction vectors of these surfaces are unit vectors that convert Smarandache curves. The Gaussian and mean curvatures of the generated ruled surfaces are then separately calculated, and the surfaces are required to be minimal or developable. We report our main conclusions in terms of the angle between normal vectors and the relationship between normal curvature and geodesic curvature. For every surface, examples are provided, and the graphs of these surfaces are produced.
1. Introduction
One of the branches of classical differential geometry that has been explored by a number of researchers is the theory of ruled surfaces. A ruled surface is one that has at least one straight line passing through each point on the surface that is also on the surface. One way to conceptualize a ruled surface is as one that is “swept out” by a moving straight line in space. Cones and cylinders are basic instances of ruled surfaces. Architects are interested in more intricate ruled surfaces, particularly when it comes to free-form building and intricate designs. In contemporary design, there are several instances of ruled surface structures, such as saddle roofs (hyperbolic paraboloid) and cooling towers (hyperboloid). A ruled surface is often described as a collection of a family of straight lines that are reliant on the parameters, called the rulings, of the ruled surface. The parametric representation of a ruled surface is , where is the base curve and specifies the rulings directions []. Many practical uses arise out of the study of developable surfaces. Studies on developable surfaces are widely available. A surface that can be developed into a plane without changing the surface metric is called a developable surface. Studies of developable surfaces of space curves from the perspective of singularity theory have been documented in the literature. The governed surface that results from the tangent lines of a space curve is called the tangent developable surface. Tantent developable surfaces are crucial to the duality theory of algebraic geometry. From differential calculus, singularity theory is a branch of mathematics that has applications in physics, astronomy, and geometry. Some of key investigations on the submanifold theory and singularity theory have been presented in [,,,]. The developability and minimalist principles of surfaces are two of its key characteristics. The investigation of ruled surfaces with various moving frames provides one of the most fascinating features (see [,,,,,]).
In Euclidean and Minkowski spaces, the Smarandache curve is the curve whose position vector is created by Frenet frame vectors on another regular curve [,,]. Recent studies on Smarandache curves in Minkowski and Euclidean spaces have been conducted by a number of researchers [].
In this study, the requirements of type- Smarandache ruled surfaces are specified by using the rotation minimizing Darboux frame in . Furthermore, we address the geometric analysis of a particular kind of ruled surface, namely the type- Smarandache ruled surfaces associated with the space curves as a basic example for studying the manifolds with the largest dimensions in Euclidean 3-space. We examine the sufficient and necessary requirements for these surfaces to be minimal and developable. We report our main conclusions in terms of the angle between normal vectors and the relationship between normal curvature and geodesic curvature. The organization of this work is as follows: In Section 2, we review the concepts of ruled surfaces in Euclidean space, Smarandache curves, Frenet frame, and Darboux frame. In Section 3, we derive some of the geometric characteristics of the type- Smarandache ruled surfaces, which we define due to rotation minimizing Darboux frame whose a director curve and base curve are Smarandache curves. We give an example in Section 4 that emphasizes the most significant findings in this work.
2. Preliminaries
Let be a unit speed regular curve lying on a surface in . Let denotes the moving Frenet frame of , then satisfying []:
where and .
For any arbitrary curve with in , the Darboux frame of is given by []:
where is the surface normal and . The geodesic curvature , normal curvature , and geodesic torsion that connects the curve on M are given as
The rotation minimizing Darboux frame (RMDF) of curve on the surface M is defined by [,]:
where the RMDF curvatures are obtained as follows:
where is the angle between vectors and . The relation matrix between frames may be expressed as:
The surface formed by a line moving depending on the parameter of a curve is called a ruled surface and its parametric expression is as follows:
The normal vector field, the Gaussian and mean curvatures of are given by
We also know that
and
where E, F, G are the first fundamental coefficients and e, f, and g are the second fundamental coefficients of .
The Smarandache curves of the curve is defined as follows:
3. Type-Π Smarandache Ruled Surfaces Due to RMDF
The curve on the surface M caused by RMDF serves as the parametric representation of type-Π Smarandache ruled surfaces in this section. In addition, we assess the prerequisites that make it possible for these surfaces to be minimum and developable.
3.1. Type-Π Smarandache Ruled Surfaces along -Smarandache Curve
Definition 1.
The type-Π Smarandache ruled surfaces due to RMDF in generated by continuously moving vector along -Smarandache curve is defined by
Theorem 1.
The type-Π Smarandache ruled surface is developable.
Proof.
Consider the type- Smarandache ruled surface given by (8), then the velocity vectors of are given by
Then, the normal vector field of the surface is obtained as
Hence, it can be easily seen that the first fundamental coefficients of are
Taking the derivative of (9) with respect to s and , we find
Hence, it can be seen that the coefficients of second fundamental form of are
As a consequence, we obtain
which concludes the proof. □
From Theorem 1, we have the following immediate results.
Corollary 1.
Let be a type-Π Smarandache ruled surface. Then, the s-parameter curves of Ψ are not geodesic but υ-parameter curves are geodesic.
Proof.
Corollary 2.
Let be a type-Π Smarandache ruled surface. Then, the s-parameter curves of Ψ are not asymptotic but υ-parameter curves are asymptotic.
Proof.
Corollary 3.
Let be a type-Π Smarandache ruled surface. Then, the s and υ-parameter curves are principal curves.
Proof.
Definition 2.
The type-Π Smarandache ruled surfaces due to RMDF (4) in generated by continuously moving vector along -Smarandache curve is defined as follows
Theorem 2.
The type-Π Smarandache ruled surface together with is developable iff one of the following conditions holds
- i.
- ,
- ii.
- .
Proof.
Consider the type- Smarandache ruled surface given by (14), then the velocity vectors of are given by:
Then, the normal vector field of the surface is obtained as:
Hence, it can be easily seen that the first fundamental coefficients of are:
Taking the derivative of (15) with respect to s and , we find
Thus, it can be seen that the coefficients of the second fundamental form of are:
As a result, we derive
Combining the above equation with (5), we conclude the proof. □
As a consequence of Theorem 2, we obtain the following results:
Corollary 4.
Let be a type-Π Smarandache ruled surface. Then, the Γ has constant Gauss curvature iff for some non-zero constant c.
Corollary 5.
Let be a type-Π Smarandache ruled surface. Then, the s-parameter curves of Γ are not asymptotic but υ-parameter curves are asymptotic curves.
Remark 1.
The proof of Corollary 4 and 5 is similar to the proof of Corollaries 1–3.
Definition 3.
The type-Π Smarandache ruled surfaces due to RMDF (4) in generated by continuously moving vector along -Smarandache curve is defined as follows
Theorem 3.
The type-Π Smarandache ruled surface defined by (20) due to RMDF (4) in . If , then Λ is developable iff one of the following conditions holds
- i.
- ,
- ii.
- .
Proof.
Consider the type- Smarandache ruled surface given by (20), then ’s velocity vectors are given by:
The normal vector field of the surface may be ascertained by taking the cross-product of the partial derivatives of the surface given by Equation (21). Then, we have
From Equation (21), we can obtain the ’s quantities of first fundamental form are:
Using (21)’s second derivative with regard to s and , we have
Then, ’s quantities of second fundamental form are:
As a result, given the facts above, we are able to and are given by:
So, from Equation (5) the proof is ended. □
In the same way, as for the proof of Corollaries 1–3, we can prove the following results:
Corollary 6.
3.2. Type- Smarandache Ruled Surfaces along -Smarandache Curve
Definition 4.
The type-Π Smarandache ruled surfaces due to RMDF (4) in generated by continuously moving vector along -Smarandache curve is defined as follows
Theorem 4.
The type-Π Smarandache ruled surface defined by (26) due to RMDF (4) in . If then Θ is developable iff one of the following conditions holds
- i.
- ,
- ii.
- .
Proof.
Consider the type- Smarandache ruled surface given by (26), then ’s velocity vectors are given by:
The normal vector field of the surface may be ascertained by taking the cross-product of the partial derivatives of the surface given by Equation (27). Then, we have
From Equation (27), we can obtain the ’s quantities of first fundamental form are:
Using (27)’s second derivative with regard to s and , we have
Then, ’s quantities of second fundamental form are:
As a result, given the facts above, we are able to see that and are given by:
So, from Equation (5) the proof is ended. □
In the same way, as for the proof of Corollary 2, we can prove the following result:
Corollary 8.
Definition 5.
The type-Π Smarandache ruled surfaces due to RMDF (4) in generated by continuously moving vector along -Smarandache curve is defined as follows
Theorem 5.
The type-Π Smarandache ruled surface defined by (32) due to RMDF (4) in is developable iff one of the following conditions holds
- i.
- ,
- ii.
- .
Proof.
Consider the type- Smarandache ruled surface given by (32), then ’s velocity vectors are given by:
The normal vector field of the surface may be ascertained by taking the cross-product of the partial derivatives of the surface given by Equation (33). Then, we have
From Equation (33), we can obtain the ’s quantities of first fundamental form are:
Using (33)’s second derivative with regard to s and , we have
Then, ’s quantities of second fundamental form are:
As a result, given the facts above, we are able to find that and are given by:
So, from Equation (5) the proof is ended. □
In the same way, as for the proof of Corollaries 1–3, we can prove the following results:
Corollary 9.
Corollary 10.
Corollary 11.
Definition 6.
The type-Π Smarandache ruled surfaces due to RMDF (4) in generated by continuously moving vector along -Smarandache curve is defined as follows
Theorem 6.
The type-Π Smarandache ruled surface defined by (38) due to RMDF (4) in . If , then Ω is developable iff one of the following conditions holds
- i.
- ,
- ii.
- .
Proof.
Consider the type- Smarandache ruled surface given by (38), then ’s velocity vectors are given by:
The normal vector field of the surface may be ascertained by taking the cross-product of the partial derivatives of the surface given by Equation (39). Then, we have
From Equation (39), we can obtain the ’s quantities of first fundamental form are:
Using (39)’s second derivative with regard to s and , we have
Then, ’s quantities of second fundamental form are:
As a result, given the facts above, we are able to and are given by:
So, from Equation (5) the proof is ended. □
In the same way, as for the proof of Corollary 2, we can prove the following result:
3.3. Type- Smarandache Ruled Surfaces along -Smarandache Curve
Definition 7.
The type-Π Smarandache ruled surfaces due to RMDF (4) in generated by continuously moving vector along -Smarandache curve is defined as follows
Theorem 7.
The type-Π Smarandache ruled surface defined by (44) due to RMDF (4) in is developable iff one of the following conditions holds
- i.
- ,
- ii.
- .
Proof.
Consider the type- Smarandache ruled surface given by (44), then ’s velocity vectors are given by:
The normal vector field of the surface may be ascertained by taking the cross-product of the partial derivatives of the surface given by Equation (45). Then, we have
From Equation (45), we can obtain the ’s quantities of first fundamental form are:
Using (45)’s second derivative with regard to s and , we have
Then, ’s quantities of second fundamental form are:
As a result, given the facts above, we are able to find that and are given by:
So, from Equation (5) the proof is ended. □
In the same way, as for the proof of Corollary 2, we can prove the following result:
Corollary 13.
Definition 8.
The type-Π Smarandache ruled surfaces due to RMDF (4) in generated by continuously moving vector along -Smarandache curve is defined as follows:
Theorem 8.
The type-Π Smarandache ruled surface defined by (50) due to RMDF (4) in is developable iff one of the following conditions holds
- i.
- ,
- ii.
- .
Proof.
Consider the type- Smarandache ruled surface given by (50), then ’s velocity vectors are given by:
The normal vector field of the surface may be ascertained by taking the cross-product of the partial derivatives of the surface given by Equation (51). Then, we have
From Equation (51), we can obtain the ’s quantities of first fundamental form are:
Using (51)’s second derivative with regard to s and , we have
Then, ’s quantities of second fundamental form are:
As a result, given the facts above, we are able to find that and are given by:
So, from Equation (5) the proof is ended. □
In the same way, as for the proof of Corollary 2, we can prove the following result:
Corollary 14.
Definition 9.
The type-Π Smarandache ruled surfaces due to RMDF (4) in generated by continuously moving vector along -Smarandache curve is defined as follows
Proof.
Consider the type- Smarandache ruled surface given by (56), then ’s velocity vectors are given by:
The normal vector field of the surface may be ascertained by taking the cross-product of the partial derivatives of the surface given by Equation (57). Then, we have
From Equation (57), we can obtain the ’s quantities of first fundamental form are:
Using (57)’s second derivative with regard to s and , we have
Then, ’s quantities of second fundamental form are:
As a result, given the facts above, we are able to find that and are given by:
Then, the proof is ended. □
In the same way, as for the proof of Corollaries 1–3, we can prove the following results:
Corollary 15.
Corollary 16.
Corollary 17.
4. Example
Let consider a surface parameterized by
The s-parameter curve lying on (see Figure 1). Then, the Darboux frame of on M are given, respectively, by

Figure 1.
The curve on .
Then, . So, we have
Given the parametric equations below and these vectors and definitions, the graphs of type- Smarandache ruled surface are shown in Figure 2, Figure 3, and Figure 4, respectively.

Figure 2.
The type- Smarandache ruled surfaces , , and along .

Figure 3.
The type- Smarandache ruled surfaces , , and along .

Figure 4.
The type- Smarandache ruled surfaces , , and along .
5. Conclusions
This study focused on the construction of type- Smarandache ruled surfaces, with their base curves being Smarandache curves derived from rotation minimizing Darboux frame vectors of the curve in . The direction vectors of these surfaces were unit vectors that transformed the Smarandache curves. The Gaussian and mean curvatures of the manufactured ruled surfaces were calculated separately with the requirement of being either minimum or developable. Graphs of these surfaces were generated, along with an example provided for each surface. Through this research, we have successfully examined and presented the characteristics and properties of these type- Smarandache ruled surfaces, contributing to the understanding and exploration of this specific class of surfaces. Further investigations and applications of these surfaces in relevant fields are encouraged.
Author Contributions
Data curation, E.S., I.A.-D., M.A.K. and M.A.; Formal analysis, E.S., I.A.-D., M.A.K. and M.A.; Funding acquisition, E.S.; Methodology, E.S., I.A.-D. and M.A.K.; Software, E.S., I.A.-D., M.A.K. and M.A.; Writing—original draft, E.S., I.A.-D., M.A.K. and M.A.; Writing—review and editing, E.S., I.A.-D., M.A.K. and M.A. All authors contributed equally. All authors have read and agreed to the published version of the manuscript.
Funding
This work is supported and funded by Imam Mohammad Ibn Saud Islamic University through grant number IMSIU-RG23085.
Data Availability Statement
Data are contained within the article.
Acknowledgments
This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-RG23085).
Conflicts of Interest
The authors declare no conflict of interest.
References
- Do Carmo, M.P. Differential Geometry of Curves and Surfaces; Revised and Updated Second Edition; Courier Dover Publications: Mineola, NY, USA, 2016. [Google Scholar]
- Zhao, Q.; Pei, D.; Wang, Y. Singularities for one-parameter developable surfaces of curves. Symmetry 2019, 11, 108. [Google Scholar] [CrossRef]
- Zhao, Q.; Yang, L.; Wang, Y. Geometry of Developable surfaces of Frenet type framed base curves from the singularity theory viewpoint. Symmetry 2022, 14, 975. [Google Scholar] [CrossRef]
- Abdel-Baky, R.A.-S.; Saad, M.K. Singularities of non-developable ruled surface with space-like ruling. Symmetry 2022, 14, 716. [Google Scholar] [CrossRef]
- Li, Y.; Nazra, S.H.; Abdel-Baky, R.A. Singularity properties of timelike sweeping surface in Minkowski 3-Space. Symmetry 2022, 14, 1996. [Google Scholar] [CrossRef]
- Almoneef, A.A.; Abdel-Baky, R.A. Timelike Ruled Surfaces with Stationary Disteli-Axis. Symmetry 2023, 15, 998. [Google Scholar] [CrossRef]
- Solouma, E.; Abdelkawy, M. Family of ruled surfaces generated by equiform Bishop spherical image in Minkowski 3-space. AIMS Math. 2022, 8, 4372–4389. [Google Scholar] [CrossRef]
- AL-Daye, I.; Solouma, E.M. Geometric Properties in Minkowski Space-Time of Spacelike Smarandache Curves. Int. J. Appl. Comput. Math. 2021, 7, 1–16. [Google Scholar]
- Aldossary, M.T.; Abdel-Baky, R.A. Surfaces Family with Bertrand Curves as Common Asymptotic Curves in Euclidean 3-Space E3. Symmetry 2023, 15, 1440. [Google Scholar] [CrossRef]
- Şenyurt, S.; Canlı, D.; Çan, E.; Mazlum, S.G. Some Special Smarandache Ruled Surfaces by Frenet Frame in E3-II. Honam Math. J. 2022, 44, 594–617. [Google Scholar]
- Li, Y.; Şenyurt, S.; Özduran, A.; Canlı, D. The Characterizations of Parallel q-Equidistant Ruled Surfaces. Symmetry 2022, 14, 1879. [Google Scholar] [CrossRef]
- Ashbacher, C. Smarandache geometries. Smarandache Notions J. 1997, 8, 212–215. [Google Scholar]
- Bishop, L.R. There is more than one way to frame a curve. Amer. Math. Monthly 1974, 82, 246–251. [Google Scholar] [CrossRef]
- Mao, L. Smarandache Geometries & Map Theory with Applications; Chinese Branch Xiquan House, Academy of Mathematics and Systems Chinese Academy of Sciences: Beijing, China, 2006. [Google Scholar]
- Solouma, E.M. Equiform spacelike Smarandache curves of anti-Eqiform Salkowski curve according to Equiform frame. Int. J. Math. Anal. 2021, 15, 43–59. [Google Scholar] [CrossRef]
- Darboux, G. Leçons sur la Theorie Generale des Surfaces; I-II-III-IV; Gauthier-Villars: Paris, France, 1896. [Google Scholar]
- Aldossary, M.T.; Abdel-Baky, R.A. Sweeping surface due to rotation minimizing Darboux frame in Euclidean 3-space E3. AIMS Math. 2022, 8, 447–462. [Google Scholar] [CrossRef]
- Dede, M.; Aslan, M.C.; Ekici, M.C. On a variational problem due to the B-Darboux frame in Euclidean 3-space. Math. Methods Appl. Sci. 2021, 43, 1–9. [Google Scholar] [CrossRef]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).