Abstract
In this article, we look at a surface associated with real-valued functions. The surface is known as a harmonic surface, and its unit normal vector and mean curvature have been used to characterize it. We use the Bishop-Darboux frame (-Darboux frame) in Euclidean 3-space to study and explain the geometric characteristics of the harmonic evolute surfaces of tubular surfaces. The characterizations of the harmonic evolute surface’s and parameter curves are evaluated, and then, they are compared. Finally, an example of a tubular surface’s harmonic evolute surface is presented, along with visuals of these surfaces.
1. Introduction
Darboux frame is a differential geometric approach for evaluating curves and surfaces. The Frenet frame is the most well-known frame field, although there are others, such as the Darboux frame. There have been several instances of frame studies of this sort, for example, see [1, 2].
The Bishop frame is a way for defining a moving frame that is well defined despite due to the vanished of curve’s second derivative [3]. Parallel transferring each element of an orthogonal frame along a curve is as easy as parallel transferring each element of the frame.
In , the geometrical position of the points at the inverse distance in terms of multiplication of the mean curvature from the surface is known as the harmonic evolute surface of a tubular surface. The harmonic evolute surface can be defined for a nonminimal surface.
Let be a surface associated with real-valued functions, and which, respectively, are the normal vector and mean curvature of . The harmonic surface has a parameterized description as follows:
Many researches on harmonic evolute surfaces have been published, some of which may be included here (see [4–7]). The geometric features of the harmonic evolute surface of a tubular surface via -Darboux frame have inspired us to study the geometric characteristics of the harmonic evolute surface of a tubular surface. As a result, the tubular surface and the harmonic evolute surface generated from this surface will be compared and interpreted.
2. Preliminaries
Consider the Euclidean 3-space . It contains the metric as follows:where ’s coordinate system.
For a regular curve lying on surface , we denote the Darboux frame on the surface by , where and is just surface’s normal [1, 8]. Then,where even the geodesic curvature , normal curvature , and relative torsion are defined as:
In matrix form, the -Darboux frame’s variation equation on the surface is as shown below [1]:where and , the -Darboux curvatures, are acquired in the following way:
Also, the relation matrix given bysuch that angle between and is acquired aroundfor any arbitrary constant . The relation among -Darboux’s curvatures and Darboux’s curvatures satisfies
Let be regular surface in , then the ’s unit normal vector can be written aswhere and . The Gaussian curvature and mean curvature were also provided by [9–11]where , , , , , and .
3. Obtaining Tubular Surface via -Darboux Frame
Let be an arc-length-parameterized curve in . Then, the tubular surface via the -Darboux frame has the parametrization [2, 12, 13]:where is constant and sphere’s radius. The velocity vectors of along arewhere . As a result, the trying to follow are the features of ’s first fundamental form:
The ’s unit surface normal vector , from the other hand, is acquired by
’s second order partial diffrentials are discovered as
The coefficients of the second fundamental form are derived using (13) as illustrated below.
Thus, the Gaussian curvature and mean curvature functions are calculated as
Theorem 1. The tubular surface M: via the -Darboux frame described by (12) is developable iff
Theorem 2. The tubular surface M: via the -Darboux frame described by (12) is minimal iff the following equation satisfies
Corollary 3. Let : be tubular surface via the -Darboux frame described by (12). The -parameter is then not geodesic curves but -parameter is geodesic curves.
Proof. Let be a tubular surface defined by Equation (12), and we get process and techniques from Equations (15) and (16)where stands for cross product. So, the proof is clear in such scenario.
Corollary 4. Let M: be tubular surface via the -Darboux frame described by (12). The -parameter is not asymptotic curves but -parameter is then asymptotic curves iff
Proof. If is a tubular surface as defined by Equation (12), from Equations (15) and (16), we have if only and only if or equivalently . But , which completes the proof.
Corollary 5. Let M: be tubular surface via the -Darboux frame described by (12). The and -parameters are then principal curves.
Proof. Let be a tubular surface defined by Equation (12), and we get process and techniques from Equations (14) and (17), then we haveThen, the proof is clear.
Corollary 6. The tubular surface M: via the -Darboux frame described by (12) is a ()-Weingarten surface.
Proof. If the Jacobi equation occurs between the Gaussian curvature and the mean curvature on a surface, it is termed a Weingarten surface (see [10]). Now, if be a tubular surface defined by Equation (12) and from Equation (18), we getIt is clear that .
Corollary 7. The tubular surface M: via the -Darboux frame defined by (12) is a ()-linear Weingarten surface iffwhere , , and are not all zero real numbers.
Proof. A surface is said to be a -linear Weingarten surface if the curvatures and of satisfy , where (see [10]). Then, one can see thatwhere , , and are not all zero real numbers.
4. Constructing the Harmonic Surface of Tubular Surface via -Darboux Frame
We now concentrate on the parametrization of harmonic surface of by using (12), (15), and (18). We define as follows:where . The ’s velocity vectors are
As a result, the features of ’s first fundamental forms
The ’s unit normal vector , from the other hand, is acquired by
’s second-order partial differentials are discovered as
The second fundamental form coefficients are computed using (29) and (30) as follows:
Thus, the Gaussian curvature and mean curvature functions are calculated as
Theorem 8. The harmonic evolute surface defined by (27) of tubular surface (12) via -Darboux frame is neither flat nor minimal.
Corollary 9. Let be harmonic evolute surface (27) of tubular surface (12) via -Darboux frame in E3. The ϱ and ς-parameters are then principal curves iff .
Proof. If and only if and , the coefficients of the first and second fundamental forms, respectively, vanish, the parameter curves of are lines of curvature. So, if is a nonzero constant, according to (29) and (32). As a result, the evidence is complete.
Corollary 10. Let be harmonic evolute surface (27) of tubular surface (12) via -Darboux frame in E3. Then, the following are satisfying.(1)’s -parameter curves not possible asymptotic curves(2)’s -parameter curves are asymptotic curves iff satisfies the 2nd-order differential equation
Proof. If the normal curvature of the parameter curves is zero everywhere, they are called asymptotic curves on the surface. If this is the case, from (30) and (32), we have(1)which means that -parameter curves are not asymptotic curves.(2)iff which means that -parameter curves are asymptotic curves.
Corollary 11. Let be harmonic evolute surface (27) of tubular surface (12) via -Darboux frame in E3. Then, the following are satisfying.(1)’s -parameter curves not possible geodesic curves(2)’s -parameter curves are geodesic curves ifffor any real constants and .
Proof. If the acceleration vector of the parameter curve on the surface is parallel to the normal vector of the surface, the parameter curve is termed a geodesic curve. If that is the case, using (30) and (32), we have(1)which means that -parameter curves are not geodesic curves.(2)Then, if only and only if . This implies that the -parameter curves are geodesic curves if the differential equation has a solution .
5. Example
Let be a circular helix parameterized as . Then, the curve’s Darboux frame and curvatures , , and along are dictated by
Now, . So, the -Darboux curvatures are calculated as
Then, the -Darboux frame are given as
As a result and taking , the parameterization of the tubular surface over the curve can be compiled in Darboux frame as (see Figure 1), then we have
The harmonic surface of via Darboux frame can be given as (see Figure 2)
From (12), the tubular surface over the curve via -Darboux frame can be given as (see Figure 3)
Using (27), then the harmonic surface of via -Darboux frame can be given as (see Figure 4)
6. Conclusion
Many researchers have recently researched curves and surfaces using the Bishop frame, similar to how they studied curves and surfaces using the Frenet frame. The concept of a -Darboux frame was recently shown, and there is a chance that the further studies may be conducted in the future. We study the characterisation of tubular surfaces using the -Darboux frame and the harmonic surface of tubular surfaces using the -Darboux frame in this paper. We provide the required and sufficient circumstances for a tubular surface to become a developable and minimum surface using the -Darboux frame. Furthermore, they demonstrate that the harmonic surface of a tubular surface is neither a developable nor a minimal surface.
Data Availability
No data is used in this study.
Conflicts of Interest
The authors declare no competing interest.
Authors’ Contributions
All authors have equal contribution and finalized the paper.
Acknowledgments
The authors extend their appreciation to the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University for funding this work through research group no. RG-21-09-04.