# Harmonic Evolute Surface of Tubular Surfaces viaB-Darboux Frame in Euclidean 3-Space

Authors: E. M. Solouma; Ibrahim AL-Dayel
Journal: Advances in Mathematical Physics
 (2021)

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## 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 (B-Darboux frame) in Euclidean 3-space E3 to study and explain the geometric characteristics of the harmonic evolute surfaces of tubular surfaces. The characterizations of the harmonic evolute surface’s ϱ and ς 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.


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## Body

## 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.InE3, 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.LetM:Ωρ,ς be a surface associated with real-valued functions, ℚρ,ς and Hρ,ς which, respectively, are the normal vector and mean curvature of Ω. The harmonic surface Γρ,ς has a parameterized description as follows:(1)Γρ,ς=Ωρ,ς+1Hρ,ςℚρ,ς.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 B-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-spaceE3. It contains the metric as follows:(2),=dε12+dε22+dε32,where ε1,ε2,ε3∈E3’s coordinate system.For a regular curveμρ lying on surface M=Ωρ,ς, we denote the Darboux frame on the surface by T,P,ℚ, where P=ℚ×Tand ℚis just surface’s normal [1, 8]. Then,(3)TρPρℚρρ=0κgρκnρ−κgρ0τgρ−κnρ−τgρ0TρPρℚρ,where even the geodesic curvature κg, normal curvature κn, and relative torsion τg are defined as:(4)τg=P′,ℚ,κn=T′,ℚ,κg=T′,P.In matrix form, theB-Darboux frame’s variation equation T,B1,B2 on the surface M is as shown below [1]:(5)TρB1ρB2ρρ=0ζ1ρζ2ρ−ζ1ρ00−ζ2ρ00TρB1ρB2ρ,where ζ1 and ζ2, the B-Darboux curvatures, are acquired in the following way:(6)ζ1=κgsinϕ+κncosϕ,ζ2=κnsinϕ−κgcosϕ.Also, the relation matrix given by(7)TρB1ρB2ρ=1000sinϕcosϕ0−cosϕsinϕTρPρℚρ,such that angle ϕ between ℚ and B1 is acquired around(8)ϕ−ϕ0=∫τgdt,for any arbitrary constant ϕ0. The relation among B-Darboux’s curvatures and Darboux’s curvatures satisfies(9)ζ12+ζ22=κg2+κn2.LetM:Ωρ,ς be regular surface in E3, then the Ω’s unit normal vector ℚ can be written as(10)ℚ=Ωρ×Ως∥Ωρ×Ως∥,where Ωρ=∂Ω/∂ρ and Ως=∂Ω/∂ς. The Gaussian curvature K and mean H curvature were also provided by [9–11](11)K=h11h22−h122g11g22−g122,H=g11h22+g22h11−2g12h122g11g22−g122,where g11=∥Ωρ∥2, g12=Ωρ,Ως, g22=∥Ως∥2, h11=Ωρρ,ℚ, h12=Ωρς,ℚ, and h22=Ωςς,ℚ.


## 3. Obtaining Tubular Surface viaB-Darboux Frame

Letμρ be an arc-length-parameterized curve in E3. Then, the tubular surface via the B-Darboux frame has the parametrization [2, 12, 13]:(12)Ωρ,ς=μρ+rcosςB1ρ+sinςB2ρ,where r is constant and sphere’s radius. The velocity vectors of Ω along μ are(13)Ωρ=1−rλρ,ςTρ,Ως=−rsinςB1ρ+rcosςB2ρ,where λρ,ς=κgρsinϕ−ς+κnρcosϕ−ς. As a result, the trying to follow are the features of Ω’s first fundamental form:(14)g11=1−rλ2,g12=0,g22=r2.TheΩ’s unit surface normal vector ℚΩ, from the other hand, is acquired by(15)ℚΩ=−cosςB1ρ−sinςB2ρ.Ω’s second order partial diffrentials are discovered as(16)Ωρρ=−rλρTρ+ζ11−rλB1ρ+ζ21−rλB2ρ,Ωρς=−rλςTρ,Ωςς=−rcosςB1ρ−rsinςB2ρ.The coefficients of the second fundamental form are derived using (13) as illustrated below.(17)h11=−λ1−rλ,h12=0,h22=r.Thus, the Gaussian curvatureKΩ and mean curvature HΩ functions are calculated as(18)KΩρ,ς=−λr1−rλ,HΩρ,ς=1−2rλ2r1−rλ.Theorem 1.
The tubular surface M:Ωϱ,ς via the B-Darboux frame described by (12) is developable iff(19)tanϕ−ς=−κnρκgρ.Theorem 2.
The tubular surface M:Ωϱ,ς via the B-Darboux frame described by (12) is minimal iff the following equation satisfies(20)1−2rκgρsinϕ−ς+κnρcosϕ−ς=0.Corollary 3.
LetM: Ωϱ,ς be tubular surface via the B-Darboux frame described by (12). The ϱ-parameter is then not geodesic curves but ς-parameter is geodesic curves.Proof.
LetΩ be a tubular surface defined by Equation (12), and we get process and techniques from Equations (15) and (16)(21)ℚΩ×Ωρρ≠0andℚΩ×Ωςς=0,where × stands for cross product. So, the proof is clear in such scenario.Corollary 4.
Let M:Ωϱ,ς be tubular surface via the B-Darboux frame described by (12). The ς-parameter is not asymptotic curves but ϱ-parameter is then asymptotic curves iff(22)κgϱ=1rsinϕ−ς.Proof.
IfΩ is a tubular surface as defined by Equation (12), from Equations (15) and (16), we have ℚΩ,Ωρρ=ζ11−rλ=0 if only and only if λ=1/r or equivalently κgρ=1/rsinϕ−ς. But ℚΩ,Ωςς=r≠0, which completes the proof.Corollary 5.
Let M:Ωϱ,ς be tubular surface via the B-Darboux frame described by (12). The ϱ and ς-parameters are then principal curves.Proof.
LetΩ be a tubular surface defined by Equation (12), and we get process and techniques from Equations (14) and (17), then we have(23)g12=h12=0.
Then, the proof is clear.Corollary 6.
The tubular surface M:Ωϱ,ς via the B-Darboux frame described by (12) is a (KΩ,HΩ)-Weingarten surface.Proof.
If the Jacobi equationKΩ,HΩ=0 occurs between the Gaussian curvature KΩ and the mean curvature HΩ on a surface, it is termed a Weingarten surface (see [10]). Now, if Ω be a tubular surface defined by Equation (12) and from Equation (18), we get(24)KΩρ=−λρr1−rλ2,KΩς=−λςr1−rλ2,HΩρ=−λρ21−rλ2,HΩς=−λς21−rλ2.
It is clear thatHΩρKΩς=HΩςKΩρ.Corollary 7.
The tubular surface M:Ωϱ,ς via the B-Darboux frame defined by (12) is a (KΩ,HΩ)-linear Weingarten surface iff(25)λ=2rc+b2a+rb−r2c,where c, c1, and c2 are not all zero real numbers.Proof.
A surfaceΩ is said to be a KΩ,HΩ-linear Weingarten surface if the curvatures KΩ and HΩ of Ω satisfy aKΩ+bHΩ=c, where a,b,c∈R (see [10]). Then, one can see that(26)λ=2rc+b2a+rb−r2c,where a, b, and c are not all zero real numbers.


## 4. Constructing the Harmonic Surface of Tubular Surface viaB-Darboux Frame

We now concentrate on the parametrization ofM∗ harmonic surface of M by using (12), (15), and (18). We define M∗ as follows:(27)Γϱ,ς=μϱ+ρϱ,ςcosςB1ϱ+sinςB2ϱ,where ρρ,ς=−r/1−2rλρ,ς. The Γ’s velocity vectors are(28)Γρ=1−λρTρ+ρρcosςB1ρ+ρρsinςB2ρ,Γς=ρςcosς−ρsinςB1ρ+ρςsinς+ρcosςB2ρ.As a result, the features ofΓ’s first fundamental forms(29)g11∗=ρρ2+1−λρ2,g12∗=ρρρς,g22∗=ρ2+ρς2.TheΓ’s unit normal vector ℚΓ, from the other hand, is acquired by(30)ℚΓ=1ρ2ρρ2+ρ2+ρς21−λρ2ρρρTρ−1−λρρςsinς+ρcosςB1ρ+1−λρρςcosς−ρsinςB2ρ.Γ’s second-order partial differentials are discovered as(31)Γρρ=−2λρρ+λρρT+ζ11−λρ+ρρρcosςB1+ζ21−λρ+ρρρsinςB2,Γρς=−λρςT+ρρςcosς−ρρsinςB1+ρρςsinς+ρρcosςB2,Γςς=ρςς−ρcosς−2ρςsinςB1+ρςς−ρsinς+2ρςcosςB2.The second fundamental form coefficients are computed using (29) and (30) as follows:(32)h11∗=1ρ2ρρ2+ρ2+ρς21−λρ21−λρ1−λρρςζ2cosς−ζ1sinς−λρ−ρρρρ−ρρρ2λρρ+λρρ,h12∗=1−λρρρρς−ρρρς−ρρρλρςρ2ρρ2+ρ2+ρς21−λρ2,h22∗=1−λρ2ρς2−ρρςς−ρρ2ρρ2+ρ2+ρς21−λρ2.Thus, the Gaussian curvatureKΓ and mean curvature HΓ functions are calculated as(33)KΓ=1−λρ1−λρρςζ2cosς−ζ1sinς−λρ−ρρρρ−ρρρ2λρρ+λρρ1−λρ2ρς2−ρρςς−ρ−1−λρρρρς−ρρρς−ρρρλρς2ρ2ρρ2+ρ2+ρς21−λρ22,HΓ=ρρ2+1−λρ21−λρ2ρς2−ρρςς−ρ+ρ2+ρς21−λρ1−λρρςζ2cosς−ζ1sinς−λρ−ρρρρ−ρρρ2λρρ+λρρ−2ρρρς1−λρρρρς−ρρρς−ρρρλρς2ρ2ρρ2+ρ2+ρς21−λρ23/2.Theorem 8.
The harmonic evolute surfaceM∗ defined by (27) of tubular surface (12) via B-Darboux frame is neither flat nor minimal.Corollary 9.
LetM∗ be harmonic evolute surface (27) of tubular surface (12) via B-Darboux frame in E3. The ϱ and ς-parameters are then principal curves iff ρ=constant.Proof.
If and only ifg12∗ and h12∗, the coefficients of the first and second fundamental forms, respectively, vanish, the parameter curves of M∗ are lines of curvature. So, g12∗=h12∗=0 if ρ is a nonzero constant, according to (29) and (32). As a result, the evidence is complete.Corollary 10.
LetM∗ be harmonic evolute surface (27) of tubular surface (12) via B-Darboux frame in E3. Then, the following are satisfying.(1)
M∗’s ϱ-parameter curves not possible asymptotic curves(2)
M∗’s ς-parameter curves are asymptotic curves iff ρ satisfies the 2nd-order differential equation(34)ρρςς−2ρς2−ρ2=0.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)
h11∗=Γρρ,ℚΓ=−1−λρλ1−λρ+ρρρ+ρρ2λρρ+λρρ/ρρ2+1−λρ2≠0,which means that ρ-parameter curves are not asymptotic curves.(2)
h22∗=Γςς,ℚΓ=2ρς2−ρρςς−ρ/ρ2+ρς2=0,iff ρρςς−2ρς2−ρ2=0 which means that ς-parameter curves are asymptotic curves.Corollary 11.
LetM∗ be harmonic evolute surface (27) of tubular surface (12) via B-Darboux frame in E3. Then, the following are satisfying.(1)
M∗’s ϱ-parameter curves not possible geodesic curves(2)
M∗’s ς-parameter curves are geodesic curves iff(35)ρ=c1cosς+c2sinς,for any real constants c1 and c2.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)
ℚΓ×Γρρ=ζ21−λρ2T+ρρζ21−λρB1+ρρρρρ+1−λρρρζ1+2λ+λρρB2/ρρ2+1−λρ2≠0,which means that ρ-parameter curves are not geodesic curves.(2)
ℚΓ×Γςς=−2ρρς+ρςρςς−ρT/ρ2+ρς2.
Then,ℚΓ×Γςς=0 if only and only if ρςρςς+ρ=0. This implies that the ς-parameter curves are geodesic curves if the differential equation ρςς+ρ=0 has a solution ρ=c1cosς+c2sinς.


## 5. Example

Letμ be a circular helix parameterized as μρ=cosρ/2,sinρ/2,ρ/2. Then, the curve’s Darboux frame and curvatures κg, κn, and τg along μρ are dictated by(36)Tρ=−12sinρ2,12cosρ2,12,Pρ=12cosρ2−12sinρ2,12sinρ2+12cosρ2,−12,ℚρ=12cosρ2+12sinρ2,12sinρ2−12cosρ2,12,κg=−122,κn=−122,τg=−12.Now,ϕ=∫0ρτgdt=∫0ρ−1/2dt=−ρ/2. So, the B-Darboux curvatures are calculated as(37)ζ1=122sinρ2−cosρ2,ζ2=122sinρ2+cosρ2.Then, theB-Darboux frame are given as(38)Tρ=12−sinρ2,cosρ2,1,B1ρ=12cosρ2cosρ2−sinρ2+12sinρ2cosρ2+sinρ2,sinρ2cosρ2−sinρ2−12cosρ2cosρ2+sinρ2,12cosρ2+sinρ2,B2ρ=12cosρ2cosρ2+sinρ2−12sinρ2cosρ2−sinρ2,sinρ2cosρ2+sinρ2−12cosρ2cosρ2−sinρ2,−12cosρ2−sinρ2.As a result and takingr=2, the parameterization of the tubular surface M1 over the curve μ can be compiled in Darboux frame as Ω1ρ,ς=μρ+rcosςPρ+sinςℚρ (see Figure 1), then we have(39)Ω1ρ,ς=cosρ21−cosς+sinς+12sinρ2cosς+sinς,sinρ21−cosς+sinς−12cosρ2cosς+sinς,12ρ+cosς+sinς.Figure 1
Tubular surfaceΩ1 due to Darboux frame.The harmonic surfaceΓ1 of Ω1 via Darboux frame Γ1ρ,ς=Ω1ρ,ς+1/HΩ1ρ,ςℚρ,ς can be given as (see Figure 2)(40)Γ1ρ,ς=1+cosς+sinςcosρ2+2−cosς−sinς1−cosς−sinςcosρ2−12sinρ2−12sinρ2cosς−sinς,1+cosς+sinςsinρ2+2−cosς−sinς1−cosς−sinςsinρ2−12cosρ2+12cosρ2cosς−sinς,ρ2−12cosς−sinς+2−cosς−sinς21−cosς−sinς.Figure 2
Harmonic surfaceΓ1 via Darboux frame.From (12), the tubular surface Ω2 over the curve μ via B-Darboux frame can be given as (see Figure 3)(41)Ω2ρ,ς=cosρ2+cosρ2+sinρ212cosςsinρ2+sinςcosρ2+cosρ2−sinρ2cosςcosρ2−12sinςsinρ2,sinρ2+cosρ2+sinρ2sinςsinρ2−12cosςcosρ2+cosρ2−sinρ2cosςsinρ2−12sinςcosρ2,12ρ+cosς−ρ2−sinς−ρ2.Figure 3
Tubular surfaceΩ2 via B-Darboux frame.Using (27), then the harmonic surface Γ2 of Ω2 via B-Darboux frame can be given as (see Figure 4)(42)Γ2ρ,ς=cosρ2−11+cosς+ρ/2−sinς+ρ/2cosρ2−sinρ2cosςcosρ2−12sinςsinρ2+cosρ2+sinρ2sinςcosρ2+12cosςsinρ2,sinρ2−11+cosς+ρ/2−sinς+ρ/2cosρ2−sinρ2cosςsinρ2−12sinςcosρ2−cosρ2+sinρ2cosςcosρ2−12sinςsinρ2,ρ2−2cosς−ρ/2−sinς−ρ/21+cosς+ρ/2−sinς+ρ/2.Figure 4
Harmonic surfaceΓ2 via B-Darboux frame.


## 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 aB-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 B-Darboux frame and the harmonic surface of tubular surfaces using the B-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 B-Darboux frame. Furthermore, they demonstrate that the harmonic surface of a tubular surface is neither a developable nor a minimal surface.



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