# Generated Surfaces via Inextensible Flows of Curves inR3

Authors: Rawya A. Hussien; Samah G. Mohamed
Journal: Journal of Applied Mathematics
 (2016)

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## Abstract

We study the inextensible flows of curves in 3-dimensional Euclidean spaceR3. The main purpose of this paper is constructing and plotting the surfaces that are generated from the motion of inextensible curves in R3. Also, we study some geometric properties of those surfaces. We give some examples about the inextensible flows of curves in R3 and we determine the curves from their intrinsic equations (curvature and torsion). Finally, we determine and plot the surfaces that are generated by the motion of those curves by using Mathematica 7.


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## Body

## 1. Introduction

The evolution of curves and surfaces has important applications in many fields such as computer vision [1, 2], computer animation [3], and image processing [4]. The motion of curves and surfaces in R3 leads to nonlinear evolution equations, which are often integrable. The connection between integrable systems and the differential geometry of curves has been studied extensively. Some integrable systems arise from invariant curve flows in certain geometries such as affine and centroaffine geometries [5–7] and similarity and projective geometries [8, 9]. Motion of curves in Minkowski space R13 is studied in [10–12]. Hashimoto [13] showed that the integrable nonlinear Schrodinger equation (NLS)(1)iϕt+ϕss+ϕ2ϕ=0is equivalent to the system for the curvatures k and τ of curves in R3:(2)kt=-τsk-2τks,τt=kks+∂∂s-τ2k+kssk,via the so-called Hashimoto transformation ϕ=kexpi∫‍τ(s,t)ds. System (2) is equivalent to the vortex filament equation (3)γt=γs×γss=kb,where b is the binormal vector of γ.In [14], Schief and Rogers studied the binormal motion of curves of constant curvature or torsion.This line of research has been extended to motions of curves in three-dimensional space forms.Recently, Abdel-All et al. [15–18] constructed new geometrical models of motion of plane curves. Also, they constructed a Hashimoto surface from its fundamental coefficients via numerical integration of Gauss-Weingarten equations and fundamental theorem of surfaces. Also, they studied kinematics of moving generalized curves in n-dimensional Euclidean space in terms of intrinsic geometries. Mohamed [19] studied the motions of inextensible curves in spherical space S3.In this paper, we will present the flows of curves inR3. The outline of this paper is as follows.In Section2, we study the geometry of curves in R3. In Sections 3 and 4, we study the motion of curves in R3 and we get the time evolution of Serret-Frenet frame and the evolution of curvatures. In Section 5, we study the geometric properties of the surfaces that are generated by the motion of the family of curves Ct. In Section 6, we give some examples of motions of inextensible curves in R3, and we plot the surfaces that are generated by the motion of those curves. For these surfaces, we study the Gaussian and Mean curvatures. Finally, Section 7 is devoted to conclusion.


## 2. Geometric Preliminaries

Consider a smooth curve in a3-dimensional space. Assume that u is the parameter along the curve in R3. Let r(u) denote the position vector of a point on the curve. The metric on the curve is(4)gu=∂r∂u,∂r∂u,where u is the parameter of the curve. The arc length along the curve is given by(5)su=∫0ugσdσ,∂∂s=1g∂∂u;we use {u,t} as coordinates of a point on the curve.Consider the orthonormal frameI={T,N,B}, such that T is the tangent vector and N,B denote the normal vectors at any point on the curve.Lemma 1.
The Frenet frame for the curve inR3 satisfies the following:(6)Fs=Q·F,where(7)F=TNB,Q=0k0-k0τ0-τ0.Lemma 2.
Consider the curveγ(s(u)) with an arbitrary parameter u∈I. Then the Serret-Frenet frame satisfies(8)Fu=gQ·F,where Q and F are given as in (6).


## 3. Curve Evolution

An evolving curve can be considered as a family of curves parametrized by time. This means that each curve in the family is a mappingγ:I×(0,1]→R2 that assigns for each space parameter u∈I and each time parameter t∈(0,1]; there is a point γ(u,t)∈R2. An evolution equation which is a differential equation that describes the evolution of γ(u,t) in time can be specified by the form(9)γ˙=dγdt=WT+UN+VB,where W,U,V are the velocities along the frame T,N,B. Consider a local motion; that is, the velocities W,U,V depend only on the local values of the curvatures {k,τ}.


## 4. Main Results

From [18], we considered the curve γ(u,t). For the curve flow (10)dγdt=∑j=1nvjej,where vj are the velocities in the direction of ej, we had the following.Lemma 3.
The evolution equation for the metricg is given by(11)g˙=2gλ,where λ=(∂v1/∂s-kv2).Theorem 4.
The evolution for the frameF=(e1,e2,…,en)t can be given in a matrix form(12)Ft=M·F,where M is the evolution matrix and it takes the form (13)M=0M12M13…M1n-M120M23…M2n-M13-M230…M3n⋮⋮⋮⋮⋮-M1n-1-M2n-1-M3n-1…Mn-1n-M1n-M2n-M3n…0,where the elements of the matrix M are given explicitly by(14)M1j=Aj=vj,s+kj-1vj-1-kjvj+1,j=2,3,…,n,Mαμ=1kμ-αMα-1μ,s+kμ-1Mα-1μ-1+kμMα-1μ+1+kα-2Mα-2μ,α=2,…,n-1,α<μ≤n,k0=kn=0.Theorem 5.
The time evolution equations for the curvatures take the form(15)k1,t=M12,s-k1λ-k2M13,kα,t=Mαμ,s-kαλ+kα-1Mα-1μ-kα+1Mαμ+1,k0=kn=0.For n=3, we can study the motion of curves in R3; we choose k1=k,k2=τ,v1=W,v2=U, and v3=V; then we have the following.Theorem 6.
The time evolution of the Serret-Frenet frame can be written in matrix form as follows:(16)Ft=M·F,where(17)F=TNB,M=0M12M13-M120M23-M13-M230,(18)M12=kW+Us-τV,M13=Vs+τU,M23=1kM13,s+τM12.Theorem 7.
The time evolution of the curvature and torsion of the curveCt can be given by(19)kτt=-gt2g-M13M13-gt2gkτ+M12,sM23,s.Definition 8.
An inextensible curve is a curve whose length is preserved; that is, it does not evolve in time:(20)∂s∂t=0,i.e.,gt=g˙=0.The necessary and sufficient conditions for inextensible flows are then given by the following theorem.Theorem 9.
The flow of the curve is inextensible if and only if∂W/∂s=kU.Lemma 10.
If the curveγ(s,t) is inextensible (gt=g˙=0), then the evolution equations for the curvature and torsion (19) are(21)kτt=0-M13M130kτ+M12,sM23,s.Then the PDE system (21) can be written explicitly in the following form:(22)kt=k2-τ2U+Uss+ksW-τsV-2τVs,τt=kVs+τU+1k-τ2V+τkW+2Us+Vss+τsU,s.


## 5. Examples of Inextensible Flows of Curves inR3

Example 1.
If(23)W=constant=a≠0,U=0,V=ks,ta,the PDE system (22) takes the form(24)kt=1a-kτs+a2-2τks,τt=kaks+τa-τ2+a2+kssk,s.One solution of this system is(25)ks,t=-2c1sech⁡c1s+c2t+c3,τs,t=aac1-c22c1,where c1, c2, and c3 are constants.
The curvature of the family of curvesCt as a function of the coordinates s and t is plotted in Figure 1.
Substitute (23) and (25) into systems (6) and (16) and solve them numerically. Then we can get the family of curves Ct=γ(s,t), so we can determine the surface that is generated by this family of curves (Figure 2).Figure 1
The curvature of the family of curvesCt for s∈[0,5], t∈[0,8], a=0.5, c1=1, c2=-0.001, and c3=0.Figure 2
The surface that is generated by the motion of the family of curvesCt for s∈[0,5], t∈[0,8], a=0.5, c1=1, c2=-0.001, and c3=0. The bold black curves in this surface represent the family of curves Ct for t=0,4,7.Example 2.
If(26)W=constant=a,U=0,and assuming that τ(s,t)=k(s,t), then PDE system (22) takes the form (27)τt=aks-τsV-2τVs,τt=kVs+1k-τ2V+aτk+Vss,s.One solution of this system is(28)τs,t=ks,t=c4tanh⁡c1s+c2t+c3,Vs,t=ac1-c2c1,where c1, c2, c3, and c4 are constants.
The curvature of the family of curvesCt as a function of the coordinates s and t is plotted in Figure 3.
Substitute (26) and (28) into systems (6) and (16) and solve them numerically. Then we can get the family of curves Ct=γ(s,t), so we can determine the surface that is generated by this family of curves (Figure 4).Figure 3
The curvature of the family of curvesCt for s∈[0,5], t∈[0,3], a=0.6, c1=1, c2=0.01, c3=0, and c4=1.Figure 4
The surface that is generated by the motion of the family of curvesCt for s∈[0,5], t∈[0,3], a=0.6, c1=1, c2=0.01, c3=0, and c4=1. The bold black curves in this surface represent the family of curves Ct for t=0,1,2.Example 3.
If(29)W=ks,t,U=ksk,V=b=constant≠0,then the PDE system (22) takes the form (30)kt=k2-τ2ksk+kskss+kks-bτs,τt=kτksk+1k-τ2b+τk2+2ksks+τsksk,s.One solution of this system is(31)ks,t=τs,t=-2c1bsech⁡c1-s+bt+bc2b,where c1 and c2 are constants.
The curvature of the family of curvesCt as a function of the coordinates s and t is plotted in Figure 5.
Substitute (29) and (31) into systems (6) and (16) and solve them numerically. Then we can get the family of curves Ct=γ(s,t), so we can determine the surface that is generated by this family of curves (Figure 6).Figure 5
The curvature of the family of curvesCt for s∈[0,8], t∈[0,3], b=1.5, c1=0.8, and c2=0.5.Figure 6
The surface that is generated by the motion of the family of curvesCt for s∈[0,8], t∈[0,3], b=1.5, c1=0.8, and c2=0.5. The bold black curves in this surface represent the family of curves Ct for t=0,1,2,2.9.


### 5.1. Examples of Binormal Motion of Inextensible Curves

Consider the binormal motion of inextensible curves inR3, so W=U=0. Then the evolution equation (9) takes the form(32)r˙=drdt=VB.Lemma 11.
Consider the binormal motion of inextensible curves inR3; then the time evolution of the Serret-Frenet frame can be written in matrix form as follows:(33)Ft=M·F,where(34)F=TNB,M=0-τVVsτV01k-τ2V+Vss-Vs-1k-τ2V+Vss0.The PDE system (22) takes the form(35)kt=-τsV-2τVs,τt=kVs+1k-τ2V+Vss,s.Example 4.
The famous example of binormal motion is the motion of the vortex filament inR3, where the binormal velocity equals the curvature of the curve V=k, and the evolution equation is (36)r˙=drdt=kB. If V=k, then the PDE system (35) takes the form(37)kt=-kτs-2τks,τt=kks+1k-τ2k+kss,s.The solution of the PDE (37) is(38)ks,t=-2c1sech⁡c1s+c2t+c3,τs,t=τ0=a2c1ac1-c2,where c1,c2, and c3 are constants.
The curvature of the family of curvesCt as a function of the coordinates s and t is plotted in Figures 7 and 8.
Substitute (38) into (6), (33) for V=k and solve them numerically. Then we can get the family of curves γ(s,t), so we can determine the surface that is generated by this family of curves. For τ=τ0=-1/2 see (Figure 9), and for τ=τ0=0 see (Figure 10).Figure 7
The curvature of the family of curvesCt for s∈[0,10], t∈[0,15], c1=1, c2=-1, and c3=0.Figure 8
The curvature of the family of curvesCt for s∈[0,5], t∈[0,6.3], c1=1, c2=0, and c3=0.Figure 9
The surface that is generated by the motion of the family of curvesCt for τ0=-1/2, s∈[0,10], t∈[0,15], c1=1, c2=-1, and c3=0. The bold black curves in this surface represent the family of curves Ct for t=0,6.Figure 10
The surface that is generated by the motion of the family of curvesCt for τ0=0, s∈[0,5], t∈[0,6.3], c1=1, c2=0, and c3=0. The bold black curves in this surface represent the family of curves Ct for t=0,1,3,5.Example 5.
IfV=constant=a, then (35) takes the form(39)kt=-aτs,τt=-aτ2k,s.The solution of the PDE system (39) is(40)ks,t=ac1c2c4+c5tanh⁡c1s+c2t+c3,τs,t=c4+c5tanh⁡c1s+c2t+c3,where c1, c2, c3, c4, and c5 are constants.
The curvature of the family of curvesCt as a function of the coordinates s and t is plotted in Figure 11.
Substitute (40) into (6), (33) and solve them numerically. Then we can get the family of curves γ(s,t), so we can determine the surface that is generated by this family of curves (Figure 12).Figure 11
The curvature of the family of curvesCt for s∈[0,8], t∈[0,3], b=1.5, c1=0.8, and c2=0.5.Figure 12
The surface that is generated by the motion of the family of curvesCt for s∈[0,8], t∈[0,3], b=1.5, c1=0.8, and c2=0.5. The bold black curves in this surface represent the family of curves Ct for t=0,1,2,2.9.


## 5.1. Examples of Binormal Motion of Inextensible Curves

Consider the binormal motion of inextensible curves inR3, so W=U=0. Then the evolution equation (9) takes the form(32)r˙=drdt=VB.Lemma 11.
Consider the binormal motion of inextensible curves inR3; then the time evolution of the Serret-Frenet frame can be written in matrix form as follows:(33)Ft=M·F,where(34)F=TNB,M=0-τVVsτV01k-τ2V+Vss-Vs-1k-τ2V+Vss0.The PDE system (22) takes the form(35)kt=-τsV-2τVs,τt=kVs+1k-τ2V+Vss,s.Example 4.
The famous example of binormal motion is the motion of the vortex filament inR3, where the binormal velocity equals the curvature of the curve V=k, and the evolution equation is (36)r˙=drdt=kB. If V=k, then the PDE system (35) takes the form(37)kt=-kτs-2τks,τt=kks+1k-τ2k+kss,s.The solution of the PDE (37) is(38)ks,t=-2c1sech⁡c1s+c2t+c3,τs,t=τ0=a2c1ac1-c2,where c1,c2, and c3 are constants.
The curvature of the family of curvesCt as a function of the coordinates s and t is plotted in Figures 7 and 8.
Substitute (38) into (6), (33) for V=k and solve them numerically. Then we can get the family of curves γ(s,t), so we can determine the surface that is generated by this family of curves. For τ=τ0=-1/2 see (Figure 9), and for τ=τ0=0 see (Figure 10).Figure 7
The curvature of the family of curvesCt for s∈[0,10], t∈[0,15], c1=1, c2=-1, and c3=0.Figure 8
The curvature of the family of curvesCt for s∈[0,5], t∈[0,6.3], c1=1, c2=0, and c3=0.Figure 9
The surface that is generated by the motion of the family of curvesCt for τ0=-1/2, s∈[0,10], t∈[0,15], c1=1, c2=-1, and c3=0. The bold black curves in this surface represent the family of curves Ct for t=0,6.Figure 10
The surface that is generated by the motion of the family of curvesCt for τ0=0, s∈[0,5], t∈[0,6.3], c1=1, c2=0, and c3=0. The bold black curves in this surface represent the family of curves Ct for t=0,1,3,5.Example 5.
IfV=constant=a, then (35) takes the form(39)kt=-aτs,τt=-aτ2k,s.The solution of the PDE system (39) is(40)ks,t=ac1c2c4+c5tanh⁡c1s+c2t+c3,τs,t=c4+c5tanh⁡c1s+c2t+c3,where c1, c2, c3, c4, and c5 are constants.
The curvature of the family of curvesCt as a function of the coordinates s and t is plotted in Figure 11.
Substitute (40) into (6), (33) and solve them numerically. Then we can get the family of curves γ(s,t), so we can determine the surface that is generated by this family of curves (Figure 12).Figure 11
The curvature of the family of curvesCt for s∈[0,8], t∈[0,3], b=1.5, c1=0.8, and c2=0.5.Figure 12
The surface that is generated by the motion of the family of curvesCt for s∈[0,8], t∈[0,3], b=1.5, c1=0.8, and c2=0.5. The bold black curves in this surface represent the family of curves Ct for t=0,1,2,2.9.


## 6. Geometric Properties of the Generated Surfaces

LetΣ=γ(s,t) be the surface that is generated by the motion of the family of curves Ct. In this section, we study some geometric properties of these surfaces.Lemma 12.
The first fundamental form of the surfaceΣ=γ(s,t) in R3 is given by (41)I=g11ds2+2g12dsdt+g22dt2,where g11,g12,g22 are the first fundamental quantities and they are given by (42)g11=γs,γs=1,g12=γs,γt=W2+U2+V2,g22=γt,γt=W.Lemma 13.
The unit normal vectorn(s,t) to the surface Σ=γ(s,t) in R3 at a point p=γ(s,t) on the surface is given by (43)ns,t=1U2+V2UB-VN.Lemma 14.
The second fundamental form of the surfaceΣ=γ(s,t) in R3 is given by (44)II=L11ds2+2L12dsdt+L22dt2,where L11,L12,L22 are the second fundamental quantities and they are given by (45)L11=γss,n=-kVU2+V2,L12=γst,n=1U2+V2-UM13+VM12,L22=γtt,n=1U2+V2UVt-UtV+WUM13-VM12+U2+V2M23,where M12, M13, and M23 are given from (18).Lemma 15.
For the surfaceΣ=γ(s,t) in R3, the Gaussian and the Mean curvatures K and H, respectively, are given by (46)K=det⁡IIdet⁡I=L11L22-L122g11g22-g122,H=12L11g22+L22g11g11g22-g122.Lemma 16.
The Gaussian curvatureK and the Mean curvature H for the surface in (Figure 2) are given by (47)Ks,t=-kssk2,Hs,t=12k3+kτ2-kssk2.Lemma 17.
The Gaussian curvatureK and the Mean curvature H for the surface in (Figure 4) are given by (48)Ks,t=0,Hs,t=-c2-ac1τac1-c2.Hence the surface in Figure 4 is developable surface.Lemma 18.
The Gaussian curvatureK and the Mean curvature H for the surface in (Figure 6) are given by (49)Ks,t=-kssk2,Hs,t=12k3+kτ2-kssk2.Lemma 19.
The Gaussian curvatureK and the Mean curvature H for the surfaces in Figures 9 and 10 are given by (50)Ks,t=-kssk,Hs,t=-12k3+kτ02-kssk2.Lemma 20.
The Gaussian curvatureK and the Mean curvature H for the surface in (Figure 12) are given by (51)Ks,t=0,Hs,t=-12k2+kτ2k.Hence the surface in (Figure 12) is developable surface.


## 7. Conclusion

In this paper we studied the inextensible flows of curves inR3. We constructed and plotted the surfaces that are generated from the motion of inextensible curves in R3. Also, we studied some geometric properties of those surfaces.



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