**University of Mazandaran, Iran**
http://cjms.journals.umz.ac.ir
**ISSN: 1735-0611**

CJMS.**8**(1)(2019), 58-73

**Evolution of Space Curves Using Type-3 Bishop Frame**

M.A. Soliman, Nassar H.Abdel-All, R. A. Hussien and

Taha Youssef1

1 _{Department of Mathematics Faculty of Science, Assiut University,}
Assiut, Egypt.

Abstract. In this paper, Firstly, we introduce a new version of
Bishop frame using a common vector ﬁeld as normal vector ﬁeld of
a regular curve and call this frame as type-3 Bishop frame.
Geomet-ric visualization for both Frenet frame and type-3 Bishop frame are
plotted. Secondly, time evolution equation for type-3 Bishop frame
and type three bishop curvatures are obtained and we plotted the
evolving curve via the numerical integration of type-3 Bishop Serret
Frenet equations. Finally, surfaces generated by evolving a regular
space curve in*R*3 _{as it evolves over time with arbitrary velocities}

attached to type-3 Bishop frame are obtained. We plotted these surfaces via numerical integration of Gauss-Weingarten equations.

Keywords: Type-3 Bishop frame, Serret-Frenet equations, Time evolution equation, Gauss-Weingarten equations.

*2000 Mathematics subject classiﬁcation:* 14H50; Secondary 53A04.

1. Introduction

Hasimoto established a connection between integrable equation and a moving curve in the context of a vortex ﬁlament motion [1] which got mapped to the nonlinear Schr¨odinger equation (NLSE). Soon afterward, Lamba [2] presented the evolution of space curves in three dimensional space by considering tow sets of Frenet-Serret equations that describe the

1_{Corresponding author: [email protected]}

Received: 13 July 2018 Accepted: 12 August 2018

evolution for the tangent, normal, and binormal vectors to the curve with
respect to arc-lenght*s*and time*t. He obtained coupled nonlinear partial*
diﬀerential equations for the curvature *κ* and torsion *τ* of the curve by
applying compatibility conditions on these vectors. Under certain
con-ditions showed that these turn out to be nonlinear Schr¨odinger equation,
sine-Gordon equation and Hirota equation. Subsequently, Doliwa and
Santini [3] established a connection between the motion of inextensible
curves and solitonic systems via an embedding in a space of constant
curvature.

In the same period, Langer and Perline [4] showed that the
dynam-ics of non-stretching vortex ﬁlament in*R*3 leads to the NLS hierarchy.
Lakshmanan et al. [5], derived the Heisenberg spin chain equation via
the spatial motion of a curve. Doliwa and Santini [6] studied geometric
characterization of the integrable motions of a curve on*s*2 _{and} * _{s}*3

_{.}

Recently, Nakayama et al.[7, 8] investigated motion of plane and space curves and obtained time evolution equation of the moving frame and the curvatures of the evolving curve.

All above authors are used Frenet frame that associated to a space
curve to mobilize the curve in three dimensional space while T. K¨orpinar
and E. Turhan [9] used type-2 Bishop Frame in*E*3to study inextensible
ﬂows of a space curve according to type-2 Bishop Frame. They obtained
time evolution equation for type-2 Bishop Frame and type-2 Bishop
curvatures which governing the evolution of the curve.

In this paper, Firstly, we introduce a new version of Bishop frame
using a common vector ﬁeld as normal vector ﬁeld of a regular curve
and call this frame as type-3 Bishop frame. Geometric visualization for
both Frenet frame and type-3 Bishop frame are plotted. Secondly, time
evolution equation for type-3 Bishop frame and type three bishop
curva-tures are obtained and we plotted the evolving curve via the numerical
integration of type-3 Bishop Serret Frenet equations. Finally, surfaces
generated by evolving a regular space curve in*R*3 as it evolves over time
with arbitrary velocities attached to type-3 Bishop frame are obtained.
We plotted these surfaces via numerical integration of Gauss-Weingarten
equations.

surfaces generated by evolving a regular space curve in*R*3 as it evolves
over time with arbitrary velocities attached to type-3 Bishop frame are
constructed and plotted.

2. Bishop frame along a space curve

The theory of space curves are mainly extended by the Frenet-Serret
theorem which expresses the derivative of a geometrically chosen basis
of *E*3 by the aid of itself is proved. Recently, L.R. Bishop [10] in 1975
established a frame that associated to a space curve using a common
vector ﬁeld as tangent vector ﬁeld of a regular space curve.

There are a catalogue of frames that associated to a space curve such as Frenet frame [12], Bishop frame [10, 13, 14], Kepler frame[15] and quasi frame [16] . In this section we, we introduce a new version of Bishop frame using a common vector ﬁeld as normal vector ﬁeld of a regular curve and call this frame as type-3 Bishop frame.

Let*⃗***r** =*⃗***r**(s) be a space curve embedded in three-dimensional space,
generated by the vector*⃗***r**(s) = (**x1**(s),**x2**(s),**x3**(s)), where*s*is the
arc-length. Let*⃗***t***, ⃗***n** and *⃗***b** denotes, respectively, the unit tangent, the unit
normal, the unit binormal vectors on the curve. The orthogonal
right-handed triad (*⃗***t***, ⃗***n***, ⃗***b**) evolving on the curve according to Serret-Frenet
equations [11]

*d*
*ds*

*⃗*_{t}*⃗*

**n**

*⃗*

**b**

=

* _{−}*0

*κ*

*κ*0

*τ*0 0

*−τ*0

*⃗*_{t}*⃗*

**n**

*⃗*

**b**

*.* (2.1)

The curvature *κ* and the torsion*τ* are function of*s*and given by
*κ*=*⃗***n***.d⃗***t**

*ds,*

*τ* =*−⃗***n***.d⃗***b**
*ds,*

(2.2)

its well known that a curve can be uniquely represented by specifying*κ*
and *τ* for a stated orientation.

If we rotate Frent frame (*⃗***t***, ⃗***n***, ⃗***b**) by angel *ρ* about the normal we
get a new frame denoted by (**b***⃗***1***, ⃗***b2***, ⃗***b3**), then the relation between two
frames is given by

*⃗*

**b1**
*⃗*

**b2**
*⃗*

**b3**

=

cos0*ρ(s)* 01 sin0*ρ(s)*
*−*sin*ρ(s)* 0 cos*ρ(s)*

*⃗*_{t}*⃗*

**n**

*⃗*

**b**

Thus,
*⃗*_{t}*⃗*
**n**
*⃗*
**b**
=

cos0*ρ(s)* 01 *−*sin0*ρ(s)*
sin*ρ(s)* 0 cos*ρ(s)*

*⃗*
**b1**
*⃗*
**b2**
*⃗*
**b3**

*.* (2.4)

Let,
*F* =
*⃗*_{t}*⃗*
**n**
*⃗*
**b**
*, A*1 =

cos0*ρ(s)* 01 sin0*ρ(s)*
*−*sin*ρ(s)* 0 cos*ρ(s)*

*, A*2=

cos0*ρ(s)* 01 *−*sin0*ρ(s)*
sin*ρ(s)* 0 cos*ρ(s)*

*,*
*B* =
*⃗*
**b1**
*⃗*
**b2**
*⃗*
**b3**
*, K* =

* _{−}*0

*κ*

*κ*0

*τ*0 0

*−τ*0

*,*

(2.5) then 2.1, 2.3 and 2.4 can be written as

*∂F*

*∂s* =*KF,*
*B* =*A*1*F,*
*F* =*A2Q.*

(2.6)

By using the above equation the rate of change of type-3 Bishop Serret-Frenet frame, is given by

*dB*
*ds* = (

*dA1*

*ds* *.B*+*A1.K.A2)B,* (2.7)
by using equation 2.5 we have

*d*
*ds*
*⃗*
**b1**
*⃗*
**b2**
*⃗*
**b3**
=

0 (κcos*ρ(s)−τ*sin*ρ(s))*

*dρ*
*ds*

*−*(κcos*ρ(s)−τ*sin*ρ(s))* 0 (κsin*ρ(s) +τ*cos*ρ(s))*
*−dρ*

*ds* *−*(κsin*ρ(s) +τ*cos*ρ(s))* 0

*⃗*
**b1**
*⃗*
**b2**
*⃗*
**b3**
*.*
(2.8)
Thus we can write the above equation in the form,

**d**
**d***s*
*⃗*
**b1**
*⃗*
**b2**
*⃗*
**b3**
=

* _{−}*0

*κ*1

*κ1*0

*κ*02 0

*−κ2*0

*⃗*
**b1**
*⃗*
**b2**
*⃗*
**b3**

*,* (2.9)

where type-3 Bishop curvatures are

*κ*1 =*κ*cos*ρ(s)−τ*sin*ρ(s),*
*κ*2 =*κ*sin*ρ(s) +τ*cos*ρ(s),*

(2.10) where

The frame (**b***⃗***1***, ⃗***b2***, ⃗***b3**) is properly oriented, We shall call the set (**b***⃗***1***, ⃗***b2***, ⃗***b3***, κ*1*, κ*2)
as type-3 Bishop invariants of the curve*⃗***r**=*⃗***r**(s).

There is a more compact way of writing the type-3 Bishop Frenet
equations which gives a better insight into the structure of these
equa-tions. The evolution of an orthonormal type-3 Bishop frame (**b***⃗***1***, ⃗***b2***, ⃗***b3**)
deﬁned along the curve may be speciﬁed by its angular velocity*⃗ω*through
the relations

*d ⃗***b1**

*ds* =*⃗ω×*
*⃗*

**b1***,*
*d ⃗***b2**

*ds* =*⃗ω×***b***⃗***3***,*
*d ⃗***b3**

*ds* =*⃗ω×*
*⃗*

**b3***.*

(2.12)

Since (**b***⃗***1***, ⃗***b2***, ⃗***b3**) comprise a basis for*R*3 we can write
*⃗*

*ω*=*ω1***b***⃗***1**+*ω2***b***⃗***2**+*ω3***b***⃗***3***,* (2.13)
thus equations 2.12 becomes

*d ⃗***b1**

*ds* =*−ω2***b***⃗***3**+*ω3***b***⃗***2***,*
*d ⃗***b2**

*ds* =*ω*1
*⃗*

**b3***−ω*3**b***⃗***1***,*
*d ⃗***b3**

*ds* =*−ω*1**b***⃗***2**+*ω*2**b***⃗***1***.*

(2.14)

Comparing equations 2.14 with equations 2.9 we have
*ω1* =*κ2,*

*ω2* = 0,
*ω*3 =*κ*1*.*

(2.15)

thus the Darbuox vector for type-3 Bishop frame is given by
*⃗*

*ω* =*κ*2**b***⃗***1**+*κ*1**b***⃗***3***.* (2.16)
Thus when the triad evolves along the curve the (**b***⃗***2***, ⃗***b3**)-couple
ro-tates around (**b***⃗***1** with an angular velocity *κ*2 and the (**b***⃗***2***, ⃗***b1**)-couple
rotates around (**b***⃗***3** with an angular velocity *κ1.*

2.1. **Example 1.** In this subsection, ﬁrst we show how to ﬁnd type-3
Bishop trihedra for a space curve,

The Frenet frame and curvatures are calculated and given by

*⃗*_{t}_{= (}cos* _{√}s−s*sin

*s*

*s*2

_{+2}

*,*

cos* _{√}s*+

*s*sin

*s*

*s*2

_{+2}

*,*

1

*√*

*s*2_{+2})
*⃗***n**= (*−*(*s√*2+4) sin*s−*(*s*3+3*s*) cos*s*

(*s*2_{+2)(}* _{s}*4

_{+5}

*2*

_{s}_{+8)}

*,*

(*s*2_{+4) sin}_{s}_{−}_{(}* _{s}*3

_{+3}

_{s}_{) cos}

_{s}*√*

(*s*2_{+2)(}* _{s}*4

_{+5}

*2*

_{s}_{+8)}

*,*

*−s*

*√*

(*s*2_{+2)(}* _{s}*4

_{+5}

*2*

_{s}_{+8)})

*⃗*

_{b}_{= (}

*s*sin

*s−*2

*s*cos

*s*

(*s*4_{+5}* _{s}*2

_{+8)}

*,−*2 sin

_{(}

*4*

_{s}_{+5}

*s*+2

*2*

_{s}_{+8)}

*s*cos

*s,*

*s*2

_{+2}(

*s*4

_{+5}

*2*

_{s}_{+8)})

*κ*=

√

8+5*s*2_{+}* _{s}*4
(2+

*s*2

_{)}

*τ*=

_{8+5}(6+

*s*22)

_{s}_{+}3

*/*24

_{s}(2.18) Type-3 Bishop frame and curvatures are calculated and given by

*⃗*

**b1**= ((cos*√s _{s}−*2

*s*

_{+2}sin

*s*) cos

*ρ*+ (

*s*sin*s−*2*s*cos*s*

(*s*4_{+5}* _{s}*2

_{+8)}) sin

*ρ,*(cos

*√s*+

*s*sin

*s*

*s*2_{+2} ) cos*ρ*+ (*−*

2 sin*s*+2*s*cos*s*

(*s*4_{+5}* _{s}*2

_{+8)}) sin

*ρ,*(

*√*1

*s*2_{+2}) cos*ρ*+ (

*s*2_{+2}

(*s*4_{+5}* _{s}*2

_{+8)}) sin

*ρ),*

*⃗*

**b2**= (cos*√s _{s}*+2

*s*

_{+2}sin

*s,*

*−*cos* _{√}s*+

*s*sin

*s*

*s*2

_{+2}

*,*0),

*⃗*

**b3**= (*s*_{(}sin* _{s}*4

_{+5}

*s−*22

_{s}*s*

_{+8)}cos

*s*) cos

*ρ*+ (

cos* _{√}s−s*sin

*s*

*s*2

_{+2}) sin

*ρ,*(

*−*2 sin

_{(}

*4*

_{s}_{+5}

*s*+2

*2*

_{s}_{+8)}

*s*cos

*s*) cos

*ρ*+ (cos

*√s*+

*s*sin

*s*

*s*2_{+2} ) sin*ρ,*
(_{(}* _{s}*4

_{+5}

*s*2+2

*2*

_{s}_{+8)}) cos

*ρ*+ (

*√*12

_{s}_{+2}) sin

*ρ),*

*κ1*=

√

8+5*s*2_{+}* _{s}*4

(2+*s*2_{)} cos*ρ*+

(6+*s*2)3*/*2
8+5*s*2_{+}* _{s}*4 sin

*ρ,*

*κ*2=

√

8+5*s*2_{+}* _{s}*4

(2+*s*2_{)} sin*ρ*+

(6+*s*2_{)}3*/*2
8+5*s*2_{+}* _{s}*4 cos

*ρ,*

(2.19)

*b*

_{1}

_{1}

*b*

_{3}

_{3}

*b*

_{1}

_{1}

*b*

_{3}

_{3}

*r*

*t*

_{n}

_{n}

*b*

*r*

*t*

*n*

*b*

*X*

*Y*

*Z*

Figure 1. Frenet frame and type-3 Bishop frame at two

point along the curve defended by the position vector
*⃗***r**(s) = (scos(s), ssin(s), s).

3. Evolution of a space curve with time by Bishop frame

In this section we study the evolution of a regular space curve using Bishop frame. We derive time evolution equation for Bishop frame and Bishop curvatures.

Let, the curve evolving in the space according to
*∂⃗***r**

where*µ1, µ2, andµ3* are functions of s and t, correspond to the normal,
binormal and tangent projections of the velocity. Below we restrict our
attention to a purely local form for these velocities as in the form

*µ1* = *µ1(κ, κs, ..., τ, τs, ...),*

*µ2* = *µ2(κ, κs, ..., τ, τs, ...),*

*µ*3 = *µ*3(κ, κ*s, ..., τ, τs, ...).*

If the curve evolves in time *t, then the general temporal evolution in*
which the triad (**b***⃗***1***, ⃗***b2***, ⃗***b3**) remains orthonormal adopts the form [18]

*∂*
*∂t*
*⃗*
**b1**
*⃗*
**b2**
*⃗*
**b3**
=

* _{−}*0

*γ*1

*γ1*0

*γ*02 0

*−γ*2 0

*⃗*
**b1**
*⃗*
**b2**
*⃗*
**b3**

*,* (3.2)

where*γ1,γ2* and*γ3* are geometric parameters which are in general
func-tions of *s*and *t. These describe the evolution in t of the type-3 Bishop*
Serret-Frenet frame (**b***⃗***1***, ⃗***b2***, ⃗***b3**) on the curve.

From section 2 the rate of change of type-3 Bishop Serret-Frenet frame, is given by

*∂*
*∂s*
*⃗*
**b1**
*⃗*
**b2**
*⃗*
**b3**
=

* _{−}*0

*κ*1

*κ1*0

*κ*02 0

*−κ2*0

*⃗*
**b1**
*⃗*
**b2**
*⃗*
**b3**

*.* (3.3)

Applying the compatibility condition

*∂*
*∂s*
*∂*
*∂t*
*⃗*
**b1**
*⃗*
**b2**
*⃗*
**b3**
= *∂*
*∂t*
*∂*
*∂s*
*⃗*
**b1**
*⃗*
**b2**
*⃗*
**b3**

*,* (3.4)

a short calculation using Eqs. 3.2, 3.3 and 3.4 leads to
*∂κ*1
*∂t* =
*∂γ*1
*∂s* *,*
*∂κ*2
*∂t* =
*∂γ*2
*∂s* *,*

0 =*γ*2*κ*1*−γ*1*κ*2*.*

(3.5)

Applying the compatibility condition,
*∂*

*∂s*
*∂*
*∂t⃗***r**=

*∂*
*∂t*

*∂*

yields,

*∂µ*1

*∂s* =*µ*2*κ*1*,*
*∂µ*2

*∂s* =*γ*1*−µ*1*κ*1+*µ*3*κ*2*,*
*∂µ*3

*∂s* =*κ*2*µ*2*.*

(3.7)

For a given velocity vector (µ1*, µ*2*, µ*3), equations 3.5 and 3.7 form a set
of 6 nonlinear ﬁrst order partial diﬀerential equations which governing
the evolution of the type-3 Bishop curvatures of the evolving curve in
the space.

Equation 3.5 together gives
*∂κ*1

*∂t* =
*∂γ*1

*∂s* *,*
*∂κ*2

*∂t* =*γ*1
*∂*
*∂s*(

*κ*2
*κ1*) +

*κ*2
*κ1*

*∂γ*1
*∂s* *.*

(3.8)

From equation 3.7*γ1* is given by

*γ*1 =
*∂µ*2

*∂s* +*µ*1*κ*1*−µ*3*κ*2*,* (3.9)

*∂γ*1
*∂s* =

*∂*2*µ*2
*∂s*2 +

*∂µ*1

*∂s* *κ*1+*µ*1
*∂κ*1

*∂s* *−*
*∂µ*3

*∂s* *κ*2*−µ*3
*∂κ*2

*∂s* *,* (3.10)
using ﬁrst and third equation in the system 3.7 and substituting in 3.10,
we have

*∂γ1*
*∂s* =

*∂*2*µ2*
*∂s*2 +*µ2κ*

2
1+*µ1*

*∂κ1*
*∂s* *−µ2κ*

2
2*−µ3*

*∂κ2*

*∂s* *.* (3.11)
By substitution from 3.9 and 3.11 into 3.8 , the time evolution equation
of type-3 Bishop curvatures*κ*1 and *κ*2 are given by,

*∂κ1*
*∂t* =

*∂*2*µ2*
*∂s*2 +*µ2κ*

2
1+*µ1*

*∂κ1*
*∂s* *−µ2κ*

2
2*−µ3*

*∂κ2*
*∂s* *,*
*∂κ*2

*∂t* = (
*∂µ*2

*∂s* +*µ*1*κ*1*−µ*3*κ*2)
*∂*
*∂s*(

*κ*2
*κ*1

) + *κ*2
*κ*1

*∂*
*∂s*(

*∂*2*µ*2
*∂s*2 +*µ*2*κ*

2
1+*µ*1

*∂κ*1
*∂s* *−µ*2*κ*

2
2*−µ*3

while the time evolution of type-3 Bishop frame (**b***⃗***1***, ⃗***b2***, ⃗***b3**) on the curve
is given by,

*∂ ⃗***b1**
*∂t* = (

*∂µ*2

*∂s* +*µ*1*κ*1*−µ*3*κ*2)
*⃗*

**b2***,*
*∂ ⃗***b2**

*∂t* =*−*(
*∂µ*2

*∂s* +*µ*1*κ*1*−µ*3*κ*2)**b***⃗***1**+
*κ*2
*κ*1

(*∂µ*2

*∂s* +*µ*1*κ*1*−µ*3*κ*2)**b***⃗***2***,*
*∂ ⃗***b3**

*∂t* =*−*
*κ*2
*κ*1

(*∂µ*2

*∂s* +*µ*1*κ*1*−µ*3*κ*2)**b***⃗***2***.*

(3.13)

3.1. **model 1.** If, the velocity vector *⃗µ*= (µ1*, µ*2*, µ*3) is given by
*µ*1 = 0,

*µ2* = 0,
*µ3* = 1,

(3.14)

, then type-3 Bishop curvatures evolves according to
*∂κ1*

*∂t* =*−*
*∂κ2*

*∂s* *,*
*∂κ2*

*∂t* =*−κ2*
*∂*
*∂s*(

*κ2*
*κ*1

)*−κ2*
*κ*1 *−*

*∂κ2*
*∂s*

(3.15)

On solution of the above system is given by
*κ*1(s, t) = *−*

*c*1
*c*2

(c4+*c*5tanh(c1*s*+*c*2*t*+*c*3)),
*κ*2(s, t) =*c*4+*c*5tanh(c1*s*+*c*2*t*+*c*3).

(3.16)

If we have the curvature and the torsion of a space curve as a
func-tions of arc-length parameter, then by integrating Serret-Frenet we can
reconstruct the curve up to its position in the space and this is an
imme-diate consequence of the of the fundamental existence theorem for space
curves[17]. Similarly, if we have the Bishob curvatures *κ1* and *κ2, then*
we can reconstruct the curve in the space via the integration of type-3
Bishop frame.

Figure 2. Curve evolving in the space determined by

*κ1* =*−*tanh(s+*t), κ2* = + tanh(s+*t)*

4. Differential geometry of surfaces

To meet the requirements in the next sections, here, the basic elements
of the theory of surfaces in the space *E*3 are brieﬂy presented a more
complete elementary treatment can be found in [21].

We consider a surface imbedded in 3–dimensional Euclidean space

**R**3, we denote local coordinates of the surface by (u1*, u*2). The surface
is speciﬁed by the position vector**r**(u1*, u*2) where *u*1 =*s,* *u*2 =*t. The*
metric on the surface is given by,

*gµν* =*⃗***r***µ·⃗***r***ν* *µ, ν* = 1,2, (4.1)

we use the Einstein’s convention for summation. Here*⃗***r***ν* is the tangent

vector to the surface,

*⃗***r***µ*=

*∂⃗***x**

*∂uµ,* *µ*= 1,2. (4.2)

We denote the inverse of*gµν*by*gµν*. At regular points where the tangent

vectors **x**1*,***x**2 are linearly independent, we can deﬁne the unit normal
vector**N**to the surface,

**N**= *⃗***x**1*∧⃗***x**2
*|⃗***x**1*∧⃗***x**2*|*

These vectors are related by the Gauss–Weingerten equations [17],
*∂*

*∂uν⃗***x***µ*=*⃗***x***λ*Γ
*λ*

*µν*+**N***⃗Lµν,*

*∂ ⃗***N**

*∂uν* =*−⃗***x***λg*
*λµ _{L}*

*µν.*

(4.4)

In the above, the Christoﬀel’s symbols Γ*λ _{µν}* and the second fundamental
form are deﬁned as

Γ*λ _{µν}* = 1
2

*g*

*λρ*_{(} *∂*

*∂uµgρν*+

*∂*
*∂uνgµρ−*

*∂*

*∂uρgµν*), (4.5)

*Lµν* =

*∂⃗***x***µ*

*∂uν* *·***N***⃗.* (4.6)

The Gaussian curvature*κg* and the mean curvature *κm* are given by

*κg*=*det(gµνLνλ*) =

*L*
*g* =

*L11L22−L*2_{12}
*g*11*g*22*−g*122

*,* (4.7)

*κm*=

1
2*tr(g*

*µν _{L}*

*νλ*) =

*L11g22−*2L12g12+*L22g11*

2(g11g22*−g*2_{12}) *.* (4.8)
We recall that a curve in*E*3is uniquely determined by two quantities,
curvature and torsion, as functions of arc length. Similarly, a surface
in*E*3 _{is uniquely determined by certain local invariant quantities called}
the ﬁrst and second fundamental forms 4.1 and 4.1.

5. Surfaces generated by the evolution of a space curve

using Bishob frame

There are a catalog of surfaces that can be described by integrable
equations such as surfaces of constant negative Gaussian curvature,
sur-faces of constant mean curvature, minimal sursur-faces, aﬃne spheres and
Hasimoto surfaces which generated by evolving a regular space curve
*⃗***r**=*⃗***r**(s, t) by

*∂⃗***r**

*∂t* =*κ⃗***b***,* (5.1)

this is an evolution of the curve in its binormal direction with velocity
equal to its curvature. Eq.5.1 Known as the vortex ﬁlament ﬂow or
local-ized induction equation (LIE). Here,*⃗***r**(s, t) is a position vector for a point
on the curve,*t*is the time,*s*is the arc-length parameter,*κ*is the
curva-ture of*⃗***x**(s, t) , *⃗***b** is the unit binormal. The authors in [19] constructed
Hasimoto surfaces via integration for Frenet–Serret equations while the
authors in [20] constructed Hasimoto surface from fundamental form
coeﬃcients via numerical integration of Gauss-Weingarten equations.

curve *⃗***r**(s, t) in *R*3 as it evolves over time according to the following
evolution equation:

*∂⃗***r**

*∂t* =*µ*1**b***⃗***1**+*µ*2**b***⃗***2**+*µ*3**b***⃗***3** (5.2)
where the set *{λ, µ, ν}* is the velocities. This our motivation in this
paper.

The tangent space to *S* at an arbitrary point *P* = *⃗***r**(s, t) of *S* is
spanned by

*∂⃗***r**

*∂s* =*⃗***t***,*
*∂⃗***r**

*∂t* =*λ1***b***⃗***1**+*λ2***b***⃗***2**+*λ3***b***⃗***3***.*

(5.3)

Thus the moving frame along the surface is calculated and given by

*∂⃗***r**
*∂s*
*∂⃗***r**
*∂s*
*⃗*
**N**
=

1 0 0

*µ*1 *µ*2 *µ*3
0 *−√µ*3

*µ*2+*µ*23

*µ*2
*√*

*µ*2+*µ*23
*⃗*
**b1**
*⃗*
**b2**
*⃗*
**b3**
(5.4)

The coeﬃcient of the ﬁrst fundamental form and fundamental metric are calculated and given by

*g*11= 1,
*g*12=*µ*1*,*

*g22*=*µ*2_{1}+*µ*2_{2}+*µ*2_{3}*,*
*g*=*µ*2_{2}+*µ*2_{3}*.*

(5.5)

A short calculation shows that the coeﬃcients of the seconde fundamen-tal form are given as

*L11*= *−µ√*3*κ*1+*µ*2*κ*2

*µ*2
2+*µ*23

*,*

*L12*= *√−γ*1*µ*3

*µ*2
2+*µ*23

*,*

*L*22=

*γ*2*µ*22*−γ*1*µ*1*µ*3+*γ*2*µ*23*−*
*∂µ*_{2}

*∂t* *µ*3+*µ*2
*∂µ*_{3}

*∂t*
*√*

*µ*2
2+*µ*23
*L*= * _{µ}*2 1

2+*µ*32

(*−γ*_{1}2*µ3*2*−γ*_{2}2*µ*2_{2}*µ*3*κ*1+*γ*1*µ*1*µ*23*κ*1*−γ*2*µ*33*κ*1*−γ*2*µ*3*κ*1
+µ2_{3}*κ*1*∂µ _{∂t}*2

*−µ*2

*µ*3

*κ*1

*∂µ*2).

_{∂t}Figure 3. surface genrated by the curve evolving in

the space with the curvatures,*κ*1 =*−*tanh(s+*t), κ*2 =
tanh(s+*t) and fundamental forms 5.8*

.

For the curve evolving in space by the velocity vector*⃗µ*= (µ1*, µ*2*, µ*3)
*µ*1 = 0,

*µ2* = 0,
*µ3* = 1,

(5.7)

the coeﬃcient of the ﬁrst and seconde fundamental forms are given by,

*g11*= 1,
*g12*= 0,
*g*22= 1,

*L11*= tanh(s+*t),*
*L12*=*−*tanh(s+*t),*
*L*22=*−*tanh(s+*t).*

(5.8)

Thus, the Gaussian curvature *κg* and the mean curvature *κm* are

*κg* =*−*2 tanh(s+*t)*2*,*

*κm*= 0,

and the principle curvatures are

*κ11*=*√*2 tanh(s+*t) =√*2κ2*,*
*κ*22=*−*

*√*

2 tanh(s+*t) =√*2κ1*.*

(5.10) The surface in ﬁgure 3 is obtained via numerical integration of the system of Gauss-Weingarten equation 4.4 with the coeﬃcient 5.8 using mathematica [23].

6. Main result

In this paper, Firstly, we introduce a new version of Bishop frame
using a common vector ﬁeld as normal vector ﬁeld of a regular curve
and call this frame as type-3 Bishop frame. Geometric visualization for
both Frenet frame and type-3 Bishop frame are plotted. Secondly, time
evolution equation for type-3 Bishop frame and type three bishop
curva-tures are obtained and we plotted the evolving curve via the numerical
integration of type-3 Bishop Serret Frenet equations. Finally, surfaces
generated by evolving a regular space curve in*R*3 as it evolves over time
with arbitrary velocities attached to type-3 Bishop frame are obtained.
We plotted these surfaces via numerical integration of Gauss-Weingarten
equations.

Acknowledgements

The authors would like to thank the referees for their helpful com-ments and suggestions.

References

[1] L.S. Da Rios, Sul moto d’un liquido indeﬁnito con un ﬁletto vorticoso,
*Rend. Circ. Mat. Palermo.***22** *(1906) 117-135.*

[2] G.L. Lamb Jr., Solitons on moving space curves,*J. Math. Phys.***18***(1977),*
*1654-1661.*

[3] A. Doliwa and P. Santini, An elementary geometric characterisation of the
integrable motions of a curve, *Phys. Lett. A***185***(1994), 373-384.*

[4] J. Langer and R. Perline, Poisson geometry of the ﬁlament equation, *J.*
*Nonlinear Sci.***1***(1991), 71-93.*

[5] M. Lakshmanan, Th.W. Ruijgrok and C.J. Thompson, On the dynamics
of a continuum spin system,*Physica A,***84***(1976), 577-590.*

[6] A. Doliwa and P.M. Santini, An elementary geometric characterization of
the integrable motions of a curve,*Phys. Lett. A***185***(1994), 373-384.*
[7] Kazuaki Nakayama, Takeshi Iizuka, and Miki Wadati, Curve Lengthening

Equation and Its Solutions,*J. Phys. Soc. Jpn.* **63***(1994), 1311-1321.*
[8] Kazuaki Nakayama and Miki Wadati, Motion of Curves in the Plane, *J.*

*Phys. Soc. Jpn.***62***(1993), 473-479.*

[10] R. L. Bishop, There is more than one way to frame a curve,*Amer. Math.*
*Monthly.***82***(1975), 246-251.*

[11] L.P. Eisenhart, A Treatise on the Diﬀerential Geometry of Curves and
Sur-faces, *Dover Pub., Inc., New York, (1909).*

[12] M. do Carmo, Diﬀerential Geometry of Curves and Surfaces,*Prentice-Hall,*
*Englewood Cliﬀs, 1976.*

[13] S. ¨ulmaz, M. Turgut, A new version of Bishop frame and an application
to spherical images,*J. Math. Anal. Appl.***371***(2010), 764-776.*

[14] J. Bloomenthal, Calculation of Reference Frames Along a Space Curve,
Graphics Gems,*Academic Press Professional, Inc., San Diego, CA. 1990.*
[15] Baur O. and E.W. Grafarend , Orbital rotations of a satellite case study,

*GOCE, Artiﬁcial Satellites.* **40***(2005), 87-107.*

[16] M. Dede, C. Ekici and H. Tozak, Directional tubular surfaces,*International*
*Journal of Algebra.***9***(2015) 527-535.*

[17] M. M. Lipschutz, Theory and problems of Diﬀerential Geometry,*Schaum’s*
*Outline Series, 1969.*

[18] C. Rogers and W. K. Schief, Backlund and Darboux Transformations
Ge-ometry and Modern Application in Soliton Theory,*Cambridge University*
*press, Cambridge, 2002.*

[19] Peter J. VassiliouIan and G. Lisle, Geometric approaches to diﬀerential
equations,*Cambridge University Press 2000.*

[20] Nassar H. Abdel-All, R. A. Hussien and Taha Youssef, Hasimoto Surfaces,
*Life Science Journal***3***(2012), 1170-1185.*

[21] J.J.Stoker, Diﬀerential Geometry,*Wiley-Interscience, New York, 1969.*
[22] A. Gray, Modern Diﬀerential Geometry of Curves and Surfaces with

Math-ematica,*CRC, New York, 1998.*