98

International Journal of Physical and Mathematical Sciences Vol 3, No 1 (2012)

ISSN: 2010-1791

International Journal of Physical and Mathematical Sciences journal homepage: http://icoci.org/ijpms

A Note on Tubes Fatih Doğan Ankara University, Department of Mathematics, 06100 Ankara, Turkey [email protected] RECEIVED DATE (2012-2-22)

Abstract. In this paper, we look into tube surface with different orthonormal frames of its center curve. Also, we give the relations between parameter curves and asymptotic curves, geodesics on the generalized tubes. Keywords. Tube, Generalized tube, Bishop frame, Darboux frame, Asymptotic curve, Geodesic

1. Introduction A canal surface is the envelope of a moving sphere with varying radius, defined by the trajectory C (t ) (center curve) of its center and a radius function r (t ) . If the center curve C (t ) is a helix and the radius function r (t ) is a constant, then the surface is called helical canal surface. If the radius function r (t ) is a constant, this time the canal surface is called a tube. Gross [5] defined that a generalized tube (briefly GT) is the surface constructed by sweeping some planar closed curve along an arbitrary 3D space curve. He classified them in two types as ZGT and CGT where ZGT refers to the center curve (the axis) that has torsion-free and CGT refers to tube that has circular cross sections (as a matter of fact tubes with Frenet frame). More recently, Doğan and Yaylı [3] studied tubes with Bishop frame. The same authors [4] also studied tubes with Darboux frame as well. Since the characteristic circles of tube lie in the plane which is orthogonal to the tangent vector of the center curve C (t ) , we can classify tubes with respect to different orthonormal frames of C (t ) . In this paper, first we examine the relations between parameter curves and asymptotic curves, geodesics on these tubes. Afterwards we research this property on the generalized tubes and give some characterizations about it.

2. Preliminaries In the first place, we present different orthonormal frames of a unit speed space curve . 1) Let be a unit speed curve with nonzero curvature . Then the Frenet frame is defined along the curve ,

Ts s

Ns

s

s

Bs Ts ns, where s is arclenght parameter, T is the tangent, N is the principal normal and B is the binormal of the curve . The Frenet-Serret equations are given by

T s sNs

N s sTs sBs

B s sNs ' 2) Let be a unit speed curve with tangent T ( s ) ( s) and let N 1 ( s ) be arbitrary orthogonal unit vector to

T (s ) . If N 2 ( s) is orthogonal to both T (s) and N 1 ( s ) , then N 2 ( s) = T ( s) N 1 ( s) (positively oriented) in other

99

International Journal of Physical and Mathematical Sciences Vol 3, No 1 (2012)

ISSN: 2010-1791

words { T (s) , N 1 ( s ) , N 2 ( s) } is an orthonormal frame. This frame is called Bishop frame. If we rotate the Bishop frame by the angle around the tangent vector T , we obtain the Frenet frame as below.

T N

1

B

0

T

0

0 cos sin

N1

0 sin cos

N2

Let k1 ( s) and k 2 ( s ) be Bishop parameters (normal development). The derivative formulas which correspond to Bishop frame and Bishop parameters are as follows.

T k 1 N1 k 2 N2

N 1 k 1 T

N 2 k 2 T k 1 cos k 2 sin

. 3) Let M be a regular surface with unit normal U and let a curve : I R M be a unit speed. Then, Darboux frame {T , Y U T , U } is well-defined along the curve and the Darboux equations are given by

T kgY k nU

Y k g T g U

U k n T g Y, where T is tangent of the curve and k g , k n and g are the geodesic curvature, the normal curvature and the geodesic torsion of , respectively. Let M be a regular surface. Then the Gaussian and mean curvature are computed by

eg f 2 EG F 2 eG 2fF gE H , 2EG F 2 K

where E , F , G respectively.

and

e , f , g are the coefficients of the first and second fundamental form of M ,

Definition 1. Let M be a regular surface and let x (u , v ) be a parametrization of M with x(0,0) p . A regular curve : I M , (t ) x (u (t ), v (t )) , ( t I ) is an asymptotic curve if and only if

II t 0 eu 2 2fu v gv 2 0, where II is the second fundamental form of M and e e(u , v ) , f f (u , v ) , g g (u, v ) are the coefficients of it. Therefore,

eu 2 2fu v gv 2 0 is called the differential equation of asymptotic curves on M [2] .

100

International Journal of Physical and Mathematical Sciences Vol 3, No 1 (2012)

ISSN: 2010-1791

Theorem 1. A necessary and sufficient condition for the parameter curves of a surface to be asymptotic curves in 2 a neighborhood of a hyperbolic point (eg f

International Journal of Physical and Mathematical Sciences Vol 3, No 1 (2012)

ISSN: 2010-1791

International Journal of Physical and Mathematical Sciences journal homepage: http://icoci.org/ijpms

A Note on Tubes Fatih Doğan Ankara University, Department of Mathematics, 06100 Ankara, Turkey [email protected] RECEIVED DATE (2012-2-22)

Abstract. In this paper, we look into tube surface with different orthonormal frames of its center curve. Also, we give the relations between parameter curves and asymptotic curves, geodesics on the generalized tubes. Keywords. Tube, Generalized tube, Bishop frame, Darboux frame, Asymptotic curve, Geodesic

1. Introduction A canal surface is the envelope of a moving sphere with varying radius, defined by the trajectory C (t ) (center curve) of its center and a radius function r (t ) . If the center curve C (t ) is a helix and the radius function r (t ) is a constant, then the surface is called helical canal surface. If the radius function r (t ) is a constant, this time the canal surface is called a tube. Gross [5] defined that a generalized tube (briefly GT) is the surface constructed by sweeping some planar closed curve along an arbitrary 3D space curve. He classified them in two types as ZGT and CGT where ZGT refers to the center curve (the axis) that has torsion-free and CGT refers to tube that has circular cross sections (as a matter of fact tubes with Frenet frame). More recently, Doğan and Yaylı [3] studied tubes with Bishop frame. The same authors [4] also studied tubes with Darboux frame as well. Since the characteristic circles of tube lie in the plane which is orthogonal to the tangent vector of the center curve C (t ) , we can classify tubes with respect to different orthonormal frames of C (t ) . In this paper, first we examine the relations between parameter curves and asymptotic curves, geodesics on these tubes. Afterwards we research this property on the generalized tubes and give some characterizations about it.

2. Preliminaries In the first place, we present different orthonormal frames of a unit speed space curve . 1) Let be a unit speed curve with nonzero curvature . Then the Frenet frame is defined along the curve ,

Ts s

Ns

s

s

Bs Ts ns, where s is arclenght parameter, T is the tangent, N is the principal normal and B is the binormal of the curve . The Frenet-Serret equations are given by

T s sNs

N s sTs sBs

B s sNs ' 2) Let be a unit speed curve with tangent T ( s ) ( s) and let N 1 ( s ) be arbitrary orthogonal unit vector to

T (s ) . If N 2 ( s) is orthogonal to both T (s) and N 1 ( s ) , then N 2 ( s) = T ( s) N 1 ( s) (positively oriented) in other

99

International Journal of Physical and Mathematical Sciences Vol 3, No 1 (2012)

ISSN: 2010-1791

words { T (s) , N 1 ( s ) , N 2 ( s) } is an orthonormal frame. This frame is called Bishop frame. If we rotate the Bishop frame by the angle around the tangent vector T , we obtain the Frenet frame as below.

T N

1

B

0

T

0

0 cos sin

N1

0 sin cos

N2

Let k1 ( s) and k 2 ( s ) be Bishop parameters (normal development). The derivative formulas which correspond to Bishop frame and Bishop parameters are as follows.

T k 1 N1 k 2 N2

N 1 k 1 T

N 2 k 2 T k 1 cos k 2 sin

. 3) Let M be a regular surface with unit normal U and let a curve : I R M be a unit speed. Then, Darboux frame {T , Y U T , U } is well-defined along the curve and the Darboux equations are given by

T kgY k nU

Y k g T g U

U k n T g Y, where T is tangent of the curve and k g , k n and g are the geodesic curvature, the normal curvature and the geodesic torsion of , respectively. Let M be a regular surface. Then the Gaussian and mean curvature are computed by

eg f 2 EG F 2 eG 2fF gE H , 2EG F 2 K

where E , F , G respectively.

and

e , f , g are the coefficients of the first and second fundamental form of M ,

Definition 1. Let M be a regular surface and let x (u , v ) be a parametrization of M with x(0,0) p . A regular curve : I M , (t ) x (u (t ), v (t )) , ( t I ) is an asymptotic curve if and only if

II t 0 eu 2 2fu v gv 2 0, where II is the second fundamental form of M and e e(u , v ) , f f (u , v ) , g g (u, v ) are the coefficients of it. Therefore,

eu 2 2fu v gv 2 0 is called the differential equation of asymptotic curves on M [2] .

100

International Journal of Physical and Mathematical Sciences Vol 3, No 1 (2012)

ISSN: 2010-1791

Theorem 1. A necessary and sufficient condition for the parameter curves of a surface to be asymptotic curves in 2 a neighborhood of a hyperbolic point (eg f