A CHARACTERIZATION OF CYLINDRICAL HELIX STRIP

In this paper, we investigate cylindrical helix strips. We give a new denition and a characterization of cylindrical helix strip. We use some charecterizations of general helix and the Terquem theorem (one of the Joachimsthal Theorems for constant distances between two surfaces).


Introduction
In 3-dimensional Euclidean Space, a regular curve is described by its curvatures k 1 and k 2 and also a strip is descibed by its curvatures k n , k g and t r : The relations between the curvatures of a strip and the curvatures of the curve can be seen in many di¤erential books and papers.We know that a regular curve is called a general helix if its …rst and second curvatures k 1 and k 2 are not constant, but k1 k2 is constant ( [2]; [7]).Also if a helix lie on a cylinder, it is called a cylindrical helix and a cylindrical helix has the strip at (s): The cylindrical helix strips provide being a helix condition and cylindrical helix condition at the point (s) of the strip by using the curvatures of helix k 1 and k 2 : 2. Preliminaries 2.1.The Theory of the Curves.De…nition 2.1.If : I R !E n is a smooth transformation, then is called a curve (from the class of C 1 ).Here I is an open interval of R ( [11]).

F • IL • IZ ERTEM KAYA;YUSUF YAYLI AND H. H • ILM • I HACISAL • IHO ¼ GLU
Figure1 The curve in E n De…nition 2.2.Let the curve E n be a regular curve coordinate neigbourhood and fV 1 (s); V 2 (s); :::; V r (s)gbe the Frenet frame at the point (s) that correspond for every s 2 I. Accordingly, We know that the function k i is called i th curvature function of the curve and the real number k i (s) is called i th curvature of the curve for each s 2 I ( [2]).The relation between the derivatives of the Frenet vectors among and the curvatures are given with a theorem as follows: De…nition 2.3.Let M E n be the curve with neigbouring (I; ).Let s 2 I be arc parameter.If k i (s) and {V 1 (s); V 2 (s); :::; V r (s)g be the i th curvature and the Frenet r-frame at the point (s), then 8 < : The equations that about the covariant derivatives of the Frenet r-frame fV 1 (s); V 2 (s); :::; V r (s)g the Frenet vectors V i (s) along the curve can be written as 2 These formulas are called Frenet Formulas ( [2]).
In special case if we take n = 3 above the last matrix equations, we obtain following matrix the equation : The …rst curvature of the curve k 1 (s) is called only curvature and the second curvature of the curve k 2 (s) is known as torsion ( [2]): or the equations are as follows, 2.2.The Strip Theory.
De…nition 2.4.Let M and be a surface in E 3 and a curve in M E 3 .We de…ne a surface element of M is the part of a tangent plane at the neighbour of the point.The locus of these surface element along the curve is called a strip or Let !t be the tangent vector …eld of the curve , ! n be the normal vector …eld of the curve and !b be the binormal vector …eld of the curve : : Strip vector …elds of a strip which belong to the curve are n ! ; ! ; !o .These vector …elds are; Strip tangent vector …eld is ).

Figure 3 Strip and curve vector …elds in E 3
Let be a curve in M E 3 : ) and ! is a unit strip vector …eld of a surface M at the point (s); than we have !j (s) = !j (s) !j (s) ([6]).
That is !j (s) is perpendicular !j (s) and also !j (s) :So we obtain n ! ; ! ; !o orthonormal vector …elds system is called strip three-bundle ([6]).We can see that ! ; ! ; !n ; !b vectors are in the same surface from the Figure 4. then we obtain the following equations The Equations of the Strip Vector Fields in type of the Frenet vector Fields.
and ' be the Frenet Vector …elds, strip vector …elds and the angle between ! and !n .We can write the following equations by the Figure 4: By the help of the Figure 4 we can write ( [4]; [8]).From last two equations we obtain, This equation is a relation between the curvature of a curve and normal curvature and geodesic curvature of a strip ([6]; [10]).By using similar operations, we obtain a new equation as follows 6]; [10]).This equation is a relation between (torsion or second curvature of ) and a; b; c curvatures of a strip that belongs to the curve : And also we can write a = ' + : The special case: ¬f ' =constant, then ' = 0: So the equation is a = : That is, if the angle is constant, then torsion of the strip is equal to torsion of the curve.De…nition 2.6.Let be a curve in M E 3 .If the geodesic curvature (torsion) of the curve is equal to zero, then the curve-surface pair ( ; M ) is called a curvature strip ( [6]).

General Helix
De…nition 3.1.Let be a curve in E 3 and V 1 be the …rst Frenet vector …eld of :U 2 (E 3 ) be a constant unit vector …eld.If ; ' and SpfU g is called an general helix, the slope angle and the slope axis ([1]; [2]).).
x(t) = cos t y(t) = sin t z(t) = t: As the parameter t increases, the point (x(t); y(t); z(t)) traces a right-handed helix of pitch 2 and radius 1 about the z-axis, in a right-handed coordinate system.In cylindrical coordinates (r; ; h), the same helix is parametrised by If the curve is a general helix, the ratio of the …rst curvature of the curve to the torsion of the curve must be constant.The ratio is called …rst Harmonic Curvature of the curve and is denoted by H 1 or H: Proof.( )) Let be a general helix.The slope axis of the curve is shown as SpfU g.Note that (s); U = cos ' = constant.If the Frenet trehold is {V 1 (s); V 2 (s); V 3 (s)} at the point (s), then we have hV 1 (s); U i = cos ': If we take derivative of the both sides of the last equation, then we have By using the last equation, we see that (() Let H(s) be constant for 8s 2 I; and = tan ', then we obtain 1) If U is a constant vector,then we have By substituting H(s) = tan ' is in the last equation, we see that and so U = constant.
2) If is an inclined curve with slope axis SpfU g.Since  De…nition 3.9.Let M be a cylinder in E 3 ; and be a helix on M: The part of the tangent plane on the cylindrical helix is called the surface element of the cylinder.The locus of the surface element along the cylindrical helix is called a strip of cylindrical helix.Theorem 3.10.Suppose that i0:Then is a strip of cylindrical helix if and only if the ratio { is constant.
Proof.Let be the angle between the tangent vector …eld and slope vector u of a strip of cylindrical helix.Since :u = cos is constant, we have 0 = ( :u) = u = :u Because {i0 and :u = 0, we see that u is perpendicular to and so u = cos : + sin : : By di¤erentiating the last equation,   By applying the similar way in the proof of Theorem II.3.11 in [6] to the strip of cylindric helix strip, we give the following theorem.Theorem 3.12.Let L and M be be a cylindrical helix and a surface in E 3 .Suppose that L and M have common tangent plane along and cylindrical helix : If the curve-surface pair ( ; M ) is a curvature strip, then the curve is a a helix strip. Proof.
Figure 7 The cylinder L and the surface M: If the curve is a helix on L, then it provides 1 1 is constant.We have to show that is a helix strip on M; that is, 2  2 =constant.By the Figure 7, we have where (s 1 ) = !m + r 1 (s 1 ): (3) By di¤erentiating both side of (3), we see that ; we obtain a 1 = 0 and b 1 = 1: Let r be the distance between gravity center of the cylinder and (s 1 ).We denote r = 1: If !m is a position vector of the gravity center of cylinder, then !m must be a constant vector.Since a 1 = 0, ( ; L) is a curvature strip.By the strips ( ; L) and ( ; M ) are curvature strips and by Theorem 17, we see that is non-zero constant.Let !v (s 1 ) be a vector in Sp n !
Since the vector !m and are constant, we obtain the following equation By (1), we see that Since the cylindric helix and the surface M have the same tangent plane along the curves and , we can write By subsitituting (6) in the last equation, we obtain cos ' = 0: By using that equation in (6), we have If we calculate the second and third derivatives of the curve , then we get Since the same result is obtained by using the other form of (7), we use the form By [6], we have ; a 1 = 0 By ( 8) and ( 9), we see that By using (10) in F 3 ; we obtain We obtain the proof of theorem from last equation.
and the curvatures of the curve are shown as k 1 = and k 2 = ,

2 .
The Equations of the Frenet Vector Fields in type of the Strip Vector Fields.

3 .
Some Relations between a; b; c invariants (Curvatures of a Strip) and , invariants (Curvatures of a Curve).We know that a curve has two curvatures and .A curve has a strip and a strip has three curvatures k n ; k g and t r .kn = b k g = c t r = a ([4];[6]).From the derivative equations we can write

De…nition 3 . 7 .
Let be a helix that lie on the cylinder.A helix which lies on the cylinder is called cylindrical helix.

Figure 5
Figure 5 Cylindrical helix De…nition 3.8.Let M be a cylinder in E 3 ; and be a helix on M: We de…ne a surface element of M as the part of a tangent plane at the neighbourhood of a point of the cylindrical helix.The locus of the surface element along the cylindrical helix is called a helix strip.
Since tan is constant, is also constant ([9]): Theorem 3.11.(Terquem Theorem) Let M 1 ; M 2 be two di¤ erent surfaces in E 3 .Let and be nonplanar curve in M 1 and a curve on M 2 .i.The points of the curves and corresponds to each other 1:1 on a plane " which rolls on the M 1 ve M 2 , such that the distance is constant between corresponding points.ii. ( ; M 1 ) is a curvature strip.iii.( ; M 2 ) is a curvature strip ([6]).Claim: Two of the three lemmas gives third ([6]).

( 7 )
in the rest of our proof.By di¤erentiating both sides of