# Involute-evolute curves in Galilean space [G.sub.3].

[section]1. Introduction and preliminaries

C. Boyer discovered involutes while trying to build a more accurate clock [1]. Later, H. H. Hacisalihoglu [2] gave the relations Frenet apparatus of involute-evolute curve couple in the space [E.sup.3]. A. Turgut [3 examined involute-evolute curve couple in [R.sup.n]. At the beginning of the twentieth century, Cayley-Klein discussed Galilean geometry which is one of the nine geometries of projective space. After that, the studies with related to the curvature theory were maintained [4>5>61 and A. O. Ogrenmis [7] and et al. studied the properties of the curves in the Galilean space were studied. In this paper, we define involute-evolute curves couple and give some theorems and conclusions, which are known from the classical differential geometry, in the three dimensional Galilean space G3. We hope these results will be helpful to mathematicians who are specialized on mathematical modeling.

The Galilean space G3 is a Cayley-Klein space equipped with the projective metric of signature (0,0,+,+), as in E. Molnar's paper [8]. The absolute figure of the Galilean Geometry consist of an ordered triple {w, f, I}, where w is the ideal (absolute) plane, f is the line (absolute line) in w and I is the fixed elliptic involution of points of f [6]. In the non-homogeneous coordinates the similarity group [H.sub.8] has the form

[bar.x] = [a.sub.11] + [a.sub.12]x,

[bar.y] = [a.sub.21] + [a.sub.22]x + [a.sub.23]y cos [phi] + [a.sub.23]z sin [phi],

[bar.z] = [a.sub.31] + [a.sub.32]x - [a.sub.23]y sin [phi] + [a.sub.23]z cos [phi]. (1)

Where [a.aub.ij] and [phi] are real numbers [5].

In what follows the coefficients [a.sub.12] and [a.sub.23] will play the special role. In particular, for [a.sub.12] = [a.sub.23] = 1, (1) defines the group [B.sub.6] [subset] [H.sub.8] of isometries of Galilean space [G.sub.3].

In [G.sub.3] there are four classes of lines:

i) (proper) non-isotropic lines- they don't meet the absolute line f.

ii) (proper) isotropic lines- lines that don't belong to the plane w but meet the absolute line f.

iii) unproper non-isotropic lines-all lines of w but f.

iv) the absolute line f.

Planes x =constant are Euclidean and so is the plane w. Other planes are isotropic.

If a curve C of the class [C.sup.r] (r [greater than or equal to]3) is given by the parametrization

r = r(x, y(x), z(x)) (2)

then x is a Galilean invariant the arc length on C.

The curvature is

[kappa] = [square root of y"[(x).sup.2] + z"[(x).sup.2]] (3)

and torsion is

[tau] = 1/[[kappa].sup.2]det(r'(x), r"(x), r'"(x)). (4)

The orthonormal trihedron is defined

T(s) = [alpha]'(s) = (1,y' (s),z' (s))

N(s) = 1/[kappa](s)(0,y"(s),z"(s))

B(s) = 1/[kappa](s)(0, -z"(s),y"(s)).

The vectors T, N, B are called the vectors of tangent, principal normal and binormal line of [alpha], respectively. For their derivatives the following Frenet formulas hold [9]

T'(s) = [kappa](s)N (s)

N'(s)= [tau](s)B(s)

B'(s) = -[tau](s)N (s). (6)

Galilean scalar product can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

Where [u.sub.1] = ([x.sub.1], [y.sub.1], [z.sub.1]) and [u.sub.2] = ([x.sub.2], [y.sub.2], [z.sub.2]). It leaves invariant the Galilean norm of the vector u = (x, y, z) defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

[section]2. Involute-evolute curves in Galilean space

In this section, we give a definition of involute-evolute curve and obtain some theorems about these curves in [G.sub.3].

Definition 2.1. Let [alpha] and [alpha]* be two curves in the Galilean space [G.sub.3]. The curve [alpha]* is called involute of the curve a if the tangent vector of the curve [alpha] at the point [alpha](s) passes through the tangent vector of the curve [alpoha]* at the point [alpha]*(s) and

(T,T*} = 0,

where {T, N,B} and {T*, N*, B*} are Frenet frames of [alpha] and [alpha[*, respectively. Also, the curve [alpha] is called the evolute of the curve [alpha]*.

This definition suffices to the define this curve mate as (see Figure 1.)

[alpha]* = [alpha] + [lambda]T.

[FIGURE 1 OMITTED]

Theorem 2.2. Let [alpha] and [alpha]* be two curves in the Galilean space [G.sub.3]. If the curve [alpha]* is an involute of the curve a, then the distance between the curves [alpha] and [alpha]* is [lambda[(s), where [lambda](s) = |c - s|.

Proof. From definition of involute-evolute curve couple, we know

[alpha]*(s) = [alpha](s) + [lambda](s)T (s). (9)

Differentiating both sides of the equation (9) with respect to s and use the Frenet formulas, we obtain

T* (s) = T(s) + d[lambda]/ds T(s) + [lambda](s)[kappa](s)N(s).

Since the curve [alpha]* is involute of [alpha], <T, T*> = 0.

Then we have

d[lambda]/ds + 1 = 0. (10)

From the last equation, we easily get

[lambda](s) = c - s. (11)

Where c is constant. Thus, the equation (9) can be written as

[alpha]*(s) - [alpha](s) = (c - s)T(s). (12)

Taking the norm of the equation (12), we reach

||[alpha]*(s) - [alpha](s)|| = |c - s|. (13)

This completes the proof.

Theorem 2.3. Let [alpha] and [alpha]* be two curves in Galilean space [G.sub.3]. [kappa], t and [kappa]*, [tau]* be the curvature functions of [alpha] and [alpha]*, respectively. If [alpha] is evolute of [alpha]* then there is a relationship

[kappa]* = [tau]/(c -s)[kappa],

where c is constant and s is arc length parameter of [alpha].

Proof. Let Frenet frames be {T, N, B} and {T, N, B} at the points [alpha](s) and [alpha]*(s), respectively. Differentiating both sides of equation (12) with respect to s and using Frenet formulas, we have following equation

T*(s) ds*/ds = (c - s)[kappa](s)N(s) (14)

where s and s are the arc length parameter of the curves [alpha] and [alpha]*, respectively. Taking the norm of the equation (14), we reach

ds*/ds =(c - s)[kappa](s) (15)

and

T* = N. (16)

By taking the derivative of equation (16) and using the Frenet formulas and equation (15), we obtain

[kappa]*N* = [tau]/(c - s)[kappa] B. (17)

From the last equation, we get

[kappa]* = [tau]/(c -s)[kappa]*.

Theorem 2.4. Let [alpha] be the non-planar evolute of curve [alpha]*, then [alpha] is a helix.

Proof. Under assumption s and s* are arc length parameter of the curves [alpha] and [alpha]*, respectively. We take the derivative of the following equation with respect to s

[alpha]*(s) = [alpha](s) + [lambda](s)T (s).

We obtain that

T* ds*/ds = [lambda][kappa]N.

{T*, N} are linearly dependent. We may define function as

f = (T,T* [disjuntction] N*)

and take the derivative of the function f with respect to s, we obtain

f = -[tau] {T, N*) . (18)

From the equation (18) and the scalar product in Galilean space, we have

f' = 0.

That is,

f = const.

The velocity vector of the curve a always composes a constant angle with the normal of the plane which consist of [alpha]* . Then the non-planar evolute of the curve [alpha]* is a helix.

Theorem 2.5. Let the curves [beta] and [gamma] be two evolutes of [alpha] in the Galilean space [G.sub.3]. If the points [P.sub.1] and [P.sub.2] correspond to the point P of [alpha], then the angle [P.sub.1][??][P.sub.2] is constant.

Proof. Let's assume that the curves [beta] and [gamma] be two evolutes of [alpha] (see Figure 2.). And let the Frenet vectors of the curves [alpha], [beta] and [gamma] be {T,N,B}, {T*,N*,B*} and {[??], [??], [??]}, respectively.

[FIGURE 2 OMITTED]

Following the same way in the proof of the theorem 2.4, it is easily seen that {T, N*} and {T, [??]} are linearly dependent.

Thus,

<T,T*> = 0 (19)

and

<T, [??]> = 0. (20)

When [theta] is an angle between tangent vector T* and [??], we define a function f. That is,

f (s) = (T*, [??]). (21)

Then, differentiating equation (21) with respect to s, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

where s, s* and [??] are arc length parameter of the curves [alpha], [beta] and [gamma], respectively. Also, [kappa], [kappa]* and [??] are the curvatures of the curves [alpha], [beta] and[gamma], respectively. Considering the equations (19), (20) and (22), we get

f '(s)=0.

This means that

f = const.

So, [theta] is constant. The proof is completed.

References

[1] C. Boyer, A history of Mathematics, New York, Wiley, 1968.

[2] Hacisalihoglu. H. H., Diferensiyel Geometri, Ankara Universitesi Fen Fakultesi, 2000.

[3] A. Turgut and E. Erdogan, Involute-evolute curve couples of higher order in Rn and their horizontal lifts in Rn, Common. Fac. Sci. Univ. Ank. Series A, 41(1992), No. 3, 125-130.

[4] Roschel. O., Die Geometrie des Galileischen Raumes, Habilitationsschrift, Leoben, 1984.

[5] B. J. Pavkovic and I. Kamenarovic, The equiform differentiel geometry of curves in the Galilean space [G.sub.3], Glasnik Matematicki, 22(1987), 449-457.

[6] Z. M. Sipus, Ruled Weingarten surfaces in the Galilean space, Periodica Mathematica Hungarica, 56(2008), No. 2, 213-225.

[7] A. O. Ogrenmis, M. Ergut and M. Bektas, On the helices in the Galilean space [G.sub.3], Iranian J. of Sci. Tec, Trans. A., 31(2007), No. 2, 177-181.

[8] E. Molnar, The projective interpretation of the eight 3-dimensional homogeneous geometries, Beitrage zur Algebra und Geometrie Contributions to Algebra and Geometry, 38(1997), No. 2, 261-288.

[9] I. Kamenarovic, Existence theorems for ruled surfaces in the Galilean space [G.sub.3], Rad Hrvatskeakad. Znan. Umj. Mat., 10(1991), 183-196.

M. Akyigit ([dagger]), A. Z. Azak ([double dagger]) and S. Ersoy (#)

Faculty of Arts and Science, Department of Mathematics, Sakarya University, 54187, Sakarya, Turkey

E-mail: makyigit@sakarya.edu.tr apirdal@sakarya.edu.tr sersoy@sakarya.edu.tr