OSCULATING SURFACES ALONG A CURVE ON A SURFACE IN EUCLIDEAN 3-SPACE

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. We define an osculating surface to a surface along a curve on the surface in Euclidean 3-space E3. Then, we analyze the necessary and sufficient condition for that surface to be ruled surface. Finally, we illustrate the convenience and efficiency of this approach by some representative examples.


INTRODUCTION
The problem of finding surfaces with a given common curve as a special curve play an important role in geometric design. The first paper related with this type of problem proposed by Wang et.al. [1]. They parameterized the surface by using the Serret-Frenet frame of the given curve and gave the necessary and sufficient condition to satisfy the geodesic requirement.
The basic idea is to regard the wanted surface as an extension from the given characteristic curve, and represent it as a linear combination of the marching-scale functions u(s,t), v(s,t), w(s,t) and the three vector functions t(s), n(s), b(s), which are the unit tangent, the principal normal and the binormal vector of the curve respectively. With the given geodesic curve and isoparametric constraints, they derived the necessary and sufficient conditions for the correct parametric representation of the surface pencil. The extension to ruled and developable surfaces is also outlined. Kasap et al. [2] generalized the marching-scale functions of Wang and gave a sufficient condition for a given curve to be a geodesic on a surface. With the inspiration of work of Wang, Li et al. [3] changed the characteristic curve from geodesic to line of curvature and defined the surface pencil with a common line of curvature. Bayram et al. [4] tackled the problem of constructing surfaces passing through a given asymptotic curve. Important contributions to surface passing through a given curve have been studied in [5,6,7,8].
However, the relevant work on surfaces through characteristic curve on a surface depending on the Darboux frame is rare. So, this led us to offer an approach for designing a surface possessing a given curve on a surface. We call it an osculating surface along the curve. Then, we analyze the necessary and sufficient condition for that surface to be an osculating ruled surface.
Moreover, we illustrate the convenience and efficiency of this approach by some representative examples.

PRELIMINARIES
In this section, we list some notions, formulas and conclusions for space curves, and ruled surfaces in Euclidean 3-space E 3 which can be found in the textbooks on differential geometry (See for details [9,10]).
Let α : I ⊆ R → E 3 be a unit speed curve; κ(s) and τ(s) denote the natural curvature and torsion of α = α(s), respectively. We assume α (s) = 0 for all s ∈ [0, L], since this would give us a straight line. In this paper, α (s) denote the derivative of α with respect to arc length parameter s. For each point of α(s), the set {t(s), n(s), b(s)} is called the Serret-Frenet frame along α(s), where t(s) = α (s) is the unit tangent, n(s) = α (s)/ α (s) is the unit principal normal, and b(s) = t(s) × n(s) is the unit binormal vector. The arc-length derivative of the Serret-Frenet frame is governed by the relations: Let M be a regular surface, and α : I ⊆ R → M is a unit speed curve on M. If we denote the Darboux frame along the curve α = α(s) by {e 1 (s), e 2 (s), e 3 (s)}; t = e 1 (s) be the unit tangent vector, e 3 = e 3 (s) is the surface unit normal restricted to α, and e 2 = e 3 ×e 1 be the unit tangent to the surface M. Then, the rotation matrix between Serret-Frenet frame and Darboux frame is Hence, we have the derivative formulae of the Darboux frame as: κ n = κ sin ϑ =< e 1 , e 3 >, We call κ g = κ g (s) a geodesic curvature, κ n = κ n (s) a normal curvature, and τ g = τ − ϑ a geodesic torsion of α(s), respectively. In terms of these quantities, the geodesics, asymptotic lines, and line of curvatures on a smooth surface may be characterized, as loci along which κ g = 0, κ n = 0, and τ g = 0, respectively. Further, we have:

OSCULATING SURFACES
In this section, we consider an osculating surface along a regular curve α = α(s) on the surface M such that the surface tangent plane is coincident with the subspace Sp{e 1 , e 2 }, that is expressing the surface in terms of the local Darboux frame along α(s) as: where u(s,t), and v(s,t) are all C 1 functions. If the parameter t is seen as the time, the functions u(s,t), and v(s,t) can then be viewed as directed marching distances of a point unit in the time t in the direction e 1 ; and e 2 , respectively, and the position vector α(s) is seen as the initial location of this point on M. It is easily checked that the two tangent vectors of M O are given by: The lower case subscript letters s, and t denote partial derivatives corresponding to the indicated variable, e.g., P s = ∂ P ∂ s , P t = ∂ P ∂t . Thus, the normal vector of M O is η 2 (s,t) = u t (uκ n + vτ g ), Our goal is to find the necessary and sufficient conditions for which the surface M O is osculating to the surface M along α(s). First, since α(s) is an isoparametric curve on the surface Secondly, when t = t 0 , i.e., along the curve α(s) of M, the surface normal is This is the reason why we call M O , the osculating surface of M along the curve α(s). Then we have the following theorem.
We will call the set of surfaces defined by Eqs. (6) and (12) isoparametric osculating surfaces, since the common curve is an isoparametric curve on these surfaces. Any osculating surface M O defined by Eq. (6) and satisfying Eqs. (12) is a member of them. For the purposes of simplification and better analysis, next we study the case when the marching-scale functions u(s,t), and w(s,t) can be written as follows: Here l(s), n(s),U(t) and V (t) are C 1 functions, not identically zero. Thus, from Theorem 1, we can get the following corollary.  (1) If we choose Thus, we can simply express the sufficient condition for which M O being an osculating surface along the curve α(s) of M as: where l(s), m(s), U(t), and V (t) are C 1 functions, a i j ∈ R (i = 1, 2; j = 1, 2, ..., p) and l(s), and n(s) are not identically zero.
(2) If we choose then, we can rewrite the condition (14) as: where l(s), n(s), U(t), V (t), f , and g are C 1 functions.
We consider a surface of revolution parameterized by M : X(s,t) = (s, e s sint, e s cost) .
(14) are satisfied. Hence, the osculating surface can be represented as follows: In

OSCULATING RULED SURFACES
A ruled surface is a surface generated by a straight line L moving along a curve. In this subsection, we will discuss the construction of osculating ruled surfaces. Suppose that M O is a ruled surface along the curve α(s) of M, then there exists t 0 such that P(s,t 0 ) = α(s).
This follows that the ruled surface can be expressed as where e(s) denotes the direction of the rulings. According to the Eq. (6), we have and thus when t = 0, i.e., along the curve α(s), the surface normal is  This surface is a ruled surface such that the base curve is Thus, α(s) is a regular curve on the surface M. Then, we obtain √ (3+cos(s))(13+3 cos(s)) The functions β (s) and γ(s) can control the shape of the surface and it is very clear that if these functions are given, then we immediately obtain an osculating surface in the family. In the following, we consider two cases: Case (1)

CONCLUSION
In the three-dimensional Euclidean space E 3 , an osculating surface to a surface along a curve on this surface has been defined. Then, the necessary and sufficient conditions for that surface to be a ruled surface have been investigated. Meanwhile, we have illustrated the convenience and efficiency of this approach by some representative examples.