The last ten years, the developable ruled surfaces are studied by many mathematicians. Developable surfaces are a type of important and fundamental surfaces universally used in industry design. Different methods have been presented for the design of developable surfaces. The use of developable surfaces in ship design is of engineering importance because they can be easily manufactured without stretching or tearing, or without the use of heat treatment. In some cases, a ship hull can be entirely designed with the use of developable surfaces. See, [4,8,9,10,11].

On the other hand, a conformal map is a function which preserves the angles. The conformal mapping is an important technique used in complex analysis and has many applications in different physical situations.

An inversion with respect to the spherewith the center and the radius r given by

This paper is organised as follows: The authors firstly tell inversions and inversions of surfaces in . Secondly, they give the fundamental forms, the curvatures(Gauss and mean), the shape operator and the Christoffel symbols of the tangent developable surface. Finally, using by these properties, we obtain these of the inverse surface of the tangent developable surface.

Let and We denote that . Then, an inversion of with the center and the radius r is the map

given by

(2.1)

Definition 2.1. [7] Let be an inversion with the center c and the radius r then, the tangent map of Ф at is the map

given by

where

Now, let us assume that is the patch of a surface. The inverse patch of X with respect to is the patch given by

(2.2)

Throughout this paper, the authors assume that Φ is an inversion of with the center c and the radius r, X is a patch in and Y is inverse patch of X with respect to Φ.

Let I_x , II_x and K_x , H_x be the first and second fundamental forms and the curvatures (Gauss and mean) of X, and let I_y , II_y , and K_y , H_y be these of Y, respectively. From[1], we have

(2.3)

(2.4)

(2.5)

(2.6)

where and

Let be a curve with arc-length s and {T,N,B} be Frenet frame along γ. Denote by κ and τ the curvature and the torsion of the curve γ, respectively. Then we have Frenet formulas

The tangent developable of γ is a ruled surface parametrized by

(3.1)

where T is unit tangent vector field of γ. As it is known, the coefficients of the first and second fundamental forms of the surface M(s,u) have following

(3.2)

and

(3.3)

The normal vector field of the surface M(s,u) is given by

(3.4)

Next the curvatures (mean and Gaussian) and the matrix of shape operator of this surface are respectively as follows

(3.5)

and

(3.6)

Finally, the Christoffel symbols of the surface M(s,u) are given by

(3.7)

where

The authors show that N is the inverse surface of the tangent developable surface M with respect to the inversion Ф. Thus the inverse surface N has following parametrization

(4.1)

Hence, if we take into account the equalities (2.3) and (2.4) then the coefficients of the first and second fundamental forms of the inverse surface N by the help of these of the surface M are given by

(4.2)

(4.3)

where E_N , F_N , G_N and e_N, f_N, g_N are the coefficients of the first and second fundamental forms of the inverse surface  respectively.

Morever, the Gauss and mean curvatures of the inverse surface N by the help of these of the surface M are respectively, using by (2.5) and (2.6),

(4.4)

(4.5)

where K_N and H_N are the Gauss and the mean curvatures of the inverse surface N, respectively

Theorem 4.1. Let N be the inverse surface of the tangent developable surface M with respect to the inversion Ф. Denote by S_N the matrix of the shape operator of the inverse surface N, then S_N is given by the help of that of M as follows

(4.6)

Proof. Let S_M be the matrix of the shape operator of surface M. By using the equalities (2.3) and (2.4) we can write

(4.7)

where I₂ is identity and Hence from (3.6) and (4.7), we obtain that the equality (4.6) is satisfied.

Theorem 4.2. Let be the Christoffel symbols of the inverse surface N. The Christoffel symbols of the inverse surface N by the help of these of the surface M are given by

Proof. Considering the equality (2.3), for i, j , k = 1, we can write

where and is the Christoffel symbol of the tangent developable surface. Thus, from the equalities (3.2) and (3.7), we obtain

Others are found in similar way.

Theorem 4.3. Let N be the inverse surface of the tangent developable surface M with respect to the inversion Ф. Then the inverse surface N is a flat surface if and only if either the normal lines to the surface M or the tangent planes of the surface M pass through the center of inversion.

Proof. Let us assume that the inverse surface N is flat, then from (4.4), we can write

(4.8)

where either

(4.9)

or

(4.10)

If the equality (4.9) is satisfied, then the tangent planes of the surface M pass through the center of inversion. If the equality (4.10) holds, then it follows

Namely, the normal lines to the surface M pass through the center of inversion.

The proof of sufficient condition is obvious.

Theorem 4.4. Let N be the inverse surface of the tangent developable surface M with respect S_c(r), to The inverse surface N is minimal if and only if the normal lines to the surface M pass through the center of inversion (Figures 1 & 2)

Proof. The proof is same with that of Theorem 4.1.

Figure 1. The helicoid given by (u cos v, u sin v, 2v)

Figure 2. The inverse surface of the helicoid with respect to unit sphere given by