Canal surfaces in 4-dimensional Euclidean space

Article History: Received 26 April 2016 Accepted 22 November 2016 Available 13 December 2016 In this paper, we study canal surfaces imbedded in 4-dimensional Euclidean space E. We investigate these surface curvature properties with respect to the variation of the normal vectors and ellipse of curvature. Some special canal surface examples are constructed in E. Furthermore, we obtain necessary and sufficient condition for canal surfaces to become superconformal in E. At the end, we present the graphs of projections of canal surfaces in E.

Given a space curve γ (u) called spine curve, a canal surface associated to this curve is defined as a surface swept by a family of spheres of varying radius r(u).If r(u) is constant, the canal surface is called a tube or a pipe surface.Apart from being used in pure mathematics, canal surfaces are widely used in many areas especially in CAGD, e.g.construction of blending surfaces, i.e. canal surface with a rational radius, shape reconstruction or robotic path planning (see, [5], [11], [12]).Greater part of the studies on canal surfaces within the CAGD context is related to the search of canal surfaces with rational spine curve and rational radius function.Canal surfaces are also useful in visualising long thin objects such as poles, 3D fonts, brass instruments or internal organs of the body in solid/surface modeling and CG/CAD.A national question is when the canal surface is developable.It is well known that, at regular points, the Gaussian curvature of a developable surface is identically zero.In [14] it has been proved that developable canal surface is either a cylinder or a cone.This study consists of 5 sections: In section 2, we explain some well-known properties of the surfaces in E 4 .In section 3, we give the canal surfaces in E 4 and some examples are presented.Section 4 investigates the ellipse of curvature of canal surfaces in E 4 .Additionally we prove necessary and sufficient condition of canal surfaces to become superconformal in E 4 .In Section 5, the visualization of canal surfaces are given with using Maple programme.

Basic concepts
Let M be a regular surface in E 4 given with the parametrization The tangent space of M at an arbitrary point p = X(u, v) is spanned by the vectors X u and X v .The first fundamental form coefficients of M are computed by where , is the scalar product of the Euclidean space.We consider the surface patch X(u, v) is regular, which implies that W 2 = EG − F 2 = 0.For the point p ∈ M , we can take the decomposition (2) where T expresses the tangential part.
Let us consider the spaces of the smooth vector fields χ(M ) and χ ⊥ (M ) which are tangent and normal to M , respectively.The second fundamental map is defined as follows: This map is well-defined, symmetric and bilinear.
If we take the orthonormal frame field {N 1 , N 2 } of M , then the shape operator which is self-adjoint and bilinear can be given by which satisfies the equation: for any X 1 , X 2 ∈ T p M .The equality (3) is known as the Gaussian equation, where Here Γ k ij are Christoffel symbols and c k ij are the coefficients of the second fundamental form.The Gaussian curvature are given by and the mean curvature are given by where If the mean curvature of M vanishes identically in E n , then M is said to be minimal [3].See also [1].

Canal surfaces in E 4
Let γ(u) = (f 1 (u), f 2 (u), f 3 (u), 0) be a curve given with arclength parameter.Then the Frenet formulae have the following form: where {e 1 (u), e 2 (u), e 3 (u), e 4 (u)} is the Frenet orthonormal basis of γ.The canal surface in E 4 has the following parametrization (see [6]): Example 1.Consider the helix γ(u) = (a cos u c , a sin u c , bu c ) in E 3 .Then the canal surface of γ in E 4 has the following parametrization Example 2. Consider the generalized helix Then the canal surface of γ in E 4 has the following parametrization The space which is tangent to M is spanned by X v = −r sin ve 3 +r cos ve 4 .
The first fundamental form coefficients become The Christoffel symbols Γ k ij are given by and they are symmetric according to the covariant indices ( [7], p.398).
If we take the second partial derivatives of X(u, v), we find: Hence, by using (3), we find the Gaussian equations; where Further using (20) Thus, using ( 14) with (17) we get Proposition 1.The Gaussian curvature of the canal surface M with the parametrization ( 11) in E 4 is given by where g = EG − F 2 .
Proof.By using the equation ( 8), we find which is the Gaussian curvature of the canal surface M .Taking into account ( 21) and ( 24) we obtain (23).
From the equations ( 22) with ( 23) we obtain; Corollary 1.The Gaussian curvature of the canal surface M with the parametrization ( 11) in E 4 is given by where Proposition 2. The mean curvature of the canal surface M with the parametrization ( 11) in E 4 is given by Proof.By considering (9) the mean curvature of the canal surface M becomes Taking into account ( 21) and ( 27) we get the result.
By the use of ( 22) and Proposition 2, we have the following results: Corollary 2. The mean curvature of the canal surface M with the parametrization (11) in E 4 is given by Corollary 3. If the base curve γ of the canal surface M is a straight line, then the Gaussian and mean curvatures of M are 2 , and

Ellipse of curvature of the canal surfaces in E 4
Let M be a regular surface given with the parametrization X (u, v) : (u, v) ∈ D ⊆ E 2 .Consider a circle given with the angle θ ∈ [0, 2π] in the tangent space T p M .The intersection of the direct sum of the tangent direction of X = cos θX 1 +sin θX 2 and the normal space T ⊥ p M with the surface M forms a curve.Such a curve is called as a normal section curve in the direction θ.Denote this curve by γ θ .Normal curvature vector η θ of γ θ lies in T ⊥ p M .When θ changes from 0 to 2π, the normal curvature vector constitutes an ellipse called as a ellipse of curvature of M at p in T ⊥ p M .Thus, the curvature ellipse of M at point p is given as follows with the second fundamental form h: To see that this shows an ellipse, it is enough to have a look at the formulas Here, are normal vectors and ) is the mean curvature vector.This implies that, the vector h(X, X) goes twice around the ellipse of curvature centered at − → H , while X goes once around the unit tangent circle [9].
From the equation (28), one can get that E(p) is a circle if and only if for some orthonormal basis of T p (M ) it holds that h General aspects of the ellipse of curvature for surfaces in E 4 studied by Wong [13].(See also [2], [8], [9] and [10])

Conclusion
In this manuscript, we considered canal surfaces in the 4dimensional Euclidean space E 4 .Most of the literature on canal surfaces within the CAGD context has been motivated by the observation that canal surfaces with rational spine curve.We have proved this property mathematically and also illustrated with some nice examples.

Figure 2 .Figure 3 .
Figure 2. The projections of canal surfaces of general helix in E 3 (11)rem 1.The canal surface M with the parametrization(11)in E 4 is superconformal if and only if the equalities