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In a recent works Liu and Wang (2008; 2007) study the Mannheim partner curves in the three dimensional space. In this paper, we extend the theory of the Mannheim curves to ruled surfaces and define two ruled surfaces which are offset in the sense of Mannheim. It is shown that, every developable ruled surface have a Mannheim offset if and only if an equation should be satisfied between the geodesic curvature and the arc-length of spherical indicatrix of it. Moreover, we obtain that the Mannheim offset of developable ruled surface is constant distance from it. Finally, examples are also given.

A surface is said to be “ruled” if it is generated by moving a
straight line continuously in Euclidean space

One important fact about ruled surfaces is that they can be generated by
straight lines. One would never know this from looking at the surface or its
usual equation in terms of

Among ruled surfaces, developable surfaces form an important subclass since they are useful in sheet metal design and processing. Every developable surface can be obtained as the envelope surface of a moving plane (under a one-parameter motion). Developable ruled surfaces are well-known and widely used in computer aided design and manufacture. A “developable” ruled surface is a surface that can be rolled on a plane, touching along the entire surface as it rolls. Such a surface has a constant tangent plane for the whole length of each ruling. Parallel geodesic loops (in a direction perpendicular to the rulings) on closed developable ruled surfaces all have the same length; such surfaces are thus “constant perimeter” surfaces.

In the past, offsets of
ruled surfaces have been the subject of some studies: Ravani and Ku [

In this paper, the Mannheim offsets of ruled surfaces are considered. It is shown that a theory similar to that of the Mannheim partner curves can be developed for ruled surfaces.

Offset curves
play an important role in areas of CAD/CAM, robotics, cam design and many
industrial applications, in particular in mathematical modeling of cutting
paths milling machines. The classic work in this area is that of Bertrand [

In plane, a curve

The theory of the Mannheim curves has
been extended in the three dimensional Euclidean space by Liu and Wang [

Let

Let

The detailed discussion concerned with the
Mannheim curves can be found in [

A ruled surface
is generated by a one-parameter family of straight lines and it possesses a
parametric representation,

The vector

The orthonormal system

For the geodesic Frenet vectors

If consecutive generators of a ruled
surface intersect, then the surface is said to be

In this paper, the striction curve of the ruled
surface

The ruled
surface

Let

If

The equation of

Let the ruled surface

Because of the last two
equation, we have

From the equality

Let the ruled surface

Let the ruled surface

Suppose that

From Theorem

The last equation implies
that

Conversely, suppose that
the equality

Let

Suppose that the developable ruled surface

Conversely, suppose that the
equality

We will prove that

By taking
the derivative of (

By taking the cross
product of (

Let the ruled surface

(a)

Let the ruled surface

As an immediate result we have the following.

(a)

The elliptic hyperboloid of one sheet is a ruled surface parametrized
by

A Mannheim
offset of this surface is

The surface

A Mannheim offset of this
surface is

Hyperboloid of one sheet and its Mannheim offset.

Developable ruled surface and its Mannheim offset.

In this paper, a generalization of
Mannheim offsets of curves for ruled surfaces has been developed.
Interestingly, there are many similarities between the theory of Mannheim
offsets in

Euclidean space of dimension three

curvature of a curve

torsion of a curve

arc-length

arc-length

unit sphere

spherical indicatrix vector

central normal

asymptotic normal

spherical indicatrix

geodesic
curvature of

arc-length of

striction curve

distribution parameter

function of distance

Riemannian metric