Mannheim Offsets of Ruled Surfaces
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 . Ruled surfaces are one of the simplest objects in geometric modeling.
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 and coordinates, but ruled surfaces can all be rewritten to highlights the generating lines. A practical application of ruled surfaces is that they are used in civil engineering. Since building materials such as wood are straight, they can be thought of as straight lines. The result is that if engineers are planning to construct something with curvature, they can use a ruled surface since all the lines are straight.
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 , studied Bertrand offsets of ruled surfaces. Pottman et al. , presented classical and circular offsets of rational ruled surfaces.
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.
2. Mannheim Offset of a Curve
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 , who studied curve pairs which have common principal normals. Such curves referred to as Bertrand curves and can be considered as offsets of one another. Another kind of associated curves is the Mannheim offsets.
In plane, a curve rolls on a straight line, the center of curvature of its point of contact describes a curve which is the Mannheim of , .
Let and be two space curves. is said to be a Mannheim partner curve of , if there exists a one to one correspondence between their points such that the binormal vector of is the principal normal vector of . Such curves are referred to as “Mannheim offsets,” .
Let be a Mannheim curve with the arc-length parameter . Then is the Mannheim partner curve of if and only if the curvature and the torsion of satisfy the following equation for some nonzero constant , .
3. Differential Geometry of Ruled Surfaces
A ruled surface is generated by a one-parameter family of straight lines and it possesses a parametric representation, where represents a space curve which is called the base curve and is a unit vector representing the direction of a straight line.
The vector traces a curve on the surface of unit sphere called spherical indicatrix of the ruled surface, .
The orthonormal system is called the geodesic Frenet thiedron of the ruled surface such that and are the central normal and the asymptotic normal direction of , respectively.
For the geodesic Frenet vectors , and , we can write where and are the arc-length of spherical indicatrix and the geodesic curvature of with respect to , respectively .
The striction point on a ruled surface is the foot of the common normal between two consecutive generators (or ruling). The set of striction points defines the striction curve given by
If consecutive generators of a ruled surface intersect, then the surface is said to be developable. The spherical indicatrix, , of a developable surface is tangent of its striction curve, .
The distribution parameter of the ruled surface is defined by The ruled surface is developable if and only if .
In this paper, the striction curve of the ruled surface will be taken as the base curve. In this case, for the parametric equation of , we can write
4. Mannheim Offsets of Ruled Surfaces
The ruled surface is said to be Mannheim offset of the ruled surface if there exists a one to one correspondence between their rulings such that the asymptotic normal of is the central normal of . In this case, is called a pair of Mannheim ruled surface.
Let and be two ruled surfaces which is given by where and are the striction curves of and , respectively.
If is a Mannheim offset of , then we can write where and are the geodesic Frenet triplies at the point and of the striction curves of and , respectively.
The equation of in terms of can therefore be written as where is distance between corresponding striction points and is the angle between corresponding rulings.
Let the ruled surface be Mannheim offset of the ruled surface . By definition, From the definition , we get .
Because of the last two equation, we have ( a scalar). Since the base curve of is its striction curve, we get .
From the equality it follows that It therefore follows that . Thus we have the following theorem.
Theorem 4.1. Let the ruled surface be Mannheim offset of the ruled surface . Then is developable if and only if is a constant.
Theorem 4.2. Let the ruled surface be Mannheim offset of the developable ruled surface . Then is developable if and only if the following relationship can be written
Proof. Suppose that is developable. Then we have where is the arc-length parameter of the striction curve of .
Then we obtain
From Theorem 4.1 and the realtion (3.2), we get
The last equation implies that
Conversely, suppose that the equality is satisfied. For the tangent of the striction curve of , we can write, Thus, is developable.
Theorem 4.3. Let be a developable ruled surface. The
developable ruled surface is a Mannheim offset of the ruled surface if and only if the following relationship is
Proof. Suppose that the developable ruled surface is a Mannheim offset of .
Because of Theorem 4.2, we get Using (4.2) and the chain rule of differentiation, we can
write From (4.14) and definition of ,
we have By taking the derivative of
(4.13) with respect to arc and using (4.15), we obtain
Conversely, suppose that the equality is satisfied. For nonzero constant scalar , we can define the ruled surface where .
We will prove that is a Mannheim offset of . Since is developable, we have where and are the arc-length parameter of the striction curves and , respectively. From the equality and (4.18), we get
By taking the derivative of (4.19) with respect to arc , we obtain From the hypothesis and the definition of , we get where is a scalar.
By taking the cross product of (4.19) with (4.21), we have Taking the cross product of (4.22) with (4.19), we obtain Thus, the developable ruled surface is a Mannheim offset of the ruled surface .
Let the ruled surface be a Mannheim offset of the ruled surface . If the ruled surfaces which is generated by the vectors and of denote by and , respectively, then we can write where and are the geodesic Frenet triplies of the striction curves of and , respectively. Therefore, from (4.24) we have the following.
Corollary 4.4. (a) is a Bertrand offset of .(b) is
a Mannheim offset of . Now, one will investigate developable of and while is developable:
Let the ruled surface be a Mannheim offset of the developable ruled surface . From (3.2), (3.4), and (4.2), it is easy to see that,
As an immediate result we have the following.
Corollary 4.5. (a) is nondevelopable while is developable.(b) is developable while is developable if and only if the relationship is satisfied.
Example 4.6. The elliptic hyperboloid of one sheet is a ruled surface parametrized
A Mannheim offset of this surface is where .
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 and the theory of Mannheim offsets of ruled surfaces in . For instance, a ruled surface can have an infinity of Mannheim offsets in the some way as a plane curve can have an infinity of Mannheim mates. Furthermore, in analogy with three dimensional curves, a developable ruled surface can have a developable Mannheim offset if a equation holds between the geodesic curvature and the arc-length of its spherical indicatrix.Table of Symbols
|:||Euclidean space of dimension three|
|:||curvature of a curve|
|:||torsion of a curve|
|:||spherical indicatrix vector|
|:||geodesic curvature of|
|function of distance|
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