# Mannheim Offsets of Ruled Surfaces

Authors: Keziban Orbay; Emin Kasap; İsmail Aydemir
Journal: Mathematical Problems in Engineering
 (2009)

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## Abstract

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.


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## Body

## 1. Introduction

A surface is said to be “ruled” if it is generated by moving a
straight line continuously in Euclidean space𝔼3.
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 ofx,y, and z 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  [1],
studied Bertrand offsets of ruled surfaces. Pottman et al. [2], 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 [3],
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 curveα rolls on a straight line, the center of
curvature of its point of contact describes a curve β which is the Mannheim of α, [4].The theory of the Mannheim curves has
been extended in the three dimensional Euclidean space by Liu and Wang [5, 6].LetC and C* be two space curves. C is said to be a Mannheim partner curve of C*,
if there exists a one to one correspondence between their points such that the
binormal vector of C is the principal normal vector of C*.
Such curves are referred to as “Mannheim
offsets,” [5].LetC*:α=α(s*) be a Mannheim curve with the arc-length
parameter s*.
Then C:β=β(s) is the Mannheim partner curve of C* if and only if the curvature κ and the torsion τ of C satisfy the following equation (2.1)τ˙=dτds=κλ(1+λ2τ2) for
some nonzero constant λ,
[5].The detailed discussion concerned with the
Mannheim curves can be found in  [5, 6].


## 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,(3.1)φ(s,v)=α(s)+ve(s), where α(s) represents a space curve which is called the base curve and e is a unit vector representing the direction of
a straight line.The vectore traces a curve on the surface of unit sphere S2 called spherical
indicatrix of the ruled surface, [1].The orthonormal system{e,t,g} is called the geodesic Frenet thiedron of the ruled surface φ such that t=es/∥es∥ and g=(e×es)/∥es∥ are the central normal and the asymptotic
normal direction of φ,
respectively.For the geodesic Frenet vectorse, t and g,
we can write (3.2)eq=ttq=γg−egq=−γt, where q and γ are the arc-length of spherical indicatrix (e) and the geodesic curvature of (e) with respect to S2,
respectively [1].The striction point on a ruled surface φ is the foot of the common normal between two
consecutive generators (or ruling). The set of striction points defines the striction curve given by (3.3)c(s)=α(s)−〈αs,es〉〈es,es〉e(s).If consecutive generators of a ruled
surface intersect, then the surface is said to bedevelopable.  The spherical indicatrix, e, of a developable surface is tangent of its
striction curve, [1].The
distribution parameter of the ruled surface φ is defined by (3.4)Pe=det(αs,e,es)∥es∥2. The ruled surface is
developable if and only if Pe=0.In this paper, the striction curve of the ruled
surfaceφ will be taken as the base curve. In this case,
for the parametric equation of φ,
we can write(3.5)φ(s,v)=c(s)+ve(s).


## 4. Mannheim Offsets of Ruled Surfaces

The ruled
surfaceφ* is said to be Mannheim offset of the ruled surface φ if there exists a one to one correspondence
between their rulings such that the asymptotic normal of φ is the central normal of φ*.
In this case, (φ,φ*) is called a
pair of Mannheim ruled surface.Letφ and φ* be two ruled surfaces which is given by (4.1)φ(s,v)=c(s)+ve(s),∥e(s)∥=1,φ*(s,v)=c*(s)+ve*(s),∥e*(s)∥=1,  where (c) and (c*) are the striction curves of φ and φ*,
respectively.Ifφ* is a Mannheim offset of φ,
then we can write (4.2)[e*t*g*]=[cosθsinθ0001sinθ−cosθ0][etg], where {e,t,g} and {e*,t*,g*} are the geodesic Frenet triplies at the point c(s) and c*(s) of the striction curves of φ and φ*,
respectively.The equation ofφ* in terms of φ can therefore be written as (4.3)φ*(s,v)=c(s)+R(s)g(s)+v[cosθe(s)+sinθt(s)], where R=R(s) is distance between corresponding striction
points and θ is the angle between corresponding rulings.Let the ruled surfaceφ* be Mannheim offset of the ruled surface φ.
By definition, (4.4)t*=g.From the definition t*,
we get t*=es*/∥es*∥.Because of the last two
equation, we havees*=λg (λ a scalar). Since the base curve of φ* is its striction curve, we get 〈cs*,es*〉=0.From the equalityes*=λg it follows that 〈(c+Rg)s,g〉=0
It therefore follows that ∥es∥Pe+Rs=0.
Thus we have the following theorem.Theorem 4.1.
Let the ruled surfaceφ* be Mannheim offset of the ruled surface φ.
Then φ is developable if and only if R is a constant.Theorem 4.2.
Let the ruled surfaceφ* be Mannheim offset of the developable ruled
surface φ.
Then φ* is developable if and only if the following
relationship can be written (4.5)sinθ+Rγqscosθ=0.Proof.
Suppose thatφ* is developable. Then we have (4.6)cs*=μe*(μascalar),where s is the arc-length parameter of the striction curve (c) of φ.
Then we obtain (4.7)cs+Rqsgq+Rsg=μ[cosθe+sinθt].
From Theorem4.1 and the realtion (3.2),  we get (4.8)e+Rqs(−γt)=μcosθe+μsinθt.
The last equation implies
that(4.9)sinθ+Rγqscosθ=0.
Conversely, suppose that
the equality(4.10)sinθ+Rγqscosθ=0 is satisfied. For the tangent of the striction curve of φ*,
we can write,  (4.11)cs*=(c+Rg)s=e−Rγqst=1cosθ[cosθe+sinθt]=1cosθe*.  Thus, φ* is developable.Theorem 4.3.
Letφ be a developable ruled surface. The
developable ruled surface φ* is a Mannheim offset of the ruled surface φ if and only if the following relationship is
satisfied:
(4.12)γs=dγds=1R(1+R2γ2qs2)−1qsγqss.Proof.
Suppose that the developable ruled surfaceφ* is a Mannheim offset of φ.
Because of Theorem 4.2, we get (4.13)Rγqs=−tanθ. Using (4.2) and the chain rule of differentiation, we can
write (4.14)es*=−sinθ(θs+qs)e+cosθ(θs+qs)t+γqssinθg. From (4.14) and definition of t*,
we have (4.15)θs=−qs. By taking the derivative of
(4.13) with respect to arc s and using (4.15), we obtain (4.16)γs=dγds=1R(1+R2γ2qs2)−1qsγqss.
Conversely, suppose that the
equalityγs=dγ/ds=(1/R)(1+R2γ2qs2)−(1/qs)γqss is satisfied. For nonzero constant scalar R,
we can define the ruled surface (4.17)φ*(s,v)=c*(s)+ve*(s), where c*(s)=c(s)+Rg(s).
We will prove thatφ* is a Mannheim offset of φ.
Since φ* is developable, we have (4.18)cs*=ds*dse*, where s and s* are the arc-length parameter of the striction
curves (c) and (c*),
respectively. From the equality c*(s)=c(s)+Rg(s) and (4.18), we get (4.19)ds*dse*=e−Rγqst.
By taking
the derivative of (4.19) with respect to arc s,
we obtain (4.20)d2s*ds2e*+ds*dses*=Rγqs2e+(qs−Rγsqs−Rγqss)t−Rγ2qs2g. From the hypothesis and the definition of t*,
we get (4.21)d2s*ds2e*+ds*dsλt*=Rγqs2e−R2γ2qs3t−Rγ2qs2g, where λ is a scalar.
By taking the cross
product of (4.19) with (4.21), we have (4.22)(ds*ds)2λg*=R2γ3qs3e+Rγ2qs2t. Taking the cross product of (4.22) with (4.19), we obtain (4.23)(ds*ds)3λt*=−(Rγ2qs2+R3γ4qs4)g. Thus, the developable ruled surface φ* is a Mannheim offset of the ruled surface φ.Let the ruled surfaceφ* be a Mannheim offset of the ruled surface φ.
If the ruled surfaces which is generated by the vectors t* and g* of φ* denote by φt* and φg*,
respectively, then we can write (4.24)e1*=g,t1*=∓t,g1*=±e,e2*=sinθe−cosθt,t2*=∓g,g2*=∓cosθe±sinθt,  where {e1*,t1*,g1*} and {e2*,t2*,g2*} are the geodesic Frenet triplies of the
striction curves of φt* and φg*,
respectively. Therefore, from (4.24) we have the following.Corollary 4.4.
(a)φt* is a Bertrand offset of φ.(b)
φg* is
a Mannheim offset of φ. Now, one will investigate developable of φt* and φg* while φ is developable:
Let the ruled surfaceφ* be a Mannheim offset of the developable ruled
surface φ.
From (3.2), (3.4), and (4.2), it is easy to see that, (4.25)Pt*=1γqs,Pg*=1γqscosθ(cosθ−Rγqssinθ).As an immediate result we
have the following.Corollary 4.5.
(a)φt* is nondevelopable while φ is developable.(b)
φg* is
developable while φ is developable
if and only if the relationship cosθ−Rγqssinθ=0 is satisfied.Example 4.6.
The elliptic hyperboloid of one sheet is a ruled surface parametrized
by(4.26)φ(s,v)=(cos(s)−22vsin(s),sin(s)+22vcos(s),22v).
A Mannheim
offset of this surface is(4.27)φ*(s,v)=(cos(s)−22sin(s)s−(1+22)vcos(s)sin(s),sin(s)−22cos(s)s+22vcos2(s)−vsin2(s),22s+22vcos(s)), where R=R(s)=s.Example 4.7.
The surface(4.28)φ(s,v)=(cos(22s)−22vsin(22s),sin(22s)+22vcos(22s),22s+22v) is a developable ruled surface.
A Mannheim offset of this
surface is(4.29)φ*(s,v)=(cos(22s)−522sin(22s)−(1+24)vsin(2s),sin(22s)−522cos(22s)+22vcos2(22s)−vsin2(22s),22s+522+22vcos(22s)), where R=R(s)=5. (See Figures 1 and 2.)Figure 1
Hyperboloid of one sheet and its Mannheim offset.Figure 2
Developable ruled surface and its Mannheim offset.


## 5. Conclusion

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 in𝔼2 and the theory of Mannheim
offsets of ruled surfaces in 𝔼3.
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.



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*Source: 160917-2009-03-29.xml*