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116 Applied calculus of variations for engineers

Γ

2

12

=

EG

u

− FE

v

2(EG − F

2

)

,

and

Γ

2

22

=

EG

v

− 2FF

v

+ FG

u

2(EG − F

2

)

.

These formulae all require that

EG − F

2

=0

which is true when a parameterization is regular.

8.2.1 Geodesics of surfaces of revolution

Another practically important special case is represented by surfaces of revo-

lution. Their generic description may be of the form

x = ucos(v),

y = usin(v),

and

z = f (u).

Here the last equation describes the meridian curve generating the surface.

The ﬁrst order fundamental terms are

E =1+f

2

(u),

F =0,

and

G = u

2

.

The solution following the discussion in Section 8.1 becomes

v = c

1

1+f

2

(u)

u

u

2

− c

2

1

du + c

2

.

A simple sub-case of this class is a unit cylinder, described as

x = cos(v),

y = sin(v),

and

z = u.

The geometric meaning of the v parameter is the rotation angle generating

the cylinder’s circumference and the u parameter is the axial direction of the

Diﬀerential geometry 117

surface. The fundamental terms are

E =1,

F =0,

and

G =1.

The equation of the geodesic curve on the cylinder following above is

v = c

1

1

1

1 − c

2

1

du + c

2

,

or

v = c

1

1

1 − c

2

1

du + c

2

.

With

c

3

= c

1

1

1 − c

2

1

and integration we obtain

v = c

3

u + c

2

.

In the general case, this is a helix on the surface of the cylinder going through

the two points. This is also a line in the parametric space. This fact is geo-

metrically easy to explain because the cylinder is a developable surface. Such

a surface may be rectiﬁed onto a plane. In such a case the geodesic curve is

a straight line on the rectifying plane. The only curvature of the helix will be

that of the cylinder.

The constants of integration may be determined from the boundary condi-

tions. For example, assume the case shown in Figure 8.1, where the starting

point is at

P

0

=(x

0

,y

0

,z

0

)=(1, 0, 0)

corresponding to parametric coordinates

u(t

0

)=0,v(t

0

)=0.

The endpoint is located at

P

1

=(x

1

,y

1

,z

1

)=(0, 1, 1)

corresponding to parametric coordinates

u(t

1

)=1,v(t

1

)=π/2.

118 Applied calculus of variations for engineers

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

FIGURE 8.1 Geodesic curve of a cylinder

Substituting the starting point yields

0=c

3

· 0+c

2

,

which results in

c

2

=0.

The endpoint substitution produces

π/2=c

3

· 1+c

2

,

andinturn

c

3

=

π

2

.

The speciﬁc solution for this case in the parametric space is

v =

π

2

u.

The Cartesian solution is obtained in the form of

x = cos(v),

Diﬀerential geometry 119

y = sin(v),

and

z =

v

π/2

.

It is easy to see that the latter equation makes the elevation change from zero

to 1, in accordance with the turning of the helix.

Since the parametric space of the cylinder is simply the rectangle of the

developed surface, it is easy to see some special sub-cases. If the two points

are located at the same rotational position (v = constant), but at diﬀerent

heights, the geodesic curve is a straight line. If the two points are on the same

height (u = constant), but at diﬀerent rotational angles, the geodesic curve is

a circular arc.

The last two sections demonstrated the diﬃculties of ﬁnding the geodesic

curves even on regular surfaces like the sphere or the cylinder. On a general

three-dimensional surface these diﬃculties increase signiﬁcantly and may ren-

der using the diﬀerential equation of the geodesic curve unfeasible.

8.3 Geodesic curvature

Let us consider the parametric curve

r

(t)=x(t)i + y(t)j + z(t)k

on the surface

S(u, v)=x(u, v)i

+ y(u, v)j + z(u, v)k.

Let n

denote the unit normal of the surface. The curvature vector of a three-

dimensional curve is deﬁned as

k

=

dt

dt

= t

,

where t

is the tangent vector computed as

t

=

dr

dt

,

and also assumed to be a unit vector (a unit speed curve) for the simplicity

of the derivation. Then the unit bi-normal vector is

b

= n × t.

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