
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 ...