8
Diﬀerential geometry
Diﬀerential geometry is a classical mathematical area that has become very
important for engineering applications in the recent decades. This importance
is based on the rise of computer-aided visualization and geometry generation
technologies.
The fundamental problem of diﬀerential geometry, the ﬁnding of geodesic
curves, has practical implications in manufacturing. Development of non-
mathematical surfaces used in ships and airplanes has serious ﬁnancial impact
in reducing material waste and improving the quality of the surfaces.
While the discussion in this chapter will focus on analytically solvable prob-
lems, the methods and concepts we introduce will provide a foundation ap-
plicable in various engineering areas.
8.1 The geodesic problem
Finding a geodesic curve on a surface is a classical problem of diﬀerential ge-
ometry. Variational calculus seems uniquely applicable to this problem. Let
us consider a parametrically given surface
r
= x(u, v)i + y(u, v)j + z(u, v)k.
Let two points on the surface be
r
0
= x(u
0
,v
0
)i + y(u
0
,v
0
)j + z(u
0
,v
0
)k,
and
r
1
= x(u
1
,v
1
)i + y(u
1
,v
1
)j + z(u
1
,v
1
)k.
The shortest path on the surface between these two points is the geodesic
curve. Consider the square of the arc length
ds
2
=(dx)
2
+(dy)
2
+(dz)
2
,
111
112 Applied calculus of variations for engineers
and compute the diﬀerentials related to the parameters.
ds
2
= E(u, v)(du)
2
+2F (u, v)dudv + G(u, v)(dv)
2
.
Here the so-called ﬁrst fundamental quantities are deﬁned as
E(u, v)=(
∂x
∂u
)
2
+(
∂y
∂u
)
2
+(
∂z
∂u
)
2
=(r
u
)
2
,
F (u, v)=
∂x
∂u
∂x
∂v
+
∂y
∂u
∂y
∂v
+
∂z
∂u
∂z
∂v
= r
u
r
v
,
and
G(u, v)=(
∂x
∂v
)
2
+(
∂y
∂v
)
2
+(
∂z
∂v
)
2
=(r
v
)
2
.
Assume that the equation of the geodesic curve in the parametric space is
described by
v = v(u).
Then the geodesic curve is the solution of the variational problem
I(v)=
u
1
u
0
E(u, v)+2F (u, v)
dv
du
+ G(u, v)(
dv
du
)
2
du = extremum
with boundary conditions
v(u
0
)=v
0
,
and
v(u
1
)=v
1
.
The corresponding Euler-Lagrange diﬀerential equation is
E
v
+2v
F
v
+ v
2
G
v
2
E(u, v)+2F (u, v)v
+ G(u, v)v
2
d
du
F + Gv
E(u, v)+2F (u, v)v
+ G(u, v)v
2
=0,
with the notation of
E
v
=
∂E
∂v
,F
v
=
∂F
∂v
,G
v
=
∂G
∂v
,
and
v
=
dv
du
.
The equation is rather diﬃcult in general, and exploitation of special cases
arising from the particular surface deﬁnitions is advised.
Diﬀerential geometry 113
When the ﬁrst fundamental quantities are only functions of the u parame-
ter, the equation simpliﬁes to
F + Gv
E(u, v)+2F (u, v)v
+ G(u, v)v
2
= c
1
.
A further simpliﬁcation is based on the practical case when the u and v
parametric lines deﬁning the surface are orthogonal. In this case
F =0,
and the equation may easily be integrated as
v = c
1
E
G
2
c
2
1
G
du + c
2
.
The integration constants may be resolved from the boundary conditions.
When the function in the integral only contains the v function explicitly,
and the F = 0 assumption still holds, then the equation becomes
Gv
2
E + Gv
2
E + Gv
2
= c
1
.
Reordering and another integration brings
u = c
1
G
E
2
c
2
1
E
dv + c
2
.
8.1.1 Geodesics of a sphere
For an enlightening example, we consider a sphere, given by
x(u, v)=Rsin(v)cos(u),
y(u, v)=Rsin(v)sin(u),
and
z(u, v)=Rcos(v).
The ﬁrst fundamental quantities encapsulating the surface information are
E = R
2
sin
2
(v),
F =0,
and
G = R
2
.
114 Applied calculus of variations for engineers
Since this is the special case consisting of only v, the equation of the geodesic
curve becomes
u = c
1
R
R
4
sin
4
(v) c
2
1
R
2
sin
2
(v)
dv + c
2
.
After the integration by substitution and some algebraic manipulations, we get
u = asin
cot(v)
(
R
c
1
)
2
1
+ c
2
.
It follows that
sin(c
2
)(Rsin(v)cos(u)) cos(c
2
)(Rsin(v)sin(u))
Rcos(v)
(
R
c
1
)
2
1
=0.
Substituting the surface deﬁnition of the sphere yields
xsin(c
2
)+ycos(c
2
)
z
(
R
c
1
)
2
1
=0
and that represents an intersection of the sphere with a plane. By substituting
boundary conditions, it would be easy to see that the actual plane contains
the origin and deﬁnes the great circle going through the two given points. This
fact is manifested in everyday practice by the transoceanic airplane routes’
well-known northern swing in the Northern Hemisphere.
8.2 A system of diﬀerential equations for geodesic curves
Let us now seek the geodesic curve in the parametric form of
u = u(t),
and
v = v(t).
The curve goes through two points
P
0
=(u(t
0
),v(t
0
)),
and
P
1
=(u(t
1
),v(t
1
)).
Diﬀerential geometry 115
Then the following variational problem provides the solution:
I(u, v)=
t
1
t
0
Eu
2
+2Fu
v
+ Gv
2
dt = extremum.
Here
u
=
du
dt
,v
=
dv
dt
.
The corresponding Euler-Lagrange system of diﬀerential equations is
E
u
u
2
+2F
u
u
v
+ G
u
v
2
Eu
2
+2Fu
v
+ Gv
2
d
dt
2(Eu
+ Fv
)
Eu
2
+2Fu
v
+ Gv
2
=0,
and
E
v
u
2
+2F
v
u
v
+ G
v
v
2
Eu
2
+2Fu
v
+ Gv
2
d
dt
2(Fu
+ Gv
)
Eu
2
+2Fu
v
+ Gv
2
=0.
In the equations the notation
E
u
=
∂E
∂u
,F
u
=
∂F
∂u
,G
u
=
∂G
∂u
was used.
A more convenient and practically useful formulation, without the explicit
derivatives, based on [8] is
u

1
11
u
2
+2Γ
1
12
u
v
1
22
v
2
=0,
and
v

2
11
u
2
+2Γ
2
12
u
v
2
22
v
2
=0.
Here
u

=
d
2
u
dt
2
,v

=
d
2
v
dt
2
.
The Γ are the Christoﬀel symbols that are deﬁned as
Γ
1
11
=
GE
u
2FF
u
+ FE
v
2(EG F
2
)
,
Γ
1
12
=
GE
v
FG
u
2(EG F
2
)
,
Γ
1
22
=
2GF
v
GG
u
FG
v
2(EG F
2
)
,
Γ
2
11
=
2EF
u
EE
v
FE
u
2(EG F
2
)
,

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