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

+Γ

1

11

u

2

+2Γ

1

12

u

v

+Γ

1

22

v

2

=0,

and

v

+Γ

2

11

u

2

+2Γ

2

12

u

v

+Γ

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

)

,

Get *Applied Calculus of Variations for Engineers, 2nd Edition* now with the O’Reilly learning platform.

O’Reilly members experience live online training, plus books, videos, and digital content from nearly 200 publishers.