4.3. PLATEAU PROBLEM 85
If
W D
e
W CG.b/ cosh a; (4.103)
then (4.102) gives rise to Laplace’s equation
r
2
e
W D 0: (4.104)
Further substitution into (4.99) shows it is left unchanged.
In order to connect solutions of the original equation (4.85) to Laplace’s equation, it is
necessary to compose the transformations (4.89), (4.93), (4.95), (4.97) and (4.99). is gives
rise to the following result.
Exact solutions to the minimal surface equation
1 C u
2
y
u
xx
2u
x
u
y
u
xy
C
1 C u
2
x
u
yy
D 0 (4.105)
are given parametrically by
x D cosh a cos b W
a
C sinh a sin b W
b
sinh a cos b W
aa
cosh a sin b W
ab
;
y D cosh a sin b W
a
sinh a cos b W
b
C cosh a cos b W
ab
C sinh a sin b W
bb
;(4.106)
u D W W
aa
;
where W satisfies Laplace’s equation
W
aa
C W
bb
D 0: (4.107)
In the next section we will show that several well-known minimal surfaces may be obtained with
simple solutions of Laplace’s equation.
4.3.2 WELL-KNOWN MINIMAL SURFACES
Catenoid and Helicoid
Probably the simplest non-zero solution to Laplace’s equation, (4.107) is given by
W D a sin ˛ C b cos ˛; (4.108)
where ˛ is constant. Substituting this into (4.106) gives rise to
x D cosh a cos b sin ˛ C sinh a sin b cos ˛;
y D cosh a sin b sin ˛ sinh a cos b cos ˛; (4.109)
u D a sin ˛ C b cos ˛:
Setting ˛ D 0 gives
x D sinh a sin b;
y D sinh a cos b; (4.110)
u D b;
86 4. POINT AND CONTACT TRANSFORMATIONS
the surface of a helicoid, while setting ˛ D
2
gives
x D cosh a cos b;
y D cosh a sin b; (4.111)
u D a;
the surface of a catenoid [68].
Enneper’s Surface
By choosing
W D
1
3
.
sinh 2a Ccosh 2a
/
cos 2b; (4.112)
as a solution of Laplace’s equation gives, (4.106) gives
x D e
a
cos b C
1
3
e
3a
cos
3
b C e
3a
sin
2
b cos b;
y D e
a
sin b C
1
3
e
3a
sin
3
b C e
3a
cos
2
b sin b; (4.113)
u D e
2a
cos
2
b sin
2
b
:
Identifying that
p D e
a
cos b; q D e
a
sin b; (4.114)
gives Enneper’s minimal surface [68]
x D p
1
3
p
3
C pq
2
;
y D q C
1
3
q
3
p
2
q; (4.115)
u D p
2
q
2
:
Henneberg’s Surface
By choosing
W D
2
3
cosh 2a cos 2b; (4.116)
as a solution of Laplace’s equation gives, (4.106) gives
x D
2
3
sinh 3a cos 3b 2 sinh a cos b;
y D
2
3
sinh 3a sin 3b C 2 sinh a sin b; (4.117)
u D 2 cosh 2a cos 2b;
known as Henneberg’s minimal surface [68].
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