Minimal surface in differential geometry
Richmond surface for m=2. In differential geometry , a Richmond surface is a minimal surface first described by Herbert William Richmond in 1904. It is a family of surfaces with one planar end and one Enneper surface -like self-intersecting end.
It has Weierstrass–Enneper parameterization
f
(
z
)
=
1
/
z
2
,
g
(
z
)
=
z
m
{\displaystyle f(z)=1/z^{2},g(z)=z^{m}}
X
(
z
)
=
ℜ
[
(
−
1
/
2
z
)
−
z
2
m
+
1
/
(
4
m
+
2
)
]
Y
(
z
)
=
ℜ
[
(
−
i
/
2
z
)
+
i
z
2
m
+
1
/
(
4
m
+
2
)
]
Z
(
z
)
=
ℜ
[
z
m
/
m
]
{\displaystyle {\begin{aligned}X(z)&=\Re \\Y(z)&=\Re \\Z(z)&=\Re \end{aligned}}}
The associate family of the surface is just the surface rotated around the z-axis.
Taking m = 2 a real parametric expression becomes:
X
(
u
,
v
)
=
(
1
/
3
)
u
3
−
u
v
2
+
u
u
2
+
v
2
Y
(
u
,
v
)
=
−
u
2
v
+
(
1
/
3
)
v
3
−
v
u
2
+
v
2
Z
(
u
,
v
)
=
2
u
{\displaystyle {\begin{aligned}X(u,v)&=(1/3)u^{3}-uv^{2}+{\frac {u}{u^{2}+v^{2}}}\\Y(u,v)&=-u^{2}v+(1/3)v^{3}-{\frac {v}{u^{2}+v^{2}}}\\Z(u,v)&=2u\end{aligned}}}
References
Jesse Douglas , Tibor Radó, The Problem of Plateau: A Tribute to Jesse Douglas & Tibor Radó, World Scientific, 1992 (p. 239-240)
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. This allows a parametrization based on a complex parameter as
