Construction for minimal surfaces
In mathematics , the Weierstrass–Enneper parameterization of minimal surfaces is a classical piece of differential geometry .
Alfred Enneper and Karl Weierstrass studied minimal surfaces as far back as 1863.
Weierstrass parameterization facilities fabrication of periodic minimal surfaces
Let
f
{\displaystyle f}
and
g
{\displaystyle g}
be functions on either the entire complex plane or the unit disk, where
g
{\displaystyle g}
is meromorphic and
f
{\displaystyle f}
is analytic , such that wherever
g
{\displaystyle g}
has a pole of order
m
{\displaystyle m}
,
f
{\displaystyle f}
has a zero of order
2
m
{\displaystyle 2m}
(or equivalently, such that the product
f
g
2
{\displaystyle fg^{2}}
is holomorphic ), and let
c
1
,
c
2
,
c
3
{\displaystyle c_{1},c_{2},c_{3}}
be constants. Then the surface with coordinates
(
x
1
,
x
2
,
x
3
)
{\displaystyle (x_{1},x_{2},x_{3})}
is minimal, where the
x
k
{\displaystyle x_{k}}
are defined using the real part of a complex integral, as follows:
x
k
(
ζ
)
=
R
e
{
∫
0
ζ
φ
k
(
z
)
d
z
}
+
c
k
,
k
=
1
,
2
,
3
φ
1
=
f
(
1
−
g
2
)
/
2
φ
2
=
i
f
(
1
+
g
2
)
/
2
φ
3
=
f
g
{\displaystyle {\begin{aligned}x_{k}(\zeta )&{}=\mathrm {Re} \left\{\int _{0}^{\zeta }\varphi _{k}(z)\,dz\right\}+c_{k},\qquad k=1,2,3\\\varphi _{1}&{}=f(1-g^{2})/2\\\varphi _{2}&{}=if(1+g^{2})/2\\\varphi _{3}&{}=fg\end{aligned}}}
The converse is also true: every nonplanar minimal surface defined over a simply connected domain can be given a parametrization of this type.[1]
For example, Enneper's surface has f (z ) = 1 , g (z ) = zm .
Parametric surface of complex variables [ edit ]
The Weierstrass-Enneper model defines a minimal surface
X
{\displaystyle X}
(
R
3
{\displaystyle \mathbb {R} ^{3}}
) on a complex plane (
C
{\displaystyle \mathbb {C} }
). Let
ω
=
u
+
v
i
{\displaystyle \omega =u+vi}
(the complex plane as the
u
v
{\displaystyle uv}
space), the Jacobian matrix of the surface can be written as a column of complex entries:
J
=
[
(
1
−
g
2
(
ω
)
)
f
(
ω
)
i
(
1
+
g
2
(
ω
)
)
f
(
ω
)
2
g
(
ω
)
f
(
ω
)
]
{\displaystyle \mathbf {J} ={\begin{bmatrix}\left(1-g^{2}(\omega )\right)f(\omega )\\i\left(1+g^{2}(\omega )\right)f(\omega )\\2g(\omega )f(\omega )\end{bmatrix}}}
where
f
(
ω
)
{\displaystyle f(\omega )}
and
g
(
ω
)
{\displaystyle g(\omega )}
are holomorphic functions of
ω
{\displaystyle \omega }
.
The Jacobian
J
{\displaystyle \mathbf {J} }
represents the two orthogonal tangent vectors of the surface:[2]
X
u
=
[
Re
J
1
Re
J
2
Re
J
3
]
X
v
=
[
−
Im
J
1
−
Im
J
2
−
Im
J
3
]
{\displaystyle \mathbf {X_{u}} ={\begin{bmatrix}\operatorname {Re} \mathbf {J} _{1}\\\operatorname {Re} \mathbf {J} _{2}\\\operatorname {Re} \mathbf {J} _{3}\end{bmatrix}}\;\;\;\;\mathbf {X_{v}} ={\begin{bmatrix}-\operatorname {Im} \mathbf {J} _{1}\\-\operatorname {Im} \mathbf {J} _{2}\\-\operatorname {Im} \mathbf {J} _{3}\end{bmatrix}}}
The surface normal is given by
n
^
=
X
u
×
X
v
|
X
u
×
X
v
|
=
1
|
g
|
2
+
1
[
2
Re
g
2
Im
g
|
g
|
2
−
1
]
{\displaystyle \mathbf {\hat {n}} ={\frac {\mathbf {X_{u}} \times \mathbf {X_{v}} }{|\mathbf {X_{u}} \times \mathbf {X_{v}} |}}={\frac {1}{|g|^{2}+1}}{\begin{bmatrix}2\operatorname {Re} g\\2\operatorname {Im} g\\|g|^{2}-1\end{bmatrix}}}
The Jacobian
J
{\displaystyle \mathbf {J} }
leads to a number of important properties:
X
u
⋅
X
v
=
0
{\displaystyle \mathbf {X_{u}} \cdot \mathbf {X_{v}} =0}
,
X
u
2
=
Re
(
J
2
)
{\displaystyle \mathbf {X_{u}} ^{2}=\operatorname {Re} (\mathbf {J} ^{2})}
,
X
v
2
=
Im
(
J
2
)
{\displaystyle \mathbf {X_{v}} ^{2}=\operatorname {Im} (\mathbf {J} ^{2})}
,
X
u
u
+
X
v
v
=
0
{\displaystyle \mathbf {X_{uu}} +\mathbf {X_{vv}} =0}
. The proofs can be found in Sharma's essay: The Weierstrass representation always gives a minimal surface.[3] The derivatives can be used to construct the first fundamental form matrix:
[
X
u
⋅
X
u
X
u
⋅
X
v
X
v
⋅
X
u
X
v
⋅
X
v
]
=
[
1
0
0
1
]
{\displaystyle {\begin{bmatrix}\mathbf {X_{u}} \cdot \mathbf {X_{u}} &\;\;\mathbf {X_{u}} \cdot \mathbf {X_{v}} \\\mathbf {X_{v}} \cdot \mathbf {X_{u}} &\;\;\mathbf {X_{v}} \cdot \mathbf {X_{v}} \end{bmatrix}}={\begin{bmatrix}1&0\\0&1\end{bmatrix}}}
and the second fundamental form matrix
[
X
u
u
⋅
n
^
X
u
v
⋅
n
^
X
v
u
⋅
n
^
X
v
v
⋅
n
^
]
{\displaystyle {\begin{bmatrix}\mathbf {X_{uu}} \cdot \mathbf {\hat {n}} &\;\;\mathbf {X_{uv}} \cdot \mathbf {\hat {n}} \\\mathbf {X_{vu}} \cdot \mathbf {\hat {n}} &\;\;\mathbf {X_{vv}} \cdot \mathbf {\hat {n}} \end{bmatrix}}}
Finally, a point
ω
t
{\displaystyle \omega _{t}}
on the complex plane maps to a point
X
{\displaystyle \mathbf {X} }
on the minimal surface in
R
3
{\displaystyle \mathbb {R} ^{3}}
by
X
=
[
Re
∫
ω
0
ω
t
J
1
d
ω
Re
∫
ω
0
ω
t
J
2
d
ω
Re
∫
ω
0
ω
t
J
3
d
ω
]
{\displaystyle \mathbf {X} ={\begin{bmatrix}\operatorname {Re} \int _{\omega _{0}}^{\omega _{t}}\mathbf {J} _{1}d\omega \\\operatorname {Re} \int _{\omega _{0}}^{\omega _{t}}\mathbf {J} _{2}d\omega \\\operatorname {Re} \int _{\omega _{0}}^{\omega _{t}}\mathbf {J} _{3}d\omega \end{bmatrix}}}
where
ω
0
=
0
{\displaystyle \omega _{0}=0}
for all minimal surfaces throughout this paper except for
Costa's minimal surface where
ω
0
=
(
1
+
i
)
/
2
{\displaystyle \omega _{0}=(1+i)/2}
.
Embedded minimal surfaces and examples [ edit ]
The classical examples of embedded complete minimal surfaces in
R
3
{\displaystyle \mathbb {R} ^{3}}
with finite topology include the plane, the catenoid , the helicoid , and the Costa's minimal surface . Costa's surface involves Weierstrass's elliptic function
℘
{\displaystyle \wp }
:[4]
g
(
ω
)
=
A
℘
′
(
ω
)
{\displaystyle g(\omega )={\frac {A}{\wp '(\omega )}}}
f
(
ω
)
=
℘
(
ω
)
{\displaystyle f(\omega )=\wp (\omega )}
where
A
{\displaystyle A}
is a constant.
[5]
Helicatenoid [ edit ]
Choosing the functions
f
(
ω
)
=
e
−
i
α
e
ω
/
A
{\displaystyle f(\omega )=e^{-i\alpha }e^{\omega /A}}
and
g
(
ω
)
=
e
−
ω
/
A
{\displaystyle g(\omega )=e^{-\omega /A}}
, a one parameter family of minimal surfaces is obtained.
φ
1
=
e
−
i
α
sinh
(
ω
A
)
{\displaystyle \varphi _{1}=e^{-i\alpha }\sinh \left({\frac {\omega }{A}}\right)}
φ
2
=
i
e
−
i
α
cosh
(
ω
A
)
{\displaystyle \varphi _{2}=ie^{-i\alpha }\cosh \left({\frac {\omega }{A}}\right)}
φ
3
=
e
−
i
α
{\displaystyle \varphi _{3}=e^{-i\alpha }}
X
(
ω
)
=
Re
[
e
−
i
α
A
cosh
(
ω
A
)
i
e
−
i
α
A
sinh
(
ω
A
)
e
−
i
α
ω
]
=
cos
(
α
)
[
A
cosh
(
Re
(
ω
)
A
)
cos
(
Im
(
ω
)
A
)
−
A
cosh
(
Re
(
ω
)
A
)
sin
(
Im
(
ω
)
A
)
Re
(
ω
)
]
+
sin
(
α
)
[
A
sinh
(
Re
(
ω
)
A
)
sin
(
Im
(
ω
)
A
)
A
sinh
(
Re
(
ω
)
A
)
cos
(
Im
(
ω
)
A
)
Im
(
ω
)
]
{\displaystyle \mathbf {X} (\omega )=\operatorname {Re} {\begin{bmatrix}e^{-i\alpha }A\cosh \left({\frac {\omega }{A}}\right)\\ie^{-i\alpha }A\sinh \left({\frac {\omega }{A}}\right)\\e^{-i\alpha }\omega \\\end{bmatrix}}=\cos(\alpha ){\begin{bmatrix}A\cosh \left({\frac {\operatorname {Re} (\omega )}{A}}\right)\cos \left({\frac {\operatorname {Im} (\omega )}{A}}\right)\\-A\cosh \left({\frac {\operatorname {Re} (\omega )}{A}}\right)\sin \left({\frac {\operatorname {Im} (\omega )}{A}}\right)\\\operatorname {Re} (\omega )\\\end{bmatrix}}+\sin(\alpha ){\begin{bmatrix}A\sinh \left({\frac {\operatorname {Re} (\omega )}{A}}\right)\sin \left({\frac {\operatorname {Im} (\omega )}{A}}\right)\\A\sinh \left({\frac {\operatorname {Re} (\omega )}{A}}\right)\cos \left({\frac {\operatorname {Im} (\omega )}{A}}\right)\\\operatorname {Im} (\omega )\\\end{bmatrix}}}
Choosing the parameters of the surface as
ω
=
s
+
i
(
A
ϕ
)
{\displaystyle \omega =s+i(A\phi )}
:
X
(
s
,
ϕ
)
=
cos
(
α
)
[
A
cosh
(
s
A
)
cos
(
ϕ
)
−
A
cosh
(
s
A
)
sin
(
ϕ
)
s
]
+
sin
(
α
)
[
A
sinh
(
s
A
)
sin
(
ϕ
)
A
sinh
(
s
A
)
cos
(
ϕ
)
A
ϕ
]
{\displaystyle \mathbf {X} (s,\phi )=\cos(\alpha ){\begin{bmatrix}A\cosh \left({\frac {s}{A}}\right)\cos \left(\phi \right)\\-A\cosh \left({\frac {s}{A}}\right)\sin \left(\phi \right)\\s\\\end{bmatrix}}+\sin(\alpha ){\begin{bmatrix}A\sinh \left({\frac {s}{A}}\right)\sin \left(\phi \right)\\A\sinh \left({\frac {s}{A}}\right)\cos \left(\phi \right)\\A\phi \\\end{bmatrix}}}
At the extremes, the surface is a catenoid
(
α
=
0
)
{\displaystyle (\alpha =0)}
or a helicoid
(
α
=
π
/
2
)
{\displaystyle (\alpha =\pi /2)}
. Otherwise,
α
{\displaystyle \alpha }
represents a mixing angle. The resulting surface, with domain chosen to prevent self-intersection, is a catenary rotated around the
X
3
{\displaystyle \mathbf {X} _{3}}
axis in a helical fashion.
A catenary that spans periodic points on a helix, subsequently rotated along the helix to produce a minimal surface.
The fundamental domain (C) and the 3D surfaces. The continuous surfaces are made of copies of the fundamental patch (R3)
Lines of curvature [ edit ]
One can rewrite each element of second fundamental matrix as a function of
f
{\displaystyle f}
and
g
{\displaystyle g}
, for example
X
u
u
⋅
n
^
=
1
|
g
|
2
+
1
[
Re
(
(
1
−
g
2
)
f
′
−
2
g
f
g
′
)
Re
(
(
1
+
g
2
)
f
′
i
+
2
g
f
g
′
i
)
Re
(
2
g
f
′
+
2
f
g
′
)
]
⋅
[
Re
(
2
g
)
Re
(
−
2
g
i
)
Re
(
|
g
|
2
−
1
)
]
=
−
2
Re
(
f
g
′
)
{\displaystyle \mathbf {X_{uu}} \cdot \mathbf {\hat {n}} ={\frac {1}{|g|^{2}+1}}{\begin{bmatrix}\operatorname {Re} \left((1-g^{2})f'-2gfg'\right)\\\operatorname {Re} \left((1+g^{2})f'i+2gfg'i\right)\\\operatorname {Re} \left(2gf'+2fg'\right)\\\end{bmatrix}}\cdot {\begin{bmatrix}\operatorname {Re} \left(2g\right)\\\operatorname {Re} \left(-2gi\right)\\\operatorname {Re} \left(|g|^{2}-1\right)\\\end{bmatrix}}=-2\operatorname {Re} (fg')}
And consequently the second fundamental form matrix can be simplified as
[
−
Re
f
g
′
Im
f
g
′
Im
f
g
′
Re
f
g
′
]
{\displaystyle {\begin{bmatrix}-\operatorname {Re} fg'&\;\;\operatorname {Im} fg'\\\operatorname {Im} fg'&\;\;\operatorname {Re} fg'\end{bmatrix}}}
Lines of curvature make a quadrangulation of the domain
One of its eigenvectors is
f
g
′
¯
{\displaystyle {\overline {\sqrt {fg'}}}}
which represents the principal direction in the complex domain.
[6] Therefore, the two principal directions in the
u
v
{\displaystyle uv}
space turn out to be
ϕ
=
−
1
2
Arg
(
f
g
′
)
±
k
π
/
2
{\displaystyle \phi =-{\frac {1}{2}}\operatorname {Arg} (fg')\pm k\pi /2}
See also [ edit ]
References [ edit ]
^ Dierkes, U.; Hildebrandt, S.; Küster, A.; Wohlrab, O. (1992). Minimal surfaces . Vol. I. Springer. p. 108. ISBN 3-540-53169-6 .
^ Andersson, S.; Hyde, S. T.; Larsson, K.; Lidin, S. (1988). "Minimal Surfaces and Structures: From Inorganic and Metal Crystals to Cell Membranes and Biopolymers". Chem. Rev . 88 (1): 221–242. doi :10.1021/cr00083a011 .
^ Sharma, R. (2012). "The Weierstrass Representation always gives a minimal surface". arXiv :1208.5689 [math.DG ].
^ Lawden, D. F. (2011). Elliptic Functions and Applications . Applied Mathematical Sciences. Vol. 80. Berlin: Springer. ISBN 978-1-4419-3090-3 .
^ Abbena, E.; Salamon, S.; Gray, A. (2006). "Minimal Surfaces via Complex Variables". Modern Differential Geometry of Curves and Surfaces with Mathematica . Boca Raton: CRC Press. pp. 719–766. ISBN 1-58488-448-7 .
^ Hua, H.; Jia, T. (2018). "Wire cut of double-sided minimal surfaces". The Visual Computer . 34 (6–8): 985–995. doi :10.1007/s00371-018-1548-0 . S2CID 13681681 .