Weierstrass–Enneper parameterization explained
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.
Let
and
be functions on either the entire complex plane or the unit disk, where
is
meromorphic and
is
analytic, such that wherever
has a pole of order
,
has a zero of order
(or equivalently, such that the product
is
holomorphic), and let
be constants. Then the surface with coordinates
is minimal, where the
are defined using the real part of a complex integral, as follows:
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, .
Parametric surface of complex variables
The Weierstrass-Enneper model defines a minimal surface
(
) on a complex plane (
). Let
(the complex plane as the
space), the
Jacobian matrix of the surface can be written as a column of complex entries:
where
and
are holomorphic functions of
.
The Jacobian
represents the two orthogonal tangent vectors of the surface:
[2] The surface normal is given by
=\frac
\begin2\operatorname g \\2\operatorname g \\| g|^2-1\end
The Jacobian
leads to a number of important properties:
,
,
,
. 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:
and the second fundamental form matrix
Finally, a point
on the complex plane maps to a point
on the minimal surface in
by
where
for all minimal surfaces throughout this paper except for
Costa's minimal surface where
.
Embedded minimal surfaces and examples
The classical examples of embedded complete minimal surfaces in
with finite topology include the plane, the
catenoid, the
helicoid, and the
Costa's minimal surface. Costa's surface involves
Weierstrass's elliptic function
:
[4] where
is a constant.
[5] Helicatenoid
Choosing the functions
and
, a one parameter family of minimal surfaces is obtained.
Choosing the parameters of the surface as
:
At the extremes, the surface is a catenoid
or a helicoid
. Otherwise,
represents a mixing angle. The resulting surface, with domain chosen to prevent self-intersection, is a catenary rotated around the
axis in a helical fashion.
Lines of curvature
One can rewrite each element of second fundamental matrix as a function of
and
, for example
And consequently the second fundamental form matrix can be simplified asOne of its eigenvectors is which represents the principal direction in the complex domain.[6] Therefore, the two principal directions in the
space turn out to be
See also
Notes and References
- Book: Dierkes . U. . Hildebrandt . S. . Küster . A. . Wohlrab . O. . Minimal surfaces . I . 108 . Springer . 1992 . 3-540-53169-6 .
- Andersson . S. . Hyde . S. T. . Larsson . K. . Lidin . S. . Minimal Surfaces and Structures: From Inorganic and Metal Crystals to Cell Membranes and Biopolymers . Chem. Rev. . 88 . 1 . 221–242 . 1988 . 10.1021/cr00083a011 .
- Sharma . R. . The Weierstrass Representation always gives a minimal surface . 1208.5689 . 2012 . math.DG .
- Book: Lawden, D. F. . Elliptic Functions and Applications . Applied Mathematical Sciences . 80 . Springer . Berlin . 2011 . 978-1-4419-3090-3 .
- Book: Abbena . E. . Salamon . S. . Gray . A. . Minimal Surfaces via Complex Variables . Modern Differential Geometry of Curves and Surfaces with Mathematica . CRC Press . Boca Raton . 2006 . 1-58488-448-7 . 719–766 .
- Hua . H. . Jia . T. . 2018 . Wire cut of double-sided minimal surfaces . The Visual Computer . 34 . 6–8 . 985–995 . 10.1007/s00371-018-1548-0 . 13681681 .