![EnneperSurfaceAnimated](https://upload.wikimedia.org/wikipedia/commons/0/06/EnneperSurfaceAnimated.gif)
In
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
and
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, the Enneper surface is a self-intersecting surface that can be described
parametrically
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
by:
It was introduced by
Alfred Enneper
Alfred Enneper (June 14, 1830, Barmen – March 24, 1885 Hanover) was a German mathematician. Enneper earned his PhD from the Georg-August-Universität Göttingen in 1856, under the supervision of Peter Gustav Lejeune Dirichlet, for his disserta ...
in 1864 in connection with
minimal surface
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below).
The term "minimal surface" is used because these surfaces originally arose as surfaces that ...
theory.
[Ulrich Dierkes, Stefan Hildebrandt, Friedrich Sauvigny (2010). Minimal Surfaces. Berlin Heidelberg: Springer. .]
The
Weierstrass–Enneper parameterization
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 f and g be functions on either ...
is very simple,
, and the real parametric form can easily be calculated from it. The surface is
conjugate to itself.
Implicitization methods of
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
can be used to find out that the points in the Enneper surface given above satisfy the degree-9
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
equation
Dually, the
tangent plane
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
at the point with given parameters is
where
Its coefficients satisfy the implicit degree-6 polynomial equation
The
Jacobian
In mathematics, a Jacobian, named for Carl Gustav Jacob Jacobi, may refer to:
*Jacobian matrix and determinant
*Jacobian elliptic functions
*Jacobian variety
*Intermediate Jacobian
In mathematics, the intermediate Jacobian of a compact Kähler m ...
,
Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point:
K = \kappa_1 \kappa_2.
The Gaussian radius of curvature is the reciprocal of .
F ...
and
mean curvature In mathematics, the mean curvature H of a surface S is an ''extrinsic'' measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space.
The ...
are
The
total curvature
In mathematical study of the differential geometry of curves, the total curvature of an immersed plane curve is the integral of curvature along a curve taken with respect to arc length:
:\int_a^b k(s)\,ds.
The total curvature of a closed curve i ...
is
.
Osserman proved that a complete minimal surface in
with total curvature
is either the
catenoid
In geometry, a catenoid is a type of surface, arising by rotating a catenary curve about an axis (a surface of revolution). It is a minimal surface, meaning that it occupies the least area when bounded by a closed space. It was formally describe ...
or the Enneper surface.
Another property is that all bicubical minimal
Bézier surfaces are, up to an
affine transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More generally, ...
, pieces of the surface.
It can be generalized to higher order rotational symmetries by using the Weierstrass–Enneper parameterization
for integer k>1.
It can also be generalized to higher dimensions; Enneper-like surfaces are known to exist in
for n up to 7.
[Jaigyoung Choe, On the existence of higher dimensional Enneper's surface, Commentarii Mathematici Helvetici 1996, Volume 71, Issue 1, pp 556-569]
References
External links
*
* https://web.archive.org/web/20130501084413/http://www.math.hmc.edu/~gu/curves_and_surfaces/surfaces/enneper.html
* https://web.archive.org/web/20160919231223/https://secure.msri.org/about/sgp/jim/geom/minimal/library/ennepern/index.html
{{Minimal surfaces
Algebraic surfaces
Minimal surfaces