Enneper surface
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In differential geometry and algebraic geometry, 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: \begin x &= \tfrac u \left(1 - \tfracu^2 + v^2\right), \\ y &= \tfrac v \left(1 - \tfracv^2 + u^2\right), \\ z & = \tfrac \left(u^2 - v^2\right). \end It was introduced by Alfred Enneper 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 tha ...
theory.Ulrich Dierkes, Stefan Hildebrandt, Friedrich Sauvigny (2010). Minimal Surfaces. Berlin Heidelberg: Springer. . The Weierstrass–Enneper parameterization is very simple, f(z)=1, g(z)=z, and the real parametric form can easily be calculated from it. The surface is conjugate to itself. Implicitization methods of algebraic geometry 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 example ...
equation \begin & 64 z^9 - 128 z^7 + 64 z^5 - 702 x^2 y^2 z^3 - 18 x^2 y^2 z + 144 (y^2 z^6 - x^2 z^6)\\ & + 162 (y^4 z^2 - x^4 z^2) + 27 (y^6 - x^6) + 9 (x^4 z + y^4 z) + 48 (x^2 z^3 + y^2 z^3)\\ & - 432 (x^2 z^5 + y^2 z^5) + 81 (x^4 y^2 - x^2 y^4) + 240 (y^2 z^4 - x^2 z^4) - 135 (x^4 z^3 + y^4 z^3) = 0. \end 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 a + b x + c y + d z = 0,\ where \begin a &= -\left(u^2 - v^2\right) \left(1 + \tfracu^2 + \tfracv^2\right), \\ b &= 6 u, \\ c &= 6 v, \\ d &= -3\left(1 - u^2 - v^2\right). \end Its coefficients satisfy the implicit degree-6 polynomial equation \begin &162 a^2 b^2 c^2 + 6 b^2 c^2 d^2 - 4 (b^6 + c^6) + 54 (a b^4 d - a c^4 d) + 81 (a^2 b^4 + a^2 c^4)\\ & + 4 (b^4 c^2 + b^2 c^4) - 3 (b^4 d^2 + c^4 d^2) + 36 (a b^2 d^3 - a c^2 d^3) = 0. \end The Jacobian, Gaussian curvature 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 \begin J &= \frac(1 + u^2 + v^2)^4, \\ K &= -\frac\frac, \\ H &= 0. \end 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 -4\pi. Osserman proved that a complete minimal surface in \R^3 with total curvature -4\pi 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 descri ...
or the Enneper surface. Another property is that all bicubical minimal Bézier surfaces are, up to an affine transformation, pieces of the surface. It can be generalized to higher order rotational symmetries by using the Weierstrass–Enneper parameterization f(z) = 1, g(z) = z^k for integer k>1. It can also be generalized to higher dimensions; Enneper-like surfaces are known to exist in \R^n 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