Enneper surface

Article summary

In differential geometry and algebraic geometry, the Enneper surface is a self-intersecting surface that can be described parametrically by: \begin{align} x &= \tfrac{1}{3} u \left(1 - \tfrac{1}{3}u^2 + v^2\right), \\ y &= \tfrac{1}{3} v \left(1 - \tfrac{1}{3}v^2 + u^2\right), \\ z & = \tfrac{1}{3} \left(u^2 - v^2\right). \end{align} It was introduced by Alfred Enneper in 1864 in connection with minimal surface theory. 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 equation \begin{align} & 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{align} Dually, the tangent plane at the point with given parameters is a + b x + c y + d z = 0,\ where \begin{align} a &= -\left(u^2 - v^2\right) \left(1 + \tfrac{1}{3}u^2 + \tfrac{1}{3}v^2\right), \\ b &= 6 u, \\ c &= 6 v, \\ d &= -3\left(1 - u^2 - v^2\right).…

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