Enneper surface

In mathematics, in the fields of differential geometry and algebraic geometry, the Enneper surface is a surface that can be described parametrically by:: $x\; =\; u(1\; -\; u^2/3\; +\; v^2)/3,$

: $y\; =\; -v(1\; -\; v^2/3\; +\; u^2)/3,$

: $z\; =\; (u^2\; -\; v^2)/3.$

It was introduced by Alfred Enneper in connection with minimal surface theory.

:::::::"Figure 1. An Enneper surface"

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: $64\; z^9\; -\; 128\; z^7\; +\; 64\; z^5\; -\; 702\; x^2\; y^2\; z^3\; -\; 18\; x^2\; y^2\; z\; +\; 144\; (y^2\; z^6\; -\; x^2\; z^6)$

: $\{\}\; +\; 162\; (y^4\; z^2\; -\; x^4\; z^2)\; +\; 27\; (y^6\; -\; x^6)\; +\; 9\; (x^4\; z\; +\; y^4\; z)\; +\; 48\; (x^2\; z^3\; +\; y^2\; z^3)$

: $\{\}\; -\; 432\; (x^2\; z^5\; +\; y^2\; z^5)\; +\; 81\; (x^4\; y^2\; -\; x^2\; y^4)\; +\; 240\; (y^2\; z^4\; -\; x^2\; z^4)\; -\; 135\; (x^4\; z^3\; +\; y^4\; z^3)\; =\; 0.$

:::"Figure 2. The Enneper surface in Figure 1 has been rotated 30° around the +z axis."

:::"Figure 3. The Enneper surface in Figure 1 has been rotated 60° around the +z axis."

Dually, the tangent plane at the point with given parameters is $a\; +\; b\; x\; +\; c\; y\; +\; d\; z\; =\; 0,$ where: $a\; =\; -(u^2\; -\; v^2)\; (1\; +\; u^2/3\; +\; v^2/3),$

: $b\; =\; 6\; u,$

: $c\; =\; 6\; v,$

: $d\; =\; -3(1\; -\; u^2\; -\; v^2).$

Its coefficients satisfy the implicit degree-6 polynomial equation: $162\; a^2\; b^2\; c^2\; +\; 6\; b^2\; c^2\; d^2\; -\; 4\; (b^6\; +\; c^6)\; +\; 54\; (a\; b^4\; d\; -\; a\; c^4\; d)\; +\; 81\; (a^2\; b^4\; +\; a^2\; c^4)$

: $\{\}\; +\; 4\; (b^4\; c^2\; +\; b^2\; c^4)\; -\; 3\; (b^4\; d^2\; +\; c^4\; d^2)\; +\; 36\; (a\; b^2\; d^3\; -\; a\; c^2\; d^3)\; =\; 0.$

Enneper's is a minimal surface. The Jacobian, Gaussian curvature and mean curvature are: $J\; =\; (1\; +\; u^2\; +\; v^2)^4/81,$

: $K\; =\; -(4/9)/J,$

: $H\; =\; 0.$