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.