Talk:Enneper surface

U/V coordinates
which u/v interval was used to create these famous surfase? --RokerHRO (talk) 10:11, 6 August 2010 (UTC)

Degree 9 polynomial
Starting with the equations for the Enneper surface,
 * $$ x = u(1 - u^2/3 + v^2)/3, y = v(1 - v^2/3 + u^2)/3, z = (u^2 - v^2)/3 $$,

we can eliminate $$u$$ and $$v$$ to produce the degree 9 polynomial
 * $$64 z^9 - 128 z^7 + 64 z^5 - 702 x^2 y^2 z^3 - 18 x^2 y^2 z + 144 (y^2 z^6 - x^2 z^6)\ $$
 * $${} + 162 (y^4 z^2 - x^4 z^2) + 27 (y^6 - x^6) + 9 (x^4 z + y^4 z) + 48 (x^2 z^3 + y^2 z^3)\ $$
 * $${} - 432 (x^2 z^5 + y^2 z^5) + 81 (x^4 y^2 - x^2 y^4) + 240 (y^2 z^4 - x^2 z^4) - 135 (x^4 z^3 + y^4 z^3) = 0.\ $$.

For example, in Wolfram Mathematica or (free, on-line) Wolfram Alpha,
 * Eliminate[{x == u*(1 - u^2/3 + v^2)/3, y == v*(1 - v^2/3 + u^2)/3, z == (u^2 - v^2)/3}, {u, v}]

produces an equation of the form $$p(x,y,z) = q(x,y,z)$$ where $$p(x,y,z) - q(x,y,z) = 0$$ is the above degree 9 polynomial. MathPerson (talk) 15:37, 21 April 2021 (UTC)