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

$64z^{9}-128z^{7}+64z^{5}-702x^{2}y^{2}z^{3}-18x^{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)