# [Maxima] Parameterization of implicit equation

Mon Oct 6 11:16:40 CDT 2008

```Bowo Prasetyo pisze:
> Hi all,
>
> Suppose, I have an implicit equation of Cayley Cubic as below:
>
> x^2 + y^2 - x^2*z + y^2*z + z^2 - 1 = 0
>
> How could I rewrite it in parametric equation of x, y and z?
> I have googled around about parameterization and found that there is
> no definitive way to do that, rather it is a trial and error process.
>
> Could we use Maxima to help doing parameterization?
>
> Thank you
> ~ bowo ~
If I'm not wrong it is 3D and  not closed curve, so it is harder then
parametrisation of 2D closed curve by angle.

What do you think about it :

(%i1) b:x^2 + y^2 - x^2*z + y^2*z + z^2 - 1;
(%o1) z^2+y^2*z-x^2*z+y^2+x^2-1
(%i2) solve(b=0,x);
(%o2)
[x=-sqrt(z^2/(z-1)+(y^2*z)/(z-1)+y^2/(z-1)-1/(z-1)),x=sqrt(z^2/(z-1)+(y^2*z)/(z-1)+y^2/(z-1)-1/(z-1))]
(%i3) solve(b=0,[x,y,z]);
(%o3)
[[x=%r1,y=%r2,z=-(sqrt(%r2^4+(-2*%r1^2-4)*%r2^2+%r1^4-4*%r1^2+4)+%r2^2-%r1^2)/2],[x=%r3,y=%r4,z=
(sqrt(%r4^4+(-2*%r3^2-4)*%r4^2+%r3^4-4*%r3^2+4)-%r4^2+%r3^2)/2]]

HTH