Here is the equation I have:
$8(x^2-\phi^4y^2)(y^2-\phi^4z^2)(z^2-\phi^4x^2)(x^4+y^4+z^4-2x^2y^2-2x^2z^2-2y^2z^2)+(3+5\phi)(x^2+y^2+z^2-1-c)^2(x^2+y^2+z^2-\phi^2-b)^2-a=0$
Where $\phi = \frac{1+\sqrt 5}{2}$.
I'm trying to form 3D-printable models of some algebraic surfaces, and the case of the Barth decic has been a little tough. As the canonical surface contains a number of double points which prevent it from printing well, I have been attempting to tweak the equation to change zones which are locally like a double cone to become locally like a hyperboloid of one sheet.
The surface has four sets of ordinary double points, three of which I can tweak as needed by varying $a$, $b$ and $c$. My problem has been that for the remaining set of points, I can't find an easy way to introduce a small change which smooths them all equivalently. After fudging the numbers a bit, I feel it must involve some manipulation of the $(x^4+y^4+z^4-2x^2y^2-2x^2z^2-2y^2z^2)$ term. For reference, I have the project hosted here.
My question is, how can I adjust this last set of double points in the manner I describe by varying a single constant, or is it not possible to do this?
