Astroidal surface and its discriminant

Let

a = - 1/(4^(1/3))*(x^2 + y^2 - 1)

(equation of a circle containing 4 cusps),

b = 27^(1/2) * x*y .

Then the cubic surface in P^3 given by the equation

z^3 - 3 a z + b = 0

is ramified along the degree 6 discriminant curve ("astroid"):

Delta = b^2 - 4 * a^3 = (x^2 + y^2 - 1)^3 + 27 * x^2*y^2.

Astroidal surface, semi-transparent:

(Surf source code)

Astroidal surface, non-transparent:

(real points of the) astroid curve:

(Surf source code):