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:
Astroidal surface, non-transparent:
(real points of the) astroid curve:
