Discriminant of a generic projection of a minimal normal surface singularity

Let $(S,0)$ be a rational complex surface singularity with reduced fundamental cycle, also known as a {\em minimal} singularity. Using a fundamental result by M. Spivakovsky, we explain how to get a minimal resolution of the discriminant curve for a generic projection of $(S,0)$ onto $({\text{C}}^2,0)$ directly from the resolution graph of $(S,0)$.