Given a parabola with parametric equations

the evolute is given by

Eliminating and gives the implicit equation

With , this can be solved for to give

which is known as the semicubical parabola.

From a point above the evolute, three normals can be drawn to the parabola, while only one normal can be drawn to the parabola from a point below the evolute.

More things to try:

Weisstein, Eric W. "Parabola Evolute." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ParabolaEvolute.html