Parametrization of a Parabola's Evolute Let $y=x^2/2$. Its parametric form is $r(t)=t\,\hat i+t^2/2\,\hat j$, and its evolute is $$ c(t)=-t^3\,\hat i+\frac{3t^2+2}{2}\,\hat j.\tag{1} $$ Visually, $ as a normal function, by letting $x=-t^3$, I get $$ y=\frac{3x^{2/3}+2}{2}, $$ but the graph of this evolute is nothing like the one above. What am I doing wrong?

I think it would be clearer to write $|x|^{2/3}$. Other than that, your formulas are both right, but the plot is wrong - the $y$ coordinate of the evolute at $x=1$ should be $5/2$, not $3/2$, and you can see with the naked eye that the curve in your plot doesn't reflect the centres of curvature of the parabola; there are normals to the parabola that don't even cross that curve on the right side of the parabola.