I want to prove that the evolute of the astroid, $(a\cos^3{t},a\sin^3{t})$, $t \in [0,2\pi)$, is another astroid turned by a $\frac{\pi}{4}$ angle.
I have tried a variable change from the parametric equation of $t=r-\frac{\pi}{4}$ and a $\frac{\pi}{4}$ angle turn but I can´t get an equation of the following form: $$(b\cos^3{r},b\sin^3{r})$$
The rotated astroid, twice the given one, has parametric equations $$\left\{\sqrt{2} a\cos ^3 t-\sqrt{2} a\sin ^3 t,\sqrt{2}a\sin ^3 t+\sqrt{2} a\cos ^3 t\right\}$$ Because the rotation matrix is $$R(\pi/4)=\left( \begin{array}{cc} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \end{array} \right)$$
In the picture below the graph for $a=1$