We shall determine the evolute of the cycloid

where the parametre $u$ is the rolling angle of the circle with radius $a$ forming the cycloid.

The parametric equations of the evolute of a curve  $x=x(u),\;y=y(u)$  are

with $\alpha$ the slope angle of the tangent and $\varrho$ the radius of curvature of the given curve in the point  $(x,\,y)$ :

In the case of the cycloid (1), we have

Now we get

and thus the radius of curvature (red in the diagram) is

We utilised the identity

(see the half angle formula of sine in the goniometric formulas).  It is easy to show that the point, where the circle touches the $x$-axis, bisects the radius of curvature (which lies on the normal line at the point  $(x,\,y)$  of the cycloid).

Using then the derivative for parametric form, we obtain

which implies

Substituting all needed expressions (1), (3), (5) into (2) and simplifying, we arrive at the result

The equations (6) represent the evolute of the given cycloid.  But it is also a cycloid, congruent to the original one, which has been translated the distance $\pi a$ to the left and the distance $2a$ downwards; this is seen when one performs in (6) the substitution  $u=v-\pi$;  then (6) reads