Evolute of Parabola

Theorem
The evolute of the parabola $y = x^2$ is the curve:
 * $27 X^2 = 16 \paren {Y - \dfrac 1 2}^3$

Proof
From Parametric Equations for Evolute: Formulation 1:


 * $\begin {cases}

X = x - \dfrac {y' \paren {1 + y'^2} } {y''} \\ Y = y + \dfrac {1 + y'^2} {y''} \end{cases}$

where:
 * $\tuple {x, y}$ denotes the Cartesian coordinates of a general point on $C$
 * $\tuple {X, Y}$ denotes the Cartesian coordinates of a general point on the evolute of $C$
 * $y'$ and $y''$ denote the derivative and second derivative respectively of $y$ $x$.

Thus we have:

and so:

and:

Then:

The parabola (blue) and its evolute (red) are illustrated below: