Parametric Equations for Evolute

Theorem

Formulation 1

Let $C$ be a curve expressed as the locus of an equation $\map f {x, y} = 0$.

The parametric equations for the evolute of $C$ can be expressed as:

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

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 $y$ with respect to $x$.

Formulation 2

Let $C$ be a curve expressed as the locus of an equation $\map f {x, y} = 0$.

The parametric equations for the evolute of $C$ can be expressed as:

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

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 $C$

$x'$ and $x' '$ denote the derivative and second derivative respectively of $x$ with respect to $t$

$y'$ and $y' '$ denote the derivative and second derivative respectively of $y$ with respect to $t$.