Equation of Witch of Agnesi/Parametric Form

Theorem

The equation of the Witch of Agnesi can be presented in paremetric form as:

$\begin {cases} x = 2 a \cot \theta \\ y = a \paren {1 - \cos 2 \theta} \end {cases}$

Let $P = \tuple {x, y}$ and $A = \tuple {d, y}$.

Let $\theta$ be the angle that $ON$ makes with the horizontal.

We have by definition of cotangent:

$\dfrac {OM} {MN} = \dfrac {2 a} x = \cot \theta$

Hence $\angle OMA = \theta$ and so:

$OA = 2 a \cos \theta$

Thus:

$2 a - y = 2 a \cos^2 \theta$

$\ds 2 a - y$

$=$

$\ds 2 a \cos^2 \theta$

$\ds \leadsto \ \ $

$\ds y$

$=$

$\ds 2 a \paren {1 - \dfrac {\cos 2 \theta + 1} 2}$

$\ds $

$=$

$\ds a \paren {2 - \paren {\cos 2 \theta + 1} }$

$\ds $

$=$

$\ds a \paren {1 - \cos 2 \theta}$

$\blacksquare$