Equation of Witch of Agnesi/Parametric Form

Theorem

 * WitchOfAgnesi.png

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


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

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

By Thales' Theorem $\angle OAM$ is a right angle.

Hence $\angle OMA = \theta$ and so:
 * $OA = 2 a \cos \theta$

Thus:
 * $2 a - y = 2 a \cos^2 \theta$