Equation of Cissoid of Diocles/Cartesian Form

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Theorem

The cissoid of Diocles can be defined by the Cartesian equation:

$x \paren {x^2 + y^2} = 2 a y^2$


Proof

\(\displaystyle r\) \(=\) \(\displaystyle 2 a \sin \theta \tan \theta\) Equation of Cissoid of Diocles: Polar Form
\(\displaystyle \leadsto \ \ \) \(\displaystyle r^2\) \(=\) \(\displaystyle 2 a r \sin \theta \tan \theta\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle x^2 + y^2\) \(=\) \(\displaystyle 2 a y \dfrac y x\) Conversion between Cartesian and Polar Coordinates in Plane
\(\displaystyle \leadsto \ \ \) \(\displaystyle x \paren {x^2 + y^2}\) \(=\) \(\displaystyle 2 a y^2\) rearranging

$\blacksquare$


Also presented as

The Cartesian equation for the cissoid of Diocles can also be defined as:

$\dfrac {x^3} {2 a - x} = y^2$

which can be derived directly from the form given by subtracting $x y^2$ from both sides and dividing by $2 a - x$.


Also see


Sources