# Equation of Cissoid of Diocles/Cartesian Form

## Theorem

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

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

## Proof

 $\ds r$ $=$ $\ds 2 a \sin \theta \tan \theta$ Equation of Cissoid of Diocles: Polar Form $\ds \leadsto \ \$ $\ds r^2$ $=$ $\ds 2 a r \sin \theta \tan \theta$ $\ds \leadsto \ \$ $\ds x^2 + y^2$ $=$ $\ds 2 a y \dfrac y x$ Conversion between Cartesian and Polar Coordinates in Plane $\ds \leadsto \ \$ $\ds x \paren {x^2 + y^2}$ $=$ $\ds 2 a y^2$ 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 $2 a - x$.