Equation of Cissoid of Diocles/Parametric Form

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Theorem

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

$\begin {cases} x = 2 a \sin^2 \theta \\ y = \dfrac {2 a \sin^3 \theta} {\cos \theta} \end {cases}$


Proof

\(\displaystyle r\) \(=\) \(\displaystyle 2 a \sin \theta \tan \theta\) Equation of Cissoid of Diocles: Polar Form
\(\displaystyle \leadsto \ \ \) \(\displaystyle r \cos \theta\) \(=\) \(\displaystyle 2 a \cos \sin \theta \paren {\dfrac {\sin \theta} {\cos \theta} }\)
\(\displaystyle r \sin \theta\) \(=\) \(\displaystyle 2 a \sin \theta \sin \theta \paren {\dfrac {\sin \theta} {\cos \theta} }\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle x\) \(=\) \(\displaystyle 2 a \sin^2 \theta\) Conversion between Cartesian and Polar Coordinates in Plane
\(\displaystyle y\) \(=\) \(\displaystyle \dfrac {2 a \sin^3 \theta} {\cos \theta}\)

$\blacksquare$


Also see


Sources