Equation of Cissoid of Diocles/Parametric Form

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

\(\ds r\)

\(=\)

\(\ds 2 a \sin \theta \tan \theta\)

Equation of Cissoid of Diocles: Polar Form

\(\ds \leadsto \ \ \)

\(\ds r \cos \theta\)

\(=\)

\(\ds 2 a \cos \sin \theta \paren {\dfrac {\sin \theta} {\cos \theta} }\)

\(\ds r \sin \theta\)

\(=\)

\(\ds 2 a \sin \theta \sin \theta \paren {\dfrac {\sin \theta} {\cos \theta} }\)

\(\ds \leadsto \ \ \)

\(\ds x\)

\(=\)

\(\ds 2 a \sin^2 \theta\)

Conversion between Cartesian and Polar Coordinates in Plane

\(\ds y\)

\(=\)

\(\ds \dfrac {2 a \sin^3 \theta} {\cos \theta}\)

$\blacksquare$

Also see

• Equation of Cissoid of Diocles/Polar Form
• Equation of Cissoid of Diocles/Cartesian Form

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 11$: Special Plane Curves: Cissoid of Diocles: $11.34$

• Weisstein, Eric W. "Cissoid of Diocles." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CissoidofDiocles.html