Equation of Cissoid of Diocles/Polar Form
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Theorem
The cissoid of Diocles can be defined by the polar equation:
- $r = 2 a \sin \theta \tan \theta$
Proof
By construction:
| \(\ds OS\) | \(=\) | \(\ds 2 a \sec \theta\) | Definition of Secant Function | |||||||||||
| \(\ds OR\) | \(=\) | \(\ds 2 a \cos \theta\) | Definition of Cosine | |||||||||||
| \(\ds OP\) | \(=\) | \(\ds RS\) | Definition of Cissoid of Diocles | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds OP\) | \(=\) | \(\ds OS - OR\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds r\) | \(=\) | \(\ds 2 a \paren {\sec \theta - \cos \theta}\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 a \sin \theta \tan \theta\) | Secant Minus Cosine |
$\blacksquare$
Also see
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): cissoid
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): cissoid
- Weisstein, Eric W. "Cissoid of Diocles." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CissoidofDiocles.html