Equation of Cardioid/Polar
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Theorem
Let $C$ be a cardioid embedded in a polar coordinate plane such that:
- its deferent of radius $a$ is positioned with its center at $\polar {a, 0}$
- there is a cusp at the origin.
The polar equation of $C$ is:
- $r = 2 a \paren {1 + \cos \theta}$
Proof
Let $P = \polar {r, \theta}$ be an arbitrary point on $C$.
Let $A$ and $B$ be the centers of the deferent and epicycle respectively.
Let $Q$ be the point where the deferent and epicycle touch.
By definition of the method of construction of $C$, we have that the arc $OQ$ of the deferent equals the arc $PQ$ of the epicycle.
Thus:
- $\angle OAQ = \angle PBQ$
and it follows that $AB$ is parallel to $OP$.
With reference to the diagram above, we have:
\(\ds r\) | \(=\) | \(\ds OR + RS + SP\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds a \cos \theta + 2 a + a \cos \theta\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 a \paren {1 + \cos \theta}\) |
and the result follows.
$\blacksquare$
Also presented as
The polar equation for the cardioid can also be seen presented as:
- $r = a \paren {1 + \cos \theta}$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 11$: Special Plane Curves: Cardioid: $11.12$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): cardioid
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): cardioid