Condition for Conchoid of Nicomedes to have Loop

Theorem

Let $\KK$ be a conchoid of Nicomedes such that:

the perpendicular distance from the directrix $\CC$ of $\KK$ to the focus point $P$ of $\KK$ is $a$

the constant distance from $\KK$ to $\CC$ measured along a straight line $\LL$ through $P$ is $b$.

Then:

$\KK$ has a loop

if and only if:

$a < b$

Proof

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Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): conchoid
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): conchoid