Conchal

CONCHAL

Curve studied by Schlömilch in 1878 and G. Huber in 1895.
From the Latin Cochlea, itself coming from the Greek kokhlias: shell, snail (cf. the cochlea in the internal ear and the cuiller, instrument used to eat snails).
K. Fladt p. 236.

Cartesian equation: , i.e. .
Circular quartic, rational if c = a.

The conchals are the loci of the points for which the product of their distances to a fixed point (here (F(–a, 0)) and to a fixed line (here x = a) is constant: MF MH = c².

The curves assume the following aspects:

When 0 < c < a, we get two infinite branches for which the line x = –a is the asymptote when and , and an oval for When c = a (O is a double point, the curve is rational): we get two infinite branches for which x = –a is the asymptote, the second having a loop, when and , For c > a, we get two infinite branches for which the line x = –a is the asymptote, when and

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When 0 < c < a, we get two infinite branches for which the line x = –a is the asymptote when and , and an oval for	When c = a (O is a double point, the curve is rational): we get two infinite branches for which x = –a is the asymptote, the second having a loop, when and ,	For c > a, we get two infinite branches for which the line x = –a is the asymptote, when and