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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 = c^{2}.
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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© Robert FERRÉOL 2017