next curve  previous curve  2D curves  3D curves  surfaces  fractals  polyhedra 
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. .
Circularquartic, 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 



next curve  previous curve  2D curves  3D curves  surfaces  fractals  polyhedra 
© Robert FERRÉOL 2017