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CLAIRAUT'S CURVE
Curves studied by Clairaut in 1726.
Other name: Clairaut's multiplier curve. 
Polar equation:
with n a real number (or ).
Cartesian equation: . Algebraic curve
if and only if n is rational.

Clairaut's curves are defined by their polar equation written above.
Examples for positive values of n (part with positive ordinate):
n = 1: circle 
n = 2: double egg 
n = 3: simple folium 
n = 1/2: curve of the dipole 
n = 3/2 
n = 5/2 
n = 1/3 
n = 2/3 
n = 4/3 
Examples for negative values of n (part with positive ordinates):
n = 1: line y = a 
n = 2: Kampyle of Eudoxus 
n = 3: duplicatrix cubic 
n = 1/2: cf. Külp's quartic 
n = 3/2 
n = 5/2 
n = 1/3 : cf witch of Agnesi 
n = 2/3 : Roche's curve. 
n = 4/3 
Clairaut's curves are the glissettes
of the sinusoidal
spirals.
More precisely, if the sinusoidal spiral of parameter n slides on a line at a fixed point, the glissette of the pole is Clairaut's curve of parameter 1/n: . Thanks to the glissette/roulette equivalence theorem (see at glissettes), Clairaut's curves are also the linear roulettes of the evolutes of the sinusoidal spirals. Examples:


The orthogonal
trajectories of various Clairaut's curves of parameter n, ,
are the Clairaut's curves of parameter 1/n, .
Opposite, the cases n = 1 and 2. 
Clairaut's curve of parameter n is also the radial
curve of the Ribaucour curve
of parameter n + 1.
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© Robert FERRÉOL 2017