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DEVIL'S CURVE

Curve studied by Cramer in 1750 and Lacroix in 1857.
The name of this curve seems to come from the fact that it represents the section of a diabolo (= devil, etymologically), or that drawing it was considered very difficult compared to the simplicity of its Cartesian equation.
Other name: Cramer curve.

 
Polar equation: .
Cartesian equation: , i.e. .
Quartic of genus 2.

The devil's curves are the curves given by the equations above.

They are composed of two infinite branches and an eight that only appears when b > a.

The two infinite branches can be constructed geometrically from the rectangular hyperbola  (in green on the figure) as follows: given a point H describing the hyperbola, draw a triangle OHN with a right angle at H with HN = b; the two branches are the loci of the points M on (OH) for which OM = ON.

right angle at H, HN = constant = b, OM = ON.

The central eight can be constructed geometrically from the same rectangular hyperbola as the locus of a point M on a right triangle at O OHM with hypotenuse of constant length (= b), when the point H describes the hyperbola.

right angle at O, HM = constant = b.
An equivalent construction consists in saying that the eight of the devil's curve is the locus of the centre of the circles with constant radius (= b) passing by the ends of the diameters (i.e. two points symmetrical about O) of a rectangular hyperbola.

When b = 0, we get, of course, the limit case of the hyperbola.

The interest of the devil's curve seems mostly to be that is the simplest curve with genus 2.

Here is how Cramer delightfully describes this curve:
"One would get the entire curve formed of an eight-figure that ties itself at the origin, and two other separated parts which, after having wound from left to right, but at some distance from the eight-figure, throw four branches at infinity, one in each of the four coordinate angles."
 
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© Robert FERRÉOL, Jacques MANDONNET 2017