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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*.

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.

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 2017