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LEMNISCATE OF GERONO, or EIGHT
Curve studied by Grégoire de St Vincent in 1647
and Cramer in 1750.
Name given by Aubry in 1895. CamilleChristophe Gerono, 1799  1891: French mathematician. 

Cartesian parametrization:
().
Cartesian equation:
or or
also .

The lemniscate
of Gerono is a special case of besace
(see this page for a construction) and of Lissajous
curve (cf. the parametrization: ).
The lemniscate of Gerono is the antihyperbolism of a circle with respect to its centre and a tangent. 

It can also be obtained by the Newton transformation from two tangent circles, as illustrated opposite: 

Another construction, due to L. I. Magnus: while describing a circle, a point P is projected on Q on a diameter, the point Q is projected on R on (OM); the lemniscate of Gerono is the locus of the point M of [PQ] such that QM = QR. 

The equation shows that it can be obtained as a polyzomal curve, median of the parabolas and . 

As are all Lissajous curves, the lemniscate of Gerono
is the projection of two sinusoidal
crowns:
1) Projection on xOy of the pancake
curve, parametrized by: .
2) Projection on xOy of the Viviani
window: .
More generally, the lemniscate of Gerono is a view of
the horse fetter,
intersection of a sphere and a tangent cylinder.
The lemniscate of Gerono can be obtained from the lemniscate of Bernoulli in the following way: trace on the sphere with centre O and radius a the curve (Viviani's curve) the south pole stereographic projection of which is the lemniscate of Bernoulli, and project this curve perpendicularly on xOy. 

A difference between the lemniscate of Gerono and that of Bernoulli: the first one has six vertices (4 maxima of curvature and two minima) as the opposite view with its evolute shows; the lemniscate of Bernoulli only has two vertices, at both ends. 

See also:
 the Tannery pear, rotation of a
halfeight around its axis.
 the Klein
bottle, a representation of which is based on an eight, as well as
the pseudo crosscap.
If the equation is changed into , we get the red curve opposite, with polar equation , and parametrization: . Its reunion with the lemniscate of Gerono gives a curve close to the bow tie.  
If is changed into we also get a curve with the shape of an eight, also parametrized by . 

See here how
to "thicken" an eight: .
See also the biaxial inverse
of an eight.
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© Robert FERRÉOL 2017