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TANNERY'S PEAR
Surface discovered by Tannery in 1892 (bulletin
des sciences mathematiques, page 190).
Jules Tannery (1848  1910): French mathematician. 
Paramétrisation cartésienne : ().
Équation cylindrique : . Première forme quadratique fondamentale : . Équation d'une géodésique donnée par 

Cartesian parametrization:
().
Cylindrical equation: . First quadratic form : Equation of a geodesic given by . 

Tannery's pear is the surface of revolution generated
by the rotation around its axis of a halflemniscate
of Gerono subject to a scaling in one direction with ratio
(the rotation of the entire figureeight is sometimes called Tannery's
hourglass).
It is remarkable that the geodesics
of this surface are closed, shaped like a curved figureeight, except for
the meridians (the surfaces for which the geodesics are closed are called
"Zoll surfaces").
Moreover, the equation above proves that they are algebraic curves, and Tannery proved that they all have the same length (and therefore the length of a meridian and the double of the length of the parallel with maximal length that is also a geodesic).


En
1903, Otto Zoll determined asurface of which all geodesics are closed
and of same length,
other than a sphere, and that is smooth, contrary to the Tannery's
pear :
Cartesian parametrization : (). First quadratic form : . 
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© Robert FERRÉOL 2020