TANNERY'S PEAR

 Surface discovered by Tannery in 1892 (bulletin des sciences mathematiques, page 190). Jules Tannery (1848 - 1910): French mathematician.

 Cartesian parametrization: . Cylindrical equation: . Equation of a geodesic given by  for . Rotation of an entire "normal" figure-eight (k = 1).

Tannery's pear is the surface of revolution generated by the rotation around its axis of a half-lemniscate of Gerono subject to a scaling in one direction with ratio k (the rotation of the entire figure-eight is sometimes called Tannery's hourglass).

 It is remarkable that for , the geodesics of this surface are closed, shaped like a curved figure-eight, 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). Like for the sphere, we see that the parallels are not geodesics, except the one with maximum length.