On 4/15/09, rwg@sdf.lonestar.org <rwg@sdf.lonestar.org> wrote:
... But it occurred to me that my rattleback (a solid plastic lifesize statue of a dead banana slug, with the puzzling ability to spin only counterclockwise on a flat surface) might be a segment of a torus, which takes seven points to determine.
If it really is bananoid, it's more likely a Dupin cyclide, with freedom 9. These are quite easy to test for: in Lie-sphere (hexaspherical) coordinates they are quadric reguli generated by three spheres. Incidentally, regarding the mysterious Berger cyclides with 6 systems of circles, I'm tempted to conclude that they were probably just some kind of myth-translation. With the exception of trivial special cases such as (double) planes and spheres, or (super-dense) lines and circles, every Dupin cyclide is either a torus, circular cone, or cylinder, or else some Moebius (conformal) inversion of one. In particular, a general Dupin cyclide also possesses just two Villarceau circle systems, along with its obvious two generator circle systems. WFL