[original to Ralph Martin; copied to math-fun] Dear Ralph, Thanks for the Mathematica session --- I still have no idea why Maple found only one pair of roots, but that's entirely within character --- for me and for Maple 9. Your engineering notation for a canonical cyclide is rather different from mine: instead of using major, minor, eccentricity parameters a,m,c and a central origin, I start from the perpendicular pair of pencils of spheres in which the cyclide is symmetric under inversion ("anallagmatic"). Considering (say) a ring cyclide, each pencil has a real axial or "limit" circle where all its spheres intersect. The circles lie in perpendicular planes (xz-plane and xy-plane), their centres [0,0,0] and [d,0,0] lying along the intersection (x-axis), at distance given by d^2 = p^2 + q^2 . [Warning: this usage differs from my earlier notation for a torus, when p,q corresponded to your a,m .] Rewritten in my reference frame (note x-translation), your nodes become [x,y,z] = [0, +/- p i, 0], [d, 0, +/- q i] . *** What this is saying is that nodes and limit circles are virtually the same thing --- it's just that a real limit circle manifests a complex node-pair, and vice-versa --- brilliant! Of course, Darboux or Maxwell or some XIX-th century mathematician probably knew all this perfectly well. As yet I haven't managed to consult for example Virgil Snyder, Annals of Math \bf 11 \rm 136--147 (1896). The nodes fix only 2 of 3 shape/size parameters. One natural choice for the other parameter is cone angle at the node, invariant under the Moebius group, but real only for spindle / horned cases (when one limit radius is imaginary). Another natural parameter is the radius of the circle in which (either) plane generator is tangent to the cyclide, invariant under the Laguerre group. Surprisingly, this is real for both ring and spindle / horned cases: explicitly in engineering parameters, its square equals a^2 - c^2 . Fred