Bi-cyclide and flat-ring cyclide coordinate surfaces: correction of two expressions

Abstract

Bi-cyclide and flat-ring cyclide coordinates are three-dimensional rotational coordinate systems based on conformal transformations using the Jacobian elliptic function sn. We have checked the previously published formulae of these systems (P. Moon and D. E. Spencer. Field Theory Handbook, Springer-Verlag, Berlin, 1971). In both cases the expression for the rotation-cyclide surfaces was incorrect: thus we present rederivations. The expressions were verified with the symbolic-algebraic computation package MACSYMA.