THE T-FAMILY AND CLP-FAMILY OF TRIPLY PERIODIC MINIMAL-SURFACES .1. DERIVATION OF PARAMETRIC EQUATIONS

The local Weierstrass representation of all members of the T and CLP families of triply periodic minimal surfaces involves integrals of the function R (tau) = 1/square-root tau-8 + lambda-tau-4 + 1 (with - infinity < lambda < -2 for T and -2 < lambda < 2 for CLP), which were previously evaluated only by numerical integration. We show that these integrals are pseudo-hyperelliptic, express them analytically in terms of the incomplete elliptic integral of the first kind, F (phi, k), and give explicit general parametric equations for coordinates of these minimal surfaces. The procedure completely obviates the need for numerical integration. The solutions for all three coordinates are intrinsically periodic. The well-known properties of elliptic integrals and their inverse functions provide new insights into the features of triply periodic minimal surfaces, and permit their systematic evaluation.