J. Phys. I France
Volume 2, Numéro 2, February 1992
Page(s) 137 - 147
DOI: 10.1051/jp1:1992129
J. Phys. I France 2 (1992) 137-147

The T and CLP families of triply periodic minimal surfaces. Part 1. Derivation of parametric equations

Djurdje Cvijovic and Jacek Klinowski

Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, G.B.

(Received 24 August 1991, revised 6 October 1991, accepted 4 November 1991)

The local Weierstrass representation of all members of the T and CLP families of triply periodic minimal surfaces involves integrals of the function $R(\tau)=1/\sqrt{\tau^8+\lambda\tau^4+1}$ (with $-\infty <\lambda<-2$ for T and $-2 <\lambda<2$ for CLP), wich were previously evaluated only by numerical integration. We show that these integrals are pseudo-hyperelliptic, expresss 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.

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