J. Phys. I France
Volume 3, Numéro 4, April 1993
Page(s) 909 - 924
DOI: 10.1051/jp1:1993172
J. Phys. I France 3 (1993) 909-924

The oCLP family of triply periodic minimal surfaces

Djurdje Cvijovic and Jacek Klinowski

Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, U.K.

(Received 12 October 1992, accepted in final form 20 December 1992)

CLP surfaces with orthorhombic distortion (oCLP for short) are a family of two-parameter triply periodic embedded minimal surfaces. We show that they correspnd to the Weierstrass function of the form $\frac{k}{\sqrt{\tau^8-(A +B)\tau^6+ (2 + AB)\tau^4-(A +B)\tau^2+1}}$ where A and B are free parameters with - 2 < A, B < 2 and A > B, $\tau$ is complex with $\mid\tau\mid\leqslant 1$ and $\kappa$ real and depends on A and B. When B = - A, the oCLP family reduces to the one-parameter CLP family with tetragonal symmetry. The Enneper-Weierstrass representation of oCLP surfaces involves pseudo-hyperelliptic integrals which can be reduced to elliptic integrals. We derive parametric equations for oCLP surfaces in terms of incomplete elliptic integrals $F (\phi, k)$ alone. These equations completely avoid integration of the Weierstrass function, thus making the use of the Enneper-Weierstrass representation unnecessary in the computation of specific oCLP surfaces. We derive analytical expressions for the normalization factor and the edge-to-length ratios in terms of the free parameters. This solves the problem of finding the oCLP saddle surface inscribed in given a right tetragonal prism, crucial for the modelling of structural data using a specific surface, and enables straightforward physical applications. We have computed exactly the coordinates of oCLP surfaces corresponding to several prescribed values of the edge-to-length ratio.


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