# The oCLP family of triply periodic minimal surfaces

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Abstract : 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 ${k}-{\sqrt{\tau^8-A +B\tau^6+ 2 + AB\tau^4-A +B\tau^2+1}}$ where A and B are free parameters with - 2 < A, B < 2 and A > B, τ is complex with |τ|≤1 and κ 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 φ, 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.

Autor: Djurdje Cvijović Jacek Klinowski

Fuente: https://hal.archives-ouvertes.fr/

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