THE OCLP FAMILY OF TRIPLY PERIODIC MINIMAL-SURFACES

CLP surfaces with orthorhombic distortion (oCLP for short) are a family of two-parameter triply periodic embedded minimal surfaces. We show that they correspond to the Weierstrass function of the form [GRAPHICS] where A and B are free parameters with - 2 < A, B < 2 and A > B, tau is complex with \tau\ less-than-or-equal-to 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.