The T and CLP families of triply periodic minimal surfaces. Part 2. The properties and computation of T surfaces

(Received 7 May 1992, accepted in final form 4 August 1992)

Abstract
The geometry of a tD minimal surface depends on the ratio c/a of tetragonal axes, and can be fully described in terms of a single free parameter. We offer a choice of three such parameters, all related to surface geometry, and derive analytical expressions for their relationships to the axes ratio and the normalization factor. The latter is crucial for the matching of specific surfaces to real structures. Parametric equations for normalized tD surfaces make it possible, for the first time, to find the surface corresponding to any given value of the c/a ratio, and to compare it with actual structural data. Straightforward physical applications of tD surfaces are therefore possible. We discuss the geometric consequences of this result and show that the z coordinate of any tD surface can be approximated using elementary functions. A rational approximation for the relationship between the free parameter and the c/a ratio is also found. We list exact coordinates of tD surfaces corresponding to several prescribed values of the c/a ratio.