Numéro |
J. Phys. I France
Volume 3, Numéro 4, April 1993
|
|
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Page(s) | 909 - 924 | |
DOI | https://doi.org/10.1051/jp1:1993172 |
J. Phys. I France 3 (1993) 909-924
The oCLP family of triply periodic minimal surfaces
Djurdje Cvijovic and Jacek KlinowskiDepartment of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, U.K.
(Received 12 October 1992, accepted in final form 20 December 1992)
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
where
A and
B are free parameters with
- 2 < A,
B < 2 and
A > B,
is complex with
and
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
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.
01.55
© Les Editions de Physique 1993