Trapezoidal discrete surfaces: geometry and integrability

Surfaces on which the lines of curvature form geodesics and parallels are d iscretized in a purely geometric manner. Discrete principal curvatures are defined and it is shown that the natural discrete Gauss equation is given b y the standard discrete Schrodinger equation with the discrete Gaussian cur vature as its potential. The subclass of discrete surfaces of revolution is considered and used to establish algebraic and geometric properties which are reminiscent of those known in the continuous case. Important connection s with integrable discrete equations are also recorded. (C) 1999 Elsevier S cience B.V. All rights reserved.