Surfaces of positive curvature in \(E_ 3\) whose characteristic curves form a Tchebychef net (Q1912002)

Let \(S\) be a surface of positive Gaussian curvature with no umbilical points in a three-dimensional Euclidean space \(E^3\). Suppose that the characteristic curves of \(S\) form a Chebyshev net. Then \(S\) is a translation surface, and the Gauss image of its characteristic curves is a rhombic net. The determination of these surfaces depend on two elliptic integrals of first kind. The authors describe translation surfaces for which these two elliptic integrals are reduced to elementary integrals.