A Chebychev net obeying Sine-Gordon equation spreads out on a surface of constant negative Gauss Curvature $K$  so that the asymptotic differential rhombic element corners lie on lines of  principal  curvature. 

[![HyperbolicGeodesicParallels_DifferentialRhombus][1]][1]

Show that principal rotation of surface normals across diagonals of rhombus are related as:

$$ d \phi_1^2 + d \phi_2^2 = - K ds^2 $$ 


  [1]: https://i.sstatic.net/Am2ez.png