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

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

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