  We further develop the formalism of arXiv:0712.0159 for approximate solution
of Nambu-Goto (NG) equations with polygon conditions in AdS backgrounds, needed
in modern studies of the string/gauge duality. Inscribed circle condition is
preserved, which leaves only one unknown function y_0(y_1,y_2) to solve for,
what considerably simplifies our presentation. The problem is to find a
delicate balance -- if not exact match -- between two different structures: NG
equation -- a non-linear deformation of Laplace equation with solutions
non-linearly deviating from holomorphic functions, -- and the boundary ring,
associated with polygons made from null segments in Minkovski space. We provide
more details about the theory of these structures and suggest an extended class
of functions to be used at the next stage of Alday-Maldacena program:
evaluation of regularized NG actions.
