Is it possible to combine further the last two of the three equations in the link given above into a single ODE / PDE to obtain constant geodesic curvature = 1/b?
Aim is to get an equation for u = constant "circles" or loops on surface of constant Gauss curvature K = 1/a^2, ( b not equal to a ) in geodesic polar coordinates in general.
Or, is the last equation alone adequate to define the boundary for all constant K surfaces?
I attempted to numerically find a solution in a procedure that computes constant geodesic curvature in case of constant K surfaces of revolution, positive and negative K, am attaching what might serve as a representative image:
( Found an error after uploading: T is actually the radius of curvature = 1/kg).
" J-circle " is nomenclature from an old Russian book that I cannot readily find again.
However, I look for a general formulation that removes restriction on surfaces of revolution.
I also like to know if the line kg= const on K = const surface is always a closed loop/ oval.