Inverse of Parameterization of Klein Bottle

Question

On wikipedia i found a parameterization of the immersion of the Klein Bottle into 3-d space (see Image).

Does anyone know how to compute its inverse? Given coordinates $(x,y,z),$ I would like to compute parameters $(u,v).$

Answer 1

Actually the inverse does not exist, because the immersion has self-intersections. I would be very surprised if you could do it in closed form (in the cases where a unique solution inverse does exist). Of course numerical methods for solving equations could be used.