I'm curious about Van der Grinten IV: finding the longitude was trivial, but the latitude eludes me. Do you know of a closed-form inverse for the latitude? I have a feeling there is no such thing, as the formula for latitude is fairly complicated.
P.S. I'm pretty sure both the American polyconic and the Rectangular polyconic have no closed-form inverse. But I would love to be corrected.
OK, I've implemented the latitude via Newton–Raphson, and it converges very rapidly so I'm happy. I'm still interested in knowing whether a closed-form exists (or if not, whether it's easy to show why).
Just as well you were able to implement an iterative result. I have no record of anyone publishing an inverse. Van der Grinten IV is quite uncommon. American polyconic also has no closed-form inverse; Snyder gives an iteration in Map Projections—A Working Manual. Nor have I seen any inverse published for rectangular polyconic.
