Inverses and n -uncial property of Jacobian elliptic functions

Abstract

The inverses of Jacobi elliptic functions possess an apparently-non-crucial property: they provide almost-everywhere-conformal maps on a hemisphere onto a torus and so, onto a parallelogram. Thus, they produce map projections on the sphere generalizing the famous quincuncial projection of Charles S. Peirce. Besides providing a general practical definition of $n$ -uncial map and proving that all the considered inverse elliptic functions are $n$ -uncial, we give operative handy formulas to calculate these maps. To the best of our knowledge, these useful formulas have not been all together published before, except for Pierce projection. We look forward to their numerical implementation. By the way, we also classify the resulting map projections according the number of singularities.