Essays on automorphic forms Stereographic

It is impossible to render paths on a sphere onto a flat surface in such a way that all distances remain the same. In drawing a map of a sphere, therefore, some compromises must be made. Most maps adopt one of two possible strategies—that areas are preserved or that angles are preserved. Stereographic projection is one way of making maps, and it preserves angles. It has been used since ancient times for this purpose, and its basic geometrical properties were known even then. The main results of this chapter are that: • stereographic projection takes circles to circles; • it is conformal; • it endows the sphere S2 with a complex structure