At a math exhibition, I learned the concept of stereographic projection for the first time. However, I am curious about the purpose of the stereographcal projection. I've learned that an area of sphere is converted into infinite area, and they said all the points of sphere and complex plane are in one-to-one correspondence. But, I think, we can make a bijection even we don't use stereographic projection.(E.G. just cut the sphere and glue it on the plane)

  1. My example is reasonable? Is that bijection?
  2. Why we use stereographic projection?
  • $\begingroup$ The sphere and the (complex) plane are not homeomorphic, i.e. there is no (both way) continuous bijection between them. Stereographic projection is a bijection of the sphere minus one point to the plane. $\endgroup$ – Quang Hoang Aug 19 '14 at 15:16

The streographic projection is important because is conformal (angle-preserving). About your example: how you cut the sphere and how you glue it on the plane? Some stretching is required because plane and sphere aren't isometric.


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