The stereographic projection maps all points of to the -dimensional Euclidean space except one. Let be this point (it is usually called the north pole). Then the stereographic projection is defined by
Here, is an arbitrary real number. If , the projection degenerates; in all other cases, however, is a smooth bijective mapping.
The image of a point under can be geometrically constructed as follows. Embed into as a hyperplane at . Unless , the straight line defined by and intersects with in precisely one point, . The most common values for are and , see figures 1 and 2.
Let be the map , then (a suitably restricted composition) maps all points of except the south pole smoothly and bijectively to . Together, and form an atlas of , so is an -dimensional smooth manifold.
|Date of creation||2013-03-22 15:18:35|
|Last modified on||2013-03-22 15:18:35|
|Last modified by||GrafZahl (9234)|