Editing Inversive geometry (section)

=== Stereographic projection as the inversion of a sphere ===
[[File:Inv-stereogr-proj.svg|250px|thumb|Stereographic projection as an inversion of a sphere]]
A [[stereographic projection]] usually projects a sphere from a point  <math>N</math> (north pole) of the sphere onto the tangent plane at the opposite point <math>S</math> (south pole). This mapping can be performed by an inversion of the sphere onto its tangent plane. If the sphere (to be projected) has the equation <math>x^2+y^2+z^2 = -z</math> (alternately written <math>x^2+y^2+(z+\tfrac{1}{2})^2=\tfrac{1}{4}</math>; center <math>(0,0,-0.5)</math>, radius <math>0.5</math>, green in the picture), then it will be mapped by the inversion at the unit sphere (red) onto the tangent plane at point <math>S=(0,0,-1)</math>. The lines through the center of inversion (point <math>N</math>) are mapped onto themselves. They are the projection lines of the stereographic projection.