Stereographic projection as an inversion of a sphere. A stereographic projection usually projects a sphere from a point  N{\displaystyle N} (north pole) of the sphere onto the tangent plane at the opposite point S{\displaystyle S} (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 x2+y2+z2=−z{\displaystyle x^{2}+y^{2}+z^{2}=-z} (alternately written x2+y2+(z+12)2=14{\displaystyle x^{2}+y^{2}+(z+{\tfrac {1}{2}})^{2}={\tfrac {1}{4}}}; center (0,0,−0.5){\displaystyle (0,0,-0.5)}, radius 0.5{\displaystyle 0.5}, green in the picture), then it will be mapped by the inversion at the unit sphere (red) onto the tangent plane at point S=(0,0,−1){\displaystyle S=(0,0,-1)}. The lines through the center of inversion (point N{\displaystyle N}) are mapped onto themselves. They are the projection lines of the stereographic projection.
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