Stereographic Projection
Map the unit sphere to the plane and back; see circles become circles/lines.
About this tool
Project the unit sphere from the north pole onto the plane. A sphere point (x, y, z) maps to (u, v) = (x/(1−z), y/(1−z)); the north pole maps to infinity. The inverse map is shown by the connecting line.
The equator maps to the unit circle. Circles through the projection pole map to lines; other circles map to circles.