Complex Inversion

This post explores the relationship between the Riemann sphere representation of the extended complex plane and the complex plane. The 'north pole' is shown as a blue point. The points of the sphere are shown in red. Three points represent: z, 1/conjugate(z), 1/z. The 2D slider moves point on sphere. The black points are the stereographic projections of the points on the sphere onto the(complex) plane z=0 On the sphere the first mapping corresponds to reflection through the plane z=0. The second mapping is complex conjugation and corresponds to reflection in the plane y=0. The composite of these two reflections corresponds to an 180 degree rotation around the real axis(illustrated by the brown arc, noting the green line intersections the real axis).