Project a sphere onto a plane from the north pole: each point P on the sphere maps to where the line (north pole → P) intersects the equatorial plane. Circles on the sphere map to circles on the plane — drag to rotate the sphere.
Sphere (drag to rotate)
Stereographic plane projection
Stereographic projection is conformal (angle-preserving) and circle-preserving: every circle/line on the sphere maps to a circle/line on the plane. Ptolemy used it for star maps (~150 CE). The Riemann sphere identifies the complex plane ℂ with the sphere via z ↦ (2Re(z), 2Im(z), |z|²−1)/(|z|²+1) — the north pole corresponds to z=∞.