stereographic projection
Stereographic projection is the precise recipe for flattening a sphere onto a plane (or, run backwards, wrapping a plane onto a sphere). Imagine a sphere resting on a flat sheet of paper and a single light source at the very top — the north pole. The shadow each point of the sphere casts onto the paper is its stereographic image. Concretely, to project a sphere point P (other than the north pole) onto the plane, draw the straight line from the north pole through P and see where it crosses the plane; that crossing point is P's shadow.
It sets up a perfect one-to-one correspondence between the plane and the sphere with the north pole removed. Points near the south pole land near the origin; points near the equator land near the unit circle; and points creeping up toward the north pole shoot off toward infinity. That last fact is exactly why the north pole is the natural home for the point at infinity: it is the one sphere point with no finite shadow, so we assign it to infinity and complete the correspondence.
Stereographic projection has a famous and useful property: it is conformal — it preserves angles. Two curves crossing at some angle on the sphere project to two curves crossing at the very same angle in the plane. (It does NOT preserve areas or distances — regions near the north pole get wildly stretched, which is why this projection distorts world maps so dramatically near the poles.) Its angle-preserving nature is why it meshes so well with conformal mapping, and its other gem is that it sends every circle on the sphere to a circle-or-line in the plane — tying directly into the unified line-and-circle family.
A circle of latitude on the sphere (a horizontal slice) projects to a circle in the plane centered at the origin; the equator goes to the unit circle, and a small circle hugging the north pole goes to a huge circle far out in the plane.
Circles map to circles (or lines), and angles are preserved — but areas are badly distorted near the pole.
Conformal does not mean distance- or area-preserving. Stereographic projection preserves angles only; the brutal area distortion near the north pole is unavoidable, which is why it makes a poor equal-area world map.