A common way of displaying a sphere (such as Earth) on a plane (such as a map)

Question:

A common way of displaying a sphere (such as Earth) on a plane (such as a map) is to use a stereo-graphic projection. Here is the two-dimensional version of the method, which maps a circle to a line. Let P be a point on the right half of a circle of radius R identified by the angle ϕ. Find the function x = F(ϕ) that gives the x-coordinate (x ≥ 0) corresponding to ϕ for 0 < ϕ ≤ π.

North Pole х