Stereographic projection

There is no isometry between the sphere and the plane; any isometry must preserve the Gauss curvature, but the unit sphere has Gauss curvature one, while the plane has Gauss curvature zero. Stereographic projection is a conformal map between the sphere and the plane, meaning it may stretch and shrink lengths but preserves angles. Mathematically speaking, $𝜑$ is conformal if ${𝜑}^{\ast }\widetilde{𝑔}={\text{e}}^{2𝑓}𝑔$ for some smooth function $𝑓$.

In this picture, stereographic projection is visualized by placing a point light source at the north pole of a spherical wireframe. It can be seen visually that the sphere is made from right-angled shapes, and their shadow form squares on the plane.