Ways of representing a sphere on a plane

To see a sphere represented on a plane, we can project the solid onto the plane using a variety of ways:

Central Projection

If we have a center point E, a plane, and an object between the two, then we use the center point as our ‘eye’ (viewed as a light source). The shadow of the object on the plane is our projection.

If we want to project a sphere onto the plane, our projection will come out as a circle or an ellipse.

The drawback of this method is that the light ray will cut through the sphere at two points, so we lose half the sphere.

Gnomonic Projection

If we move the light source to the center of the sphere, then we can project each hemisphere separately and then put them together to get the projection of the whole sphere.

In this method, every great circle becomes a line, except the equator, which goes to infinity. Then great circle geometry becomes plane geometry.

Stereographic Projection

If we are given a sphere, then we can put a light source at the top of the sphere (picture the North Pole). Then each ray only intersects the sphere at one point.

If we envision our sphere as a globe, then the circles of latitude become circles with center at the South Pole, and circles of longitude become lines through the South Pole.

This form of projection is good because in the projection, angles are preserved (conformal) and all circles on the sphere transform to circles or lines. Also, the antipodal points are constructed easily.

The drawback to this method is that the points closest to the North Pole stretch far away and become distorted.

*There are some good visualizations of these projections in the very beginning of the Transformational Geometry book from 444.