Lotusland of Mathematical Dreams

This is an alternate representaion for Hyperbolic geometry. Wikipedia says that this done by projecting a point on a sphere onto a plane tangential to the sphere at the point antinodal to the center of projection (that is the point diametrically opposite to the center of projection).

With this projection, any circle that crosses through the center of projection becomes a straight line.Any other circle (that does not touch center of projection), can become circles (possibly ellipses in the case of inclined circles).

For an example, think that we are projecting Earth. The center of projection is absolute North Pole. The plane of projection is parallel to the equitorial circle. Now consider the projection of a longitude. It will be a straight line.

Consider the case of a latitude. This becomes a circle. An extreme case will be the equator. It will be a bounding circle. It is called primitive of the projection. Any other latitude becomes a circle concentric to the primitive and inside it. Infact those latitudes in southern hemisphere will have projection with radius greater than the primitive (Still haven’t figured out how).

A beautiful description is also available from International Union of Crystallography. Stereography is used in crystallography it explains.