Parallel (geometry) - Extension To Non-Euclidean Geometry

Extension To Non-Euclidean Geometry

In non-Euclidean geometry it is more common to talk about geodesics than (straight) lines. A geodesic is the path that a particle follows if no force is applied to it. In non-Euclidean geometry (spherical or hyperbolic) the three Euclidean definitions are not equivalent: only the second one is useful in other non-Euclidean geometries. In general, equidistant lines are not geodesics so the equidistant definition cannot be used. In the Euclidean plane, when two geodesics (straight lines) are intersected with the same angles by a transversal geodesic (see image), every (non-parallel) geodesic intersects them with the same angles. In both the hyperbolic and spherical plane, this is not the case. For example, geodesics sharing a common perpendicular only do so at one point (hyperbolic space) or at two (antipodal) points (spherical space).

In general geometry it is useful to distinguish the three definitions above as three different types of lines, respectively equidistant lines, parallel geodesics and geodesics sharing a common perpendicular.

While in Euclidean geometry two geodesics can either intersect or be parallel, in general and in hyperbolic space in particular there are three possibilities. Two geodesics can be either:

1. intersecting: they intersect in a common point in the plane
2. parallel: they do not intersect in the plane, but do in the limit to infinity
3. ultra parallel: they do not even intersect in the limit to infinity

In the literature ultra parallel geodesics are often called parallel. Geodesics intersecting at infinity are then called limit geodesics.