Hyperbolic Geometry - Non-intersecting Lines

Non-intersecting Lines

An interesting property of hyperbolic geometry follows from the occurrence of more than one line parallel to R through a point P, not on R: there are two classes of non-intersecting lines. Let B be the point on R such that the line PB is perpendicular to R. Consider the line x through P such that x does not intersect R, and the angle θ between PB and x counterclockwise from PB is as small as possible; i.e., any smaller angle will force the line to intersect R. This is called an asymptotic line in hyperbolic geometry. Symmetrically, the line y that forms the same angle θ between PB and itself but clockwise from PB will also be asymptotic. x and y are the only two lines asymptotic to R through P. All other lines through P not intersecting R, with angles greater than θ with PB, are called ultraparallel (or disjointly parallel) to R. Notice that since there are an infinite number of possible angles between θ and 90°, and each one will determine two lines through P and disjointly parallel to R, there exist an infinite number of ultraparallel lines.

Thus we have this modified form of the parallel postulate: In hyperbolic geometry, given any line R, and point P not on R, there are exactly two lines through P which are asymptotic to R, and infinitely many lines through P ultraparallel to R.

The differences between these types of lines can also be looked at in the following way: the distance between asymptotic lines shrinks toward zero in one direction and grows without bound in the other; the distance between ultraparallel lines (eventually) increases in both directions. The ultraparallel theorem states that there is a unique line in the hyperbolic plane that is perpendicular to each of a given pair of ultraparallel lines.

In Euclidean geometry, the "angle of parallelism" is a constant; that is, any distance between parallel lines yields an angle of parallelism equal to 90°. In hyperbolic geometry, the angle of parallelism varies with the Π(p) function. This function, described by Nikolai Ivanovich Lobachevsky, produces a unique angle of parallelism for each distance p = . As the distance gets shorter, Π(p) approaches 90°, whereas with increasing distance Π(p) approaches 0°. Thus, as distances get smaller, the hyperbolic plane behaves more and more like Euclidean geometry. Indeed, on small scales compared to, where K is the (constant) Gaussian curvature of the plane, an observer would have a hard time determining whether the environment is Euclidean or hyperbolic.