Plane Geometry - Bruce E. Shapiro

Section 47Perpendicular Lines inHyperbolic GeometryIn Euclidean geometry, if two lines l and m are parallel, and if for any pointP ∈ l we have d(P, m) = D then for every point Q ∈ l, d(Q, m) = D. Inother words, parallel lines are a “fixed distance” apart. Contrary to ourusual intuitive feeling for what parallelism means, this result is not true inhyperbolic geometry; in fact, there are at most two such points on l thatare the same distance from m!Theorem 47.1 If l is a line, P ∉ l is a point, and m is a line such thatP ∈ m, then there exists at most one point Q ∈ m, Q ≠ P , such thatd(Q, l) = d(P, l).Figure 47.1: In hyperbolic geometry, parallel lines are not a fixed distanceapart; there are at most two points on any given line that are the samedistance from any other (distinct) line.271