Hyperparallels and Horoparallels

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In hyperbolic geometry, two parallel lines which share a common perpendicular are called hyperparallel. Above, the blue lines l and m are hyperparallel; their common perpendicular is the segment PQ (light green).

You can adjust line m by sliding P along l, or moving Q closer or further away, and adjust line l by moving A or B. Slide C along line m to measure the distance at different points: the dark green segment CD is forced to be perpendicular to l at D (so QPDC is a Lambert quadrilateral). Notice that |PQ| is always the shortest distance between the two lines, and no other segment can have this length.

It is possible for lines to be parallel but not hyperparallel. Such pairs of lines are called either horoparallel (or sometimes limiting parallel) lines. At right is an example of horoparallels. It is possible to show that given a line l and a point Q not on that line, a line n which realizes the angle of parallelism for Q and l is a horoparallel; all other lines parallel to l that contain Q must be hyperparallels. (There are two such lines through Q; only one is shown at right. The other makes the same angle with PQ, but on the other side, and limits on the other "end" of AB.)

A proof of this can be found, for example, in David Royster's notes on Neutral and Non-Euclidean Geometries (See here for the full set of notes in PDF.)

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Java image created using Cinderella by Scott Sutherland on March 31, 2004 .