Hyperparallels and Horoparallels
This page requires a java-enabled browser for correct functioning. You can drag the points labelled A, B, C, P, and Q around with the mouse, and the other points will move accordingly.
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.
Java image created using Cinderella by Scott Sutherland on March 31, 2004 .