Hyperparallels and HoroparallelsThis page requires a javaenabled 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 .