Non-Euclidean Constructions in

- Any line that intersects one of two parallel lines intersects the other.
- More specifically, any transversal perpendicular to one of two parallel lines is perpendicular to the other.
- Parallel lines are everywhere equidistant.

- In hyperbolic geometry, some parallel lines have no common perpendicular.
- In hyperbolic geometry, if two parallel lines have a common perpendicular, then it is unique.
- In hyperbolic geometry, parallel lines are not everywhere
equidistant. If the lines have a common perpendicular, that segment is
the shortest distance between the lines, while if the lines have no common
perpendicular, then there is no "shortest distance."

The hyperbolic models provide a means to visualize these strange results. Below are demonstration

Geometer's Sketchpadfiles for the three models, with each model providing a slightly different perspective on the issue of common perpendiculars. The demonstration(s) could be used either as an introduction to hyperbolic geometry and properties of parallel lines, or as an illustration of some of the principles.In each model, two parallel hyperbolic lines are constructed. The first is defined by points

AandB, with a perpendicular dropped from pointAto the second line.A'is defined to be the foot of that perpendicular on that second line. The length of segmentAA'is given, as is the measure of angleBAA'.In conducting a classroom demonstration, several questions for students arise:

- How do we measure the distance between a point and a line?
- How do we measure the distance between two lines? Is this a well-defined notion?
- Are parallel lines everywhere equidistant?
- What segment joining the two parallel lines gives the shortest distance?
Other natural questions also arise as a result of this demonstration that can lead to further explorations in the models. For example:

- Do all pairs of parallel lines have a shortest segment joining them? If not, can you describe a pair that don't?
- Given a point on one of two of the (divergently) given parallel lines, is there another point on that same line which is equidistant from the second line? How do we find it?
With students directing the constructions to be done, these questions can be explored in depth, and a lively discussion can take place.

**Common Perpendiculars in the Poincaré Disk Model**:

**Common Perpendiculars in the the Poincaré Half-Plane Model**:

**Common Perpendiculars in the the Klein Disk Model**:

One particular attraction of the Klein model is the ease in common perpendiculars may be constructed. In this model, one can hide and show the common perpendicular between two divergently parallel lines. Drag the free point

Aon lineABuntil segmentAA'is as short as possible, then compare segmentAA'to the common perpendicular by clicking on "Show Common Perpendicular."

The dynamic figures on this page were produced using