## Example 6.4:  Common Perpendiculars

In Euclidean geometry, parallel lines (i.e. lines that do not intersect) have certain "obvious" properties:
• 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.
These properties lead us to naturally think of parallel lines as looking like "railroad tracks."  However, each of  the properties just listed is logically equivalent to Euclid's fifth postulate.  In other words, none of the properties holds  in hyperbolic geometry.  In hyperbolic geometry, parallel lines are very different objects than in Euclidean geometry!  Overcoming Euclidean preconceptions about parallel lines can be a challenge for beginning geometry students.  In particular, many of the abstract hyperbolic theorems appear counterintuitive:
• 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."

Classroom Demonstration Example:

The hyperbolic models provide a means to visualize these strange results.  Below are demonstration Geometer's Sketchpad files 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 A and B, with a perpendicular dropped from point A to the second line. A' is defined to be the foot of that perpendicular on that second line.  The length of segment AA' is given, as is the measure of angle BAA'.

In conducting a classroom demonstration, several questions for students arise:

1. How do we measure the distance between a point and a line?
2. How do we measure the distance between two lines?  Is this a well-defined notion?
3. Are parallel lines everywhere equidistant?
4. 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:

1. Do all pairs of parallel lines have a shortest segment joining them?  If not, can you describe a pair that don't?
2. 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 A on line AB until segment AA' is as short as possible, then compare segment AA' to the common perpendicular by clicking on "Show Common Perpendicular."