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The Hyperbolic Toolbox - Background: Credit where Credit is Due

Author(s): 
Stephen Szydlik

Using dynamic geometry software such as Geometer's Sketchpad to investigate non-Euclidean geometry is by no means a new idea.  This section provides a brief description of, and pointers to the work of others in this area.

Mike Alexander (with modifications by Bill Finzer)  wrote Geometer's Sketchpad scripts to perform ten typical constructions in the Poincaré disk model:

  1. Construct a non-Euclidean line, given two points on the line.
  2. Construct a non-Euclidean line segment, given the endpoints of the segment.
  3. Measure the length of a non-Euclidean line segment.
  4. Calculate the measure of an angle.
  5. Construct the bisector of a given angle.
  6. Construct a perpendicular to a given line through a given point on the line.
  7. Construct a perpendicular to a given line through a given point not on the line.
  8. Construct the perpendicular bisector of a non-Euclidean line segment.
  9. Construct a circle, given its center and a point on the circle.
  10. Construct a circle, given its center and two points determining the radius of the circle.

These tools have been available for some time:  Alexander and Finzer presented them in a MAA session on Geometer's Sketchpad at the AMS-MAA meetings in Orlando in 1996.   Their Poincaré disk scripts may be currently found at

 http://mathforum.org/sketchpad/maa96/alexander/index.html.

A smaller subset of these construction tools is now included as part of the standard package of sample files in the current version (3.0) of Geometer's Sketchpad.

The ten constructions listed above provide a useful package of tools for the Poincaré disk.  As such, they will somewhat arbitrarily be referred to as the "standard constructions".  It is the purpose of this document to present the corresponding tools for the standard constructions in the other two well-known models of hyperbolic geometry, the Beltrami-Klein disk and Poincaré half-plane models.   Each of these constructions use the Euclidean tools of Geometer's Sketchpad, which are, in essence, computer versions of the straightedge and compass. (It should be noted, of course, that items #3 and #4 above technically are not constructions, since they require the notion of measurement.)

Although Alexander and Finzer's scripts form the standard toolbox for the Poincaré disk and provided the inspiration for this project, others have also developed non-Euclidean Sketchpad construction tools.  Dan Bennett has created scripts for the first four constructions (lines, segments, length and angle measure) for the Poincaré half-plane model, as an accompaniment to a text by Sibley.  I do not duplicate Bennett's constructions, and encourage interested users to obtain Bennett's scripts, which may be found at 

http://www.keypress.com/sketchpad/general_resources/
advanced_sketch_gallery/half_plane_model/index.php
.

Peil has also written several scripts for the half-plane model, which may be found at

http://web.mnstate.edu/peil/projects/geo.html.

Although several of this author's constructions given in the next section do appear to duplicate Peil's work, there are distinct differences. For example, the midpoint and circle constructions described here use only straightedge and compass constructions, while Peil's scripts involve coordinate geometry as well. 

The next section describes how to obtain, install, and use the hyperbolic scripts, with the goal of establishing a toolbox of the ten "standard" constructions for each of the three models.  In addition, while creating the constructions the models, it proved useful in some cases to develop additional "intermediate" tools.  For example, constructing the perpendicular bisector of a given line segment becomes much simpler once we have algorithms for constructing midpoints and raising perpendiculars. These additional constructions are included as well.  To the best of our knowledge, the Klein scripts are original. That is, though others have certainly demonstrated the same constructions, they are an original compilation in the form of  Geometer's Sketchpad scripts. The same may be said for constructions 5--10 in the Poincaré half-plane model. 

Though the main purpose of this article is to discuss constructions in hyperbolic geometry, tools for constructions in elliptic geometry have been developed as well.  Brad Findell presented tools for many standard constructions in a model of elliptic geometry at the 1996 AMS-MAA meetings.  These tools are currently available at

http://mathforum.org/sketchpad/maa96/findell/index.html.

Stephen Szydlik, "The Hyperbolic Toolbox - Background: Credit where Credit is Due," Loci (January 2005)

JOMA

Journal of Online Mathematics and its Applications

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