I had my first exposure to the Geometer's Sketchpad at the St. Olaf
conference on using technology in mathematics courses other than calculus,
November 1994. Our geometry course at Wabash College covers hyperbolic
as well as Euclidean geometry, and so I wondered if there was a similar
interactive program for hyperbolic geometry. Of course, since the
models for hyperbolic geometry are *in* Euclidean geometry, one can
program Geometer's Sketchpad to perform hyperbolic constructions. While
this is an excellent means for students to understand how hyperbolic geometry
"sits inside" Euclidean geometry, the Euclidean constructions are too cumbersome
for all but the most elementary hyperbolic constructions, and the dynamic
aspect of Geometer's Sketchpad is significantly slowed by the numerous
dependencies among the objects of the constructions.

At the same conference, Dana Mackenzie demonstrated the program *Poincaré*,
which allows the user to construct points and lines in the upper-half plane
model of hyperbolic geometry. Another program, *Non-Euclid*, does
the same for the Poincaré disk model. Both of these programs,
however, are static---points and lines can be plotted and drawn, but once
placed, their locations are fixed.

During the summer of 1995, Wabash student Nathan Fouts and I developed
the program *PoincaréDraw*. Nathan is a physics major, mathematics
and computer science double minor, and at the time had just finished his
sophomore year.

*PoincaréDraw* allows the user to do "compass and straight
edge" constructions in the Poincaré* *disk model of the hyperbolic
plane. The program is dynamic (in the spirit of Geometer's Sketchpad):
once a construction is made, the user can move parts of it with the mouse
and the objects on the screen will move subject to the construction that
gave rise to them. For example, suppose the user plots points *A*,
*B*, and *C*, the line *L* passing through *A* and
*B*, and the line *N* passing through *C* perpendicular
to *L*. If the user moves the point *A*, then the two lines move
so that *L* continues to pass through *A* and *B*, and *N*
continues to pass through *C* and to be perpendicular to *L*.
Points *B* and *C* do not move. Some constructions unique
to the hyperbolic plane are also available, namely constructions
involving ideal points and common perpendiculars (parallel lines that do
not share an ideal point have a unique common perpendicular).

In addition to constructions, the user can also perform hyperbolic motions. Translations in the hyperbolic plane differ from those in the Euclidean plane. When the Euclidean plane undergoes a translation, all points move the same distance and each point moves along a line. In the hyperbolic plane this is impossible. If two points move along different lines at the same speed, the distance between them necessarily changes, even if the lines are parallel. Therefore a translation cannot displace every point the same distance. A hyperbolic translation has a "center," which is a line. Points on the center are all moved the same distance along the center. Points off the center are moved farther, and not along lines. Instead they are moved along curves that are equidistant from the center.

Continuous translation with the mouse allows the user to experience
holonomy, which is one of the more striking features of *PoincaréDraw*.
A continuous translation can be thought of as a composition of small translations.
In Euclidean geometry, a composition of translations is a translation.
A continuous translation that moves some point around a closed loop moves
every point around a congruent loop, and all points return to their original
positions. In hyperbolic geometry this is not true. If a continuous
translation moves the point *P* in a closed loop, all of the other
points will move farther than *P*, resulting in a net rotation, which
is called holonomy. The Gauss-Bonnet Theorem states that the holonomic
rotation is proportional to the hyperbolic area enclosed by the loop. When
the rotation is measured in radians, the constant of proportionality is
-1. The fact that this constant is negative means that the rotation
due to holonomy is in the opposite direction of the rotation of the loop.

Many of the computations in the program use a matrix-based "hyperbolic
analytic geometry" due to Mackenzie. The hyperbolic plane is represented
by the open unit disk in the complex plane. Hyperbolic points and lines
are represented by certain 2x2 matrices in the group of hyperbolic motions
and its Lie algebra. If *MA* and *MB* are matrices for the points
*A* and *B*, then [*MA*,*MB*] is the matrix for the
line *AB*, and *MA* + *MB* (suitably renormalized) is the
matrix for the midpoint of *AB*. Similarly, if *ML* and
*MN* are matrices for the lines *L* and *N*, then [*ML*,*MN*]
is the matrix for the point where *L* and *N* intersect if they
do, otherwise it is the matrix for the line that is the common perpendicular
of *L* and *N*. For details, see Mackenzie's paper.

*PoincaréDraw* is still very much in a test stage (complete
with bugs!), however it is certainly usable for class demonstrations and
for students to get a dynamic feel of hyperbolic geometry, as long
as the users are patient.

The hardware requirements consist of an IBM PC or compatible with EGA/VGA graphics, a mouse, and a math co-processor. The program does not require extended memory, so it should run on any PC that has the standard 640K of memory. It was developed on a 486, so it may be a bit sluggish on slower machines. It is not a Windows program.

*PoincaréDraw* is available from the author. Geometer's
Sketchpad is available from Key Curriculum Press. A demo version
is available from the Mathematics
Archives. *Poincaré* is available from George Parker, 1702
West Taylor, Carbondale, IL 62901. *NonEuclid* is available from the
Department of Mathematics, Rice University, and from the Mathematics
Archives.

Mackenzie, Dana, "Hyperbolic Isometries and Hexanometry,"* Expositiones
Mathematicae, *V13, pp.337-357, 1995.

Wabash College

Department of Mathematics & Computer Science,

Crawfordsville, IN 47933

footer@wabash.edu

This paper was written for the proceedings of the conference "Exploring Undergraduate Algebra & Geometry with Technology" held at DePauw University, June 1996, organized by Ellen Parker of DePauw University and Clifton Corzatt of St. Olaf College, and funded by the National Science Foundation, Division of Undergraduate Education. The same NSF grant, DUE-9554636, funded further development of

This page created 17 December 1997