Hyperbolic Geometry using Cabri
This page and links maintained
by Tim Lister, firstname.lastname@example.org
|A tessellation of the hyperbolic plane H2
(the Poincaré unit disc model) by (2, i, i) triangles, that
is, with angles (90, 0, 0). Every triangle has two vertices on the disc
boundary (at infinity). The diagram was built up from an initial triangle
with a vertex at the centre, by reflection (inversion in a disc line) about
one of its sides. A line in this model is the arc of circle orthogonal
to the disc boundary. Parts of the tessellation are shown in varying degrees
of layers in each of the quarters.
Full screen version
|During the summer of 97 I had great fun playing
with some marvelous software, Cabri
, and devising constructions for use in teaching the basic ideas of a geometry
course put on by the Open University. These started with some figures to
demonstrate the transformations of Inversive Geometry, and progressed to
figures for the Arbelos, the inversors of Peucellier and Hart, Coaxial
Circles and so on, much of which was driven by the discovery of a Dover
edition of a small pearl of a book "Advanced
Euclidean Geometry (Modern Geometry) An elementary Treatise on the Geometry
of the triangle and the Circle" (to give its full title) written
by Roger A. Johnson and first published in 1929. It had languished on my
bookshelves, having been bought years ago for 20 cents (South African)
in some sale or other.
||I can recommend it as a fascinating read, or
just for taking in the breathtaking complexity of the many hand crafted
diagrams to be found on its pages.
Compared to these mechanic and static drawings the beauty of a Cabri
figure is that you can "grab and drag" various objects in it and this movement
of the figure often gives a clear or most certainly new insight into the
underlying result or concept. The Cabri icon shown above, with its "grabbing
hand", perfectly captures the major strength of this program. It was with
this in mind that I started to construct a series of Cabri macros and an
ever growing menu for constructions for hyperbolic (or non-Euclidean) geometry
in the Poincaré disc model. The menu commands can be used to draw
figures that illustrate some of the fascinating results and figures to
be found in the hyperbolic plane.
The Cabri Java Handbook
|Download the hyperbolic geometry menu, plus notes for its use.
||The latest innovation is Cabri Java.
Fully interactive Cabri figures displayed on the page via a Java applet.
||d-lines in the Poincare disc, parallels and ultra parallels
||Tiling the disc with regular triangles
||non Euclidean distance
||Talk given at CabriWorld 2001, Montreal
A collection of figures, animated gifs and downloadable Cabri (Fig) files.
|Circumscribed Circle. A result of both Euclidean
and Hyperbolic geometry.
|Isometries of H2
|Tessellation of H2 using regular triangles.