trigrp(l) (Date: 92/08/26) trigrp(l)
NAME
trigrp - explore triangle groups
SYNOPSIS
trigrp
DESCRIPTION
Trigrp runs in conjuction with geomview to compute and
display tesselations arising from reflections in the sides
of triangles. In particular, it deals with so-called '23n'
triangles, those whose vertices have angle measurement Pi/2,
Pi/3, and Pi/n, where n is one of 3,4,5,6, or 7. The first
3 cases yield tesselations of the 2-dimensional sphere, the
case n=6 yields a tesselation of the Euclidean plane, and
the final case takes the user into the hyperbolic plane.
Trigrp is a demonstration program which shows off the
features of OOGL and of the geomview viewer while at the
same time illustrating the three fundamental 2-dimensional
geometries.
DESCRIPTION OF THE PROGRAM
Trigrp has its own graphics window where is displayed the
image of a 236, or Euclidean, triangle. There is a
distinguished point P in the interior of this triangle.
Perpendiculars are dropped from this point to the three
sides of the triangle, determining 3 quadrilaterals which
are colored three different colors. That containing the Pi/2
vertex is colored tan; that containing the Pi/3 vertex is
colored green; and that containing the Pi/n vertex is
colored purple. This pattern of 3 quadrilaterals is then
replicated in geomview as if there were mirrors along the
sides of the triangle. The tan quadrilateral is tesselated
to form another quadrilateral; the green is tesselated to
form a hexagon; and the purple forms a 2n-gon.
P can be moved around manually by the mouse by clicking
button 1, or it can be made to move automatically by
choosing 'auto' off the menu (keyboard stroke 'a'). Then
the point will move in a straight line but will bounce off
the sides of the triangle so it will stay within the figure.
The user can choose value for n off the menu or by entering
it as a keystroke. Values less than 6 yield spherical
triangles; values greater yield hyperbolic, as explained
above. Since only the 236 triangle is Euclidean, there has
to be a conversion from P as shown in the trigrp window,
into the actual curved triangle which will be tesselated in
geomview. This is done via barycentric coordinates: P is
converted into a sum aV1 + bV2 + cV3 = P, with a+b+c=1,
where V1, V2, and V3 are the three vertices of the Euclidean
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trigrp(l) (Date: 92/08/26) trigrp(l)
triangle. (a,b,c) are the barycentric coordinates of P.
Then the values of V1, V2, and V3 for the actual curved
triangle are substituted back into the expression and this
value is used for the position of P in the geometry sent to
geomview.
MENU
The menu allows the user to choose any of the 5 groups
described above. He can also use it to toggle automatic
movement of P. Finally, it is possible to print out the
barycentric coordinates of P.
FILES
Source code is in ${GEOM}/src/bin/trigrp, where e.g., GEOM =
/u/gcg/ngrap.
SEE ALSO
group(5)
AUTHORS
Charlie Gunn | The Geometry Center |
gunn@geom.umn.edu
| 1300 S. 2nd St.
| Minneapolis, MN 55454
BUGS
The program currently only runs on SGI machines.
It would be nice to consider the orientation-preserving
subgroups, too. Also other triangle groups besides these 5.
Trigrp should notify the viewer to switch between hyperbolic
and euclidean mode depending on the triangle group.
Page 2 (printed 12/22/98)
