trigrp1
Explore triangle groups
Usage
trigrp
Overview
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 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.
Files
Source code is in ${GEOM}/src/bin/trigrp, where e.g., GEOM = /u/gcg/ngrap.