Last Update 05/ 08/ 2005
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Here is a cube oriented with one of its tetrad axis coincident with the North South axis of the projection sphere. At what angle shall this cube be rotated in order to have one of its triad axis coincident with the North South axis of the projection sphere?
The figure shows on the upper left corner the parallel projection of the cube with its tetrad axis parallel to the screen. On the lower left corner the parallel projection of the same polyhedron is presented with its tetrad axis perpendicular to the screen. On the right side the proper stereographic projection is shown.
By a push and drag mouse action on the green square from zero to 90o the projections may be rotated at different q angles of spherical coordinate around the four fold axis of the polyhedron. The result appears on the three projections.
A click on s will rotate the cube 45o. At this angle the upper left figure shows the intersection of the segment AB and the segment CD at O. The geometrical definitions of the main segments can be observed in table 1. With some trigonometry it will be possible to calculate the answer for the question above.
Table 1. Geometrical definitions
| Segment | Definition |
| AB | projection of the cube edge |
| CD | cube diagonal |
| BC | face diagonal/2 |
Exercise
Define the location of the axis used to rotate the cube from the original position to the final position, with a three fold axis coincident with the North South axis of the projection sphere.
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