Writing Mathlets II: A Call to Math Professionals - The Puzzle

The applet shown here displays a surface in transparent wireframe form. The surface is a hyperbolic paraboloid, given by the equation z = (x²  - y²)/8, and with axis bounds -10 to 10 for all three axes. The graph can be rotated in the applet by clicking and dragging the mouse on the graph.

Note the effects of left-right and up-down mouse-dragging motions on the graph, and how the rotations affect the coordinate axes. In particular, while there seems to be much freedom in the motions of the x- and y-axes, the motions of the z-axis are much more constrained: It can be "rolled" forward or backward (relative to the viewer), but it cannot be rotated clockwise or counterclockwise. Why not? This question is the puzzle that motivates this article.

The answer comes from the mathematics needed to establish reasonable three-dimensional graphics, and in particular the projection of three dimensions to two (for the purpose of displaying on the computer screen). Certain compromises must be made to achieve natural-seeming mouse click-and-drag rotations in the applet.