A Simple Viewing Geometry

Sometimes, it is handy to be able to set up a simple viewing geometry. For example, packages without matrix libraries, quick preview hacks, and simple ray tracers all can benefit from a simple viewing construction. Here is one that I have found handy.

The input is a viewpoint E, a gaze direction and distance G, and up vector U, and viewing half-angles theta and phi. The output is a screen midpoint M and two vectors, H and V, which are used to sweep the screen. Figure 1 shows the setup.

First, create vector A by A = G x U. (I assume right-handed cross products.) Then find B from B = A x G. Vector B is coplanar with U and G, but it is orthogonal to A and G. The midpoint of the screen is found from M = E += G. (Note that the length of G tells you how far away the viewscreen is located.) Vectors A and B span the viewplane, but they are the wrong size. Find the vector H by scaling A by the horizontal half-angle: H = ( A |G| tan(theta)) / |A|. Similarly, the vertical vector is found by V = ( B |G| tan(phi)/|B|.

Assuming your origin is in the lower left, as shown, then any point S on the screen may be specified by (sx, sy), both numbers between 0 and 1. (For example, if your frame buffer is 640 pixels wide y 480 pixels high, then (30, 250) would map to 30/639, 250/479)). The point P on the image plane then is P = M + (2*sx - 1)*H + (2*sy - 1)*V. If you oprefer to have your origin in the upper left, then generate points with the equation P = M + (2*sx - 1)*H + (1 - 2*sy)*V.

If you are ray tracing, the ray equation would be R = E + (P - E)*t. (You may with to normalize (P-E) before actually shooting the ray.)