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### Conical Mirrors

In Solution (16), if we set c=0 and , we get a conical mirror with circular cross section:
 (20)
See Figure 3 for an illustration of this solution. The angle at the apex of the cone is where:
 (21)
This might seem like a reasonable solution, but since c=0 the pinhole of the camera must be at the apex of the cone. This implies that the only rays of light entering the pinhole from the mirror are the ones which graze the cone and so do not originate from (finite extent) objects in the world (see Figure 3.) Hence, the cone with the pinhole at the vertex is a degenerate solution of no practical value.

The cone has been used for wide-angle imaging a number of times [Yagi and Kawato, 1990] [Yagi and Yachida, 1991] [Bogner, 1995]. In these implementations the pinhole is placed quite some distance from the apex of the cone. It is easy to show that in such cases the viewpoint is no longer a single point [Nalwa, 1996]. If the pinhole lies on the axis of the cone at a distance e from the apex of the cone, the locus of the effective viewpoint is a circle. The radius of the circle is easily seen to be:
 (22)
If , the circular locus lies inside (below) the cone, if the circular locus lies outside (above) the cone, and if the circular locus lies on the cone.

Next: Spherical Mirrors Up: Specific Solutions of the Previous: Planar Mirrors
Simon Baker
1/22/1998