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### Spherical Mirrors

In Solution (17), if we set c=0 and k>0, we get the spherical mirror:
 (23)
Like the cone, this is a solution with little practical value. Since the viewpoint and pinhole coincide at the center of the sphere, the observer sees itself and nothing else, as is illustrated in Figure 4.

The sphere has also been used to enhance the field of view several times [Hong, 1991] [Bogner, 1995] [Murphy, 1995]. In these implementations, the pinhole is placed outside the sphere and so there is no single effective viewpoint. The locus of the effective viewpoint can be computed in a straightforward manner using a symbolic mathematics package. Without loss of generality, suppose that the radius of the mirror is 1.0. The first step is to compute the direction of the ray of light which would be reflected at the mirror point and then pass through the pinhole. This computation is then repeated for the neighboring mirror point . Next, the intersection of these two rays is computed, and finally the limit is taken while constraining by .The result of performing this derivation is that the locus of the effective viewpoint is:
 (24)
as r varies from to .The locus of the effective viewpoint is plotted for various values of c in Figure 5.

As can be seen, for all values of c the locus lies within the mirror and is of comparable size to it. Spheres have also been used in stereo applications [Nayar, 1988] [Nene and Nayar, 1998], but as described before, multiple viewpoints are a requirement for stereo.

Next: Ellipsoidal Mirrors Up: Specific Solutions of the Previous: Conical Mirrors
Simon Baker
1/22/1998