Spherical Mirrors

In Solution (17), if we set c=0 and k>0, we get the spherical mirror:

$$z^{2} + r^{2} \ = \ \frac{k}{2}.$$ (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.

  

Figure 4: The spherical mirror satisfies the fixed viewpoint constraint when the pinhole lies at the center of the sphere. (Since c=0 the viewpoint also lies at the center of the sphere.) Like the conical mirror, the sphere is of little practical value because the observer can only see itself; rays of light emitted from the center of the sphere are reflected back at the surface of the sphere directly towards the center of the sphere.

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 $(r,z) = (r, \sqrt{1-r^2})$ and then pass through the pinhole. This computation is then repeated for the neighboring mirror point $(r+\mathrm{d}r, z+\mathrm{d}z)$. Next, the intersection of these two rays is computed, and finally the limit $\mathrm{d}r \rightarrow 0$ is taken while constraining $\mathrm{d}z$ by $(r+\mathrm{d}r)^2 + (z+\mathrm{d}z)^2=1$.The result of performing this derivation is that the locus of the effective viewpoint is:

$$\left( \frac{c \left[ 1 + c (1 + 2 r^2) \sqrt{1-r^2} \right]... ...1-r^2}}, \frac{2 c^2 r^2}{1 + 2 c^2 - 3 c \sqrt{1-r^2}} \right)$$ (24)

as r varies from $-\sqrt{1 - \frac{1}{c^2}}$ to $\sqrt{1 - \frac{1}{c^2}}$.The locus of the effective viewpoint is plotted for various values of c in Figure 5.

 

Figure 5: The locus of the effective viewpoint of a circular mirror of radius 1.0 (which is also shown) plotted for c = 1.1 (a), c = 1.5 (b), c = 3.0 (c), and c = 100.0 (d). For all values of c, the locus lies within the mirror and is of comparable size to the mirror.  

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.