Hyperboloidal Mirrors

In Solution (16), when k>2 and c>0, we get the hyperboloidal mirror:

$$\frac{1}{a^{2}_{h}} \left( z - \frac{c}{2} \right)^{2} - \frac{1}{b^{2}_{h}} r^{2} \ = \ 1$$ (27)

where:

$$a_{h} = \frac{c}{2} \sqrt{\frac{k-2}{k}} \ \ \ \mathrm{and} \ \ \ b_{h} = \frac{c}{2}\sqrt{\frac{2}{k}}.$$ (28)

As seen in Figure 7, the hyperboloid also yields a realizable solution. The curvature of the mirror and the field of view both increase with k. In the other direction (in the limit $k \rightarrow 2$) the hyperboloid flattens out to the planar mirror of Section 2.3.1.

Rees [Rees, 1970] appears to have been first to use a hyperboloidal mirror with a perspective lens to achieve a large field of view camera system with a single viewpoint. Later, Yamazawa et al. [Yamazawa et al., 1993] [Yamazawa et al., 1995] also recognized that the hyperboloid is indeed a practical solution and implemented a sensor designed for autonomous navigation.

 

Figure 7: The hyperboloidal mirror satisfies the fixed viewpoint constraint when the pinhole and the viewpoint are located at the two foci of the hyperboloid. This solution does produce the desired increase in field of view. The curvature of the mirror and hence the field of view increase with k. In the limit $k \rightarrow 2$, the hyperboloid flattens to the planar mirror of Section 2.3.1.