Conical Mirrors

  

Figure 3: The conical mirror is a solution of the fixed viewpoint constraint equation. Since the pinhole is located at the apex of the cone, this is a degenerate solution of little practical value. If the pinhole is moved away from the apex of the cone (along the axis of the cone), the viewpoint is no longer a single point but rather lies on a circular locus. If $2 \tau$ is the angle at the apex of the cone, the radius of the circular locus of the viewpoint is $e \cdot \cos 2 \tau$, where e is the distance of the pinhole from the apex along the axis of the cone. If $\tau \gt 60^{\circ}$, the circular locus lies inside (below) the cone, if $\tau < 60^{\circ}$ the circular locus lies outside (above) the cone, and if $\tau = 60^{\circ}$ the circular locus lies on the cone.

In Solution (16), if we set c=0 and $k \geq 2$, we get a conical mirror with circular cross section:

$$z \ = \ \sqrt{\frac{k-2}{2} r^{2}}.$$ (20)

See Figure 3 for an illustration of this solution. The angle at the apex of the cone is $2 \tau$ where:

$$\tan \tau \ = \ \sqrt{\frac{2}{k-2}}.$$ (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:

$$e \cdot \cos 2 \tau.$$ (22)

If $\tau \gt 60^{\circ}$, the circular locus lies inside (below) the cone, if $\tau < 60^{\circ}$ the circular locus lies outside (above) the cone, and if $\tau = 60^{\circ}$ the circular locus lies on the cone.