Specific Solutions of the Constraint Equation

Together, the two relations in Equations (16) and (17) represent the entire class of mirrors that satisfy the fixed viewpoint constraint. A quick glance at the form of these equations reveals that the mirror profiles form a 2-parameter (c and k) family of conic sections. Hence, the 3-D mirrors themselves are swept conic sections. However, as we shall see, although every conic section is theoretically a solution of one of the two equations, a number of them prove to be impractical and only some lead to realizable sensors. We will now describe each of the solutions in detail in the following order:

$\bullet$ Planar Solutions: Equation (16) with k=2 and c > 0.

$\bullet$ Conical Solutions: Equation (16) with $k \geq 2$ and c=0.

$\bullet$ Spherical Solutions: Equation (17) with k > 0 and c=0.

$\bullet$ Ellipsoidal Solutions: Equation (17) with k > 0 and c>0.

$\bullet$ Hyperboloidal Solutions: Equation (16) with k > 2 and c>0.

There is one conic section which we have not mentioned: the parabola. Although, the parabola is not a solution of either Equation (16) or Equation (17) for finite values of c and k, it is a solution of Equation (16) in the limit that $c \rightarrow \infty$, $k \rightarrow \infty$,and $\frac{c}{k} = h$, a constant. Under these limiting conditions, Equation (16) tends to:

$$z \ = \ \frac{h^{2} - r^{2}}{2h}.$$ (16)

As shown in [Nayar, 1997b], this limiting case corresponds to orthographic projection. Moreover, in that setting the parabola does yield a practical omnidirectional sensor with a number of advantageous properties. In this tutorial, we restrict attention to the perspective case and refer the reader to [Nayar, 1997b] for a discussion of the orthographic case. However, most of the results in this tutorial can either be extended to or applied directly to the orthographic case.