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:

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

**Conical Solutions:**
Equation (16) with and *c*=0.

**Spherical Solutions:**
Equation (17) with *k* > 0 and *c*=0.

**Ellipsoidal Solutions:**
Equation (17) with *k* > 0 and *c*>0.

**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 , ,and , a constant. Under these limiting conditions,
Equation (16) tends to:

(16) |