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: