The net result of the classification in Chapter 15 is a list of nine basic classes of conics, all of which (with the sole exception of the repeated line) involve moduli. The listing raises two natural questions, providing the material for this chapter. The first is whether any of the lists overlap: can a conic be congruent to normal forms in two different classes? As we shall see, that cannot happen, but it requires proof. Together with the zero sets, the invariants enable us to distinguish all nine classes. The net result is a simple, efficient recognition technique. Section 16.2 is in the nature of an extended example, illustrating the application of these ideas to the classical Greek construction of conics, as plane sections of a fixed cone. The second question is whether there is duplication within a class: can a conic be congruent to two normal forms within the same class? Again, that cannot happen, but it does require proof.

Distinguishing Classes

The normal forms for the nine main conic classes, their associated invariants and the cardinal of their zero sets, are listed in Table 16.1. Of the four non-degenerate classes, just one (the hyperbola) has δ < 0, just one (the parabola) has δ = 0, whilst two (the real and virtual ellipses) have δ > 0. However, by definition, the real and virtual ellipses are distinguished by their zero sets.