Invariants of a Pair of Conics Revisited

Abstract

The invariants of a pair of quadratic forms and a pair of coplanar conics are revisited following [1, 5, 2]. For a given pair of conies, with their associated matrices C₁ and C₂, we show that Trace (C^-1₂C₁), Trace (C^-1₁C₂) and ∣C₁∣/∣C₁∣ are only invariants of associated quadratic forms, but not invariants of the conies. Two of true invariants of the conies are $\frac{{Trace(C_2^{ - 1} C_1 )}}{{(Trace(C_2^{ - 1} C_1 ))^2 }}\frac{{\left| {C_2 } \right|}}{{\left| {C_1 } \right|}}and\frac{{Trace(C_2^{ - 1} C_1 )}}{{(Trace(C_2^{ - 1} C_1 ))^2 }}\frac{{\left| {C_1 } \right|}}{{\left| {C_2 } \right|}}$. Then the true invariants of the conies are geometrically interpreted, in terms of cross ratios, through the common self-polar triangle of the two conies.