Invariants of a Pair of Conies Revisited

The invariants of a pair of quadratic forms and a pair of coplanar conies are revisited following [1, 5, 2]. For a given pair of conies, with their associated matrices Ci and C2, we show that Trace(CjCi), Trac^C^ C2) and |Ci |/|C21 are only invariants of associated quadratic forms, but not invariants of the conies. Two of true invariants of the conies are Trace(C~Ci) |C->I . Trace |C, | ants of the conies are geometrically interpreted, in terms of cross ratios, through the common self-polar triangle of the two conies.