Invariants of a pair of conics revisited

摘要

The invariants of a pair of quadratic forms and a pair of coplanar conics are revisited following Forsyth et al.1,2and Semple and Kneebone 3. For a given pair of conics, with their associated matrices C1 and C2, we show that Trace(C−12 C1), Trace(C−11 C2) and ¦¦¦¦¦C1¦¦C2¦ are only invariants of associated quadratic forms, but not invariants of the conics. Two of true invariants of the conics are ¦¦¦¦Trace(C2−1C1)(Trace(C11C2))2¦C2¦¦C1¦ and ¦¦¦¦Trace(C11C2)(Trace(C12C1))2¦C1¦¦C2¦ Then the true invariants of the conics are geometrically interpreted, in terms of cross ratios, through the common self-polar triangle of the two conics.