Cross-ratio of Conics on a pencil of conics

Let $C_1, C_2, C_3$ be the three split conics in a simple pencil $L$ of conics on $\mathbb{P}_2$, with $a,b,c,d$ as its four base points. For an arbitrary conic $C \in L$, how can I compute the cross-ratio $[C_1, C_2, C_3, C]$ on $L$. And how will that compare to the cross-ratio $[a,b,c,d]$ on $C$?

Solution 1:

The paper Thomas, Geometric Characterizations Of The Cross Ratio In A Pencil Of Conics talks about this. There are several ways to compute a cross ratio of conics, but I picked out this one (pg 11):

Corollary 1. The cross ratio of four conics in a pencil is given by the cross ratio of the four tangents at a base point.

If conics $C_1,C_2,C_3$ are the degenerate conics of the pencil, I found that

$$ [C_1, C_2, C_3, C] = [a,b,c,d]_C $$

This notion of taking cross ratios in a pencil of conics goes back at least to the 19th century. E.g. see Chasles, Sections Coniques, 1865, Art 325, where the cross ratio ("rapport anharmonique") of the four polars of a point is used. (See also Milne, Cross Ratio Geometry, Art 220.)

The general expression for a pencil of entities (points, lines, conics, whatever) is $A+\lambda B$, where $A,B$ are members of the pencil.

The cross ratio of $A+kB,A+lB,A+mB,A+nB$ is $\dfrac{(n-l)(m-k)}{(n-m)(l-k)}$. For details see Salmon, Conic Sections, Art. 58.