InvolutionsProduct

Involutions product

The context is that of homographies (in particular involutive) preserving a conic. Here we study the product of two involutions F(A, a) and G(B, b). In parenthesis are the center (Fregier point) of the involution and its homography axis. The property is:
The product H = G*F of the involutions is a homography with homography axis the line c = AB, which is the polar of intersection point C of lines a and b.

The proof is a simple case of Pascal's theorem. In fact, apply this to the hexagon of X₁, X₂, Y₁, Y₂, Z₁, Z₂. The intersection points are A(X₁Y₁, X₂Y₂), D(X₁Z₂,X₂Z₁), B(Y₁Z₁,Y₂Z₂). D is on line AB. Analogously E is on line AB, thus AB coincides with the homography axis of the composition G*F. Note that F*G is the inverse homography having the same homography axis.
Note that in general a homography preserving a conic can be represented as a product of at most three involutions. This follows directly from the representation of a line homography as a product of at most three involutions. To see it conjugate if necessary the line homography F with an involution I_x, so that the product F'=I_x*F fixes the point at infinity. Then this is a translation or a similarity which decomposes trivialy in a product of involutions. Everything transfers to conic homographies via good parametrizations.
See the file InvolutionsProductGeneral.html for a discussion of the product of involutive homographies of the projective plane.

Involutions product

See Also