Such homographies are studied in InvolutionsProduct.html . Here we review the practical determination of the image point H(X) = Z of such a product H = G*F. G, F are determined by two points/lines and their polars/poles, therefore the notation G(A,a), F(B,b) in which the pair denotes the pole and its polar with respect to the given conic (c). H (and H-1 = F*G too) have line c = AB as homography axis and the intersection point C of lines a and b as pole of (c).
Following the general pattern for homographies of conics we use the homography axis (c) of H and the knowledge of a particular pair (X1, Z1=H(X1)). To find the image H(X2) of a general point X2, find the intersection point D of line [X2Z1] and (c) and draw line [DX1]. Its second intersection point with the conic (other than X1) is Z2.
Although this works for general homgraphies here the direct method of finding the intermediate Y2 = F(X2) and then Z2 = G(Y2) involves the same number of calculations but simpler in that it does not involve intersections with the conic but only line intersections (of [AX2] with a and [Y2B] with b). Using these intersections Y2 and Z2 are found by corresponding harmonic-conjugate point calculations.
![[alogo]](alogo.png){kind=small}
Products of involutive homographies on conics ![[0_0]](FregierInvolutionsProduct_files/FregierInvolutionsProduct_0_0_0.png)
![[0_1]](FregierInvolutionsProduct_files/FregierInvolutionsProduct_0_0_1.png)