The context is that of homographies (in particular involutive). Here we study the product of two involutions F(A, a) and G(B, b). In parenthesis are the center of the involution and its axis. The property is:
The composition H = GF of the involutions is a homography for which line c = AB is invariant and the intersection point C of the two axes a and b is a fixed point of H.
The statements on the invariance/constancy of c/C follow immediately from the definitions. Remark-1 The composition H induces on line (c) a homographic relation with no fixed points, except the case in which some of the axes {a, b} pass through the fixed point {B,A} of the other involution. The case in which both axes have this property is discussed in FourPoints.html . Remark-2 In the generic case there is a remarkable family of conics resulting from H, which is invariant with respect to H as well as with respect to F and G. The conics of the family result as orbits of the action of H on arbitrary points of the plane. An orbit results by taking X1 arbitrary, then X2=H(X1), X3=H(X2), ... etc.. The orbits partition the plane in mutually disjoint sets. Selecting five points from an orbit and passing a conic through them we construct conics invariant by the maps {F, G, H}. The pair of line+point (AB,C) is a polar+pole pair for every conic constructed in this way. The pairs (a,A), (b,B) are also polar+pole pairs for these conics and {F,G} are the respective conjugations with respect to these polars.
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Involutions product ![[0_0]](InvolutionsProductGeneral_files/InvolutionsProductGeneral_0_0_0.png)
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