F is called the Fregier point of the involutive homography. The polar line L of F with respect to the conic is called the homography axis of the involutive homography.

Another aspect is through the use of projective coordinates along the conic and more precisely of a

The obvious definition is to associate to each point X of c' the other intersection point X' of XF with c'. If D is the intersection point of FX with e then, by the definition of the polar, the four points will make a harmonic division: (X,X',F,D) = -1. Projectivities of this kind are called

Since their definition depends only on F and line e, we expect to find many families of conics, like {c'} invariant under this projectivity. Indeed, every conic c'' having (F, e) as pole-polar respectively, generates together with e a family of conics invariant under this projectivity (see Harmonic_Perspectivity.html for a discussion).

[2] When e passes through the center, F being then at infinity the involution being then a

[3] The so-called

Once shown that this map is indeed an involution (see Fregier.html ), one can easily show that the Fregier point F of the involution is on the normal at W. Indeed the involution interchanges points {W,W'}, where W' the other intersection point with the normal at W. Take also the bisectors {WA, WA'} of the right angle at W, {A,A'} being their other intersection points with the conic. F coincides with the intersection point of lines {AA', WW'}.

Fregier.html

Fregier_Involutive.html

GoodParametrization.html

Harmonic_Perspectivity.html

HomographyAxis.html

InversionAsInvolution.html

Polar2.html

Produced with EucliDraw© |