The Fregier involution results by turning a right angle about a point P lying on a conic. The poins X, Y, intersected on the conic by the sides of the right angle are related by an involutive homographic relation and consequently lines XY pass through a fixed point FP, the Fregier point of the involution located on the normal to the conic at P (see Fregier.html ).
Here we study the polar line (e) of FP, which is the locus of points P' of intersection of tangents at X and Y.
[1] Let Q be the intersection point of XY with the tangent (t) at P and S be the intersection point of PP' with XY. Points Q, S are harmonic conjugate to X, Y.
[2] angle(QPX) = angle(XPP').
[3] The polar (e) of FP is parallel to tangent t', symmetric of t with respect to an axis of the conic and passes through point R, defined by the intersection of the the normal at P and the tangent at X0, such that angle(QPX0) = 45 degrees.
[1] In fact, Q is contained in the polar XY of P', hence (by the duality of polarity) P' is also contained in the polar of Q. Since P is also contained in the polar of Q later coincides with line PP'.
[2] Since Q, S are harmonic conjugate to X, Y and angle(XPY) is a right one, this follows from the properties of harmonic bundles of lines (see Harmonic_Bundle.html ).
[3] When P' is at infinity the tangents at X, Y are parallel and PP' is parallel to them too. Then, by [2], angle(P''PX) = angle(XPP') and because of the parallelity of lines PP' and P''X , later is equal to angle(P''XP). This shows that t' is parallel to (e). The other statement follows directly from [2].
The remarks here apply to the homography of "rotation" about P by a fixed angle omega handled in the file RotationOnConics.html .
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Fregier polar ![[0_0]](FregierPolar_files/FregierPolar_0_0_0.png)
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