Here we study the polar line (e) of F

[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 F

[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 .

Harmonic_Bundle.html

RotationOnConics.html

ReflexionOnConics.html

InvolutiveHomography.html

Harmonic_Perspectivity.html

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