Given a conic (c) and a point P on it consider the transformation of "rotating" the points of (c) about P by a fixed angle omega, defined as follows: For a point A0 on the rotate line PA0 about P and define A1 = H(A0) to be the intersection point of (c) with the rotated line. H is a homography preserving the conic and has the following properties:
[1] Line PA1 and the tangent at A0 intersect at X on a line (e), independent of the measure of omega.
[2] Line (e) is the polar of the Fregier point FP of P, defined by the homography of turning a right angle about P (see Fregier.html ).
That H is a homography can be seen by reducing it to the product of two reflexions at two lines passing through P and forming there an angle of half omega. This kind of homography was studied in ReflexionOnConics.html .
Take A0 to be on the line which is the rotation of the tangent at P by omega. This concrete A0 is the image of P under the transformation H. All the statements follow then from the discussion in FregierPolar.html .
Notice that for Y = H(X) and X on the conic (c) the line XY is tangent to member of the family of conics generated by (c) and (e). Thus, by selecting omega to be of the form pi/n, where n a natural number n>2, we easily construct examples of polygons inscribed in (c) and circumscribing another conic (c') (see Poncelet.html ).
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