This defines a map f of the plane (with the exception of O and its polar) into itself. When {c, O} is correspondingly a circle and its center the map f coincides with the classical inversion.

In particular f is involutive (f

This is similar to the behavior of the inversion which maps every point at infinity to the center of the inversion-circle.

In order to construct such an homography, construct a triangle ABC inscribed in the conic and having point O as its pivot (see TriangleConics.html and TrianglesGivenPivot.html ).

Given the conic and the point O, there is for each point A on the conic a unique triangle ABC having this property. Take then an equilateral inscribed in the circle and define the homography g by the properties:

1) g maps the conic c to the circle c

2) g maps the three vertices of ABC to the vertices of the equilateral.

As a consequence, g maps then point O to the center of the circle.

Further g*f*g

The picture shows a case of such a generalized inversion and a circle orthogonal to the circle c

TrianglesGivenPivot.html

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