InversionGeneralization

1. Generalized inversion

Consider a conic c and a point O, not lying on the conic. For each point X different from O and not lying on the polar of O, let {C₁, C₂} be the intersection points of line OX with c. Let also Y = f(X) be the harmonic conjugate of X with respect to {C₁, C₂}.
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 (f² = 1) leaves invariant all lines through O and also leaves the conic c pointwise fixed.
Remark f is singular on the polar p_O of O, since it maps every point on this line to O.
This is similar to the behavior of the inversion which maps every point at infinity to the center of the inversion-circle.

2. Reduction to ordinary inversion

More properties result by considering the reduction of this general case to the particular case of the inversion with respect to a circle. The reduction is done via a homography mapping the conic c to a circle c₀ and the center O to the center Q of c₀.
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 c₀.
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^-1 = f' is the usual inversion with respect to the circle c₀.

3. Further consequences

Since the inversion permutes the concentric to c₀ circles, and g maps the bitangent bundle defined by the conic c and the polar of O to this family of circles, it follows that the generalized inversion permutes the members of this bitangent family of conics.Further, any conic which is invariant under f, maps via g to a conic which is invariant under theordinary inversion. But this is true only for the circles that are orthogonal to c₀. Thus the set of all circles orthogonal to c₀ maps to conics which are invariant under f.Another important property of inversion is the transformation of lines to circles through the center of inversion. This via g corresponds to conics passing through the center O of the generalized inversion.

The picture shows a case of such a generalized inversion and a circle orthogonal to the circle c₀, which maps to a conic invariant under the generalized inversion w.r. to O.The picture shows further the image d' under g of the radical axis d of the two circles.d' intersects the polar p_O of O at a point F on the tangent of the conic at O. The conic k is invariant under the harmonic perspectivity with center at F and axis the polar p_F of F with respect to the conic of reference.

1. Generalized inversion

2. Reduction to ordinary inversion

3. Further consequences

See Also