Affine image of a circle
The affinity will map every conic onto a conic because it preserves the collinearity. Also, for the same reason, the number of intersections of a given line and a conic will be the equal to the number of intersections of their affine images. Since the affine image of the points at infinity are again points at infinity, the following can be concluded:
The affinity maps a hyperbola onto a hyperbola, a parabola onto a parabola and an ellipse onto an ellipse.
Therefore the classification in section 1.2. is called affine classification of conics.
The following properties are the consequences of fact that the affinity preserves parallelism and segment ratios:
Center, diameters and conjugate diameters of a conic are invariant under the affinity of the affinity.
Specially it follows:
Affine image of a circle is an ellipse.
The center of a circle is mapped to the center of its affine image.
Every pair of perpendicular diameters of a circle is mapped to a pair of conjugate diameters of its affine image.
An affinity (o, S_{1}, S_{2}) and circle c_{1} with the center S_{1} are given in Figure 32. Construct the affine image c_{2} of the given circle c_{1}.
Figure 32


1st step: A pair of perpendicular diameters M_{1}N_{1} and P_{1}Q_{1} of a circle is chosen.
2nd step: Line segments M_{2}N_{2} and P_{2}Q_{2} are the pair of conjugated diameters of the ellipse c_{2}.
3rd step: The ellipse c_{2} is inscribed in the parallelogram determined by the affine images of the tangents at the endpoints of the circle diameters.
4th step: The ellipse c_{2} is the affine image of the circle c_{1}.
 Move the point M_{1} and notice that the properties above do not depend on the chosen perpendicular diameters of the circle.
 Change the chosen affinity by moving the point S_{2} and notice that the properties are valid for any affinity.
 The radius of the circle can be changed by moving the point X. Observe how the affine image of a circle changes when the circle radius is changed.

Created by Sonja Gorjanc and Ema Jurkin, translated by Helena Halas and Iva Kodrnja  3DGeomTeh  Developing project of the University of Zagreb, made with GeoGebra
