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, S1, S2) and circle c1 with the center S1 are given in Figure 32. Construct the affine image c2 of the given circle c1.

    Figure 32

    1st step: A pair of perpendicular diameters M1N1 and P1Q1 of a circle is chosen.

    2nd step: Line segments M2N2 and P2Q2 are the pair of conjugated diameters of the ellipse c2.

    3rd step: The ellipse c2 is inscribed in the parallelogram determined by the affine images of the tangents at the endpoints of the circle diameters.

    4th step: The ellipse c2 is the affine image of the circle c1.

    - Move the point M1 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 S2 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