In the affine plane the circle with the centre in the origin and radius can be represented by the point function
(1) |
Affine image of the circle with centre O under the general axial affinity determined by the axis o, directions s and a pair of corresponding points is an ellipse with centre O'.
Figure 1: Affine image of circle |
Pairs of lines correspending in axial affinity meet in the invariant points on the axis of affinity. A square subscribed to the circle k is mapped into the parallelogram subscribed to the corresponding ellipse e.
Lemma 1. Any pair of perpendicular diameters of the circle that is mapped under the affine mapping into an ellipse is mapped into the pair of conjugate diameters of the respective ellipse.
Lemma 2. Superposition of the line and the circle is invariant under the affine mappings, in which the circle is mapped to the ellipse in the same position to the image of the line as the original circle.
Corollary 1. Tangent lines to the circle are mapped to the tangent lines of the ellipse that is the affine image of the circle under the respective affine mapping.
Corollary 2. Intersection points of the line and the circle are mapped to the intersection points of the affine image of the line and the ellipse that is the image of the circle under the respective affine mapping.
The last two corollaries can be applied in the construction of a tangent line to the ellipse in a given direction or through a given point, and to the construction of intersection points of the line and the ellipse.