The affine invariance of the conics

In contrast to the isometric and similar transformations, which respectively preserve the congruency and shape of transformed figures, affine transformations in general do not preserve angle size or length of line segments. Under an affine transformation the following properties of a plane geometric configuration remains invariant (unchanged):

* incidence of corresponding points and lines

* collinearity of corresponding points

* parallelism of corresponding lines

* ratio into which a corresponding point divides a corresponding line

The sketch below demonstrates that a specific conic is an affine invariant. The blue conic on the left has been mapped with a general affine transformation to the green conic as its image. Drag any of the points to check for different conics that the affine image of a specific conic remains that conic. For example, the affine image of an ellipse always remains an ellipse, etc.

Please enable Java for an interactive construction (with Cinderella).

Can you explain why (prove) the result is true? If stuck, have a look at my paper at The affine invariance and line symmetries of the conics of lees my artikel in Afrikaans by Die affiene invariansie and lyn symmetriee van die kegelsnedes

Created with Cinderella

Michael de Villiers, 24 Oct 2010.