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 remain 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 dynamic Cinderella sketch above 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, while that of a hyperbola or a parabola respectively remain hyperbola or parabola.
Challenge
Can you explain why (prove) the result is true? If stuck, have a look at my 1993 paper in the Australian Senior Mathematics Journal at The affine invariance and line symmetries of the conics. Of lees my 1992 artikel in Afrikaans in Pythagoras by Die affiene invariansie and lyn symmetriee van die kegelsnedes
Related Links
All cubic polynomials are affine equivalent
An area preserving transformation: shearing
All parabola are similar - i.e. have the same shape
Some Transformations of Graphs
Three Circle Geometry Theorem Proofs by Transformations
Miscellaneous Dynamic Transformations (of Geometric Figures & Graphs)
International Mathematical Talent Search (IMTS) Problem Generalized
External Links
Cubic function at Wikipedia.
Affine transformation at Wikipedia.
Affine Transformation by Robert Fisher, Simon Perkins, Ashley Walker, Erik Wolfart.
Created with Cinderella
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Created by Michael de Villiers, 24 Oct 2010; updated 29 Jan 2024.