Affine transformations preserve the ratios of segments along straight lines (extending to midpoints of lines), but can this be extended further to say that circles will always map to circles? Not trying to formally prove, I am just curious.
2026-03-27 04:04:52.1774584292
Is circularity preserved by affine transformations?
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No. For example, scaling up by a factor of 2 in the $x$ direction will transform circles into ellipses. What you can say is that ellipses will be mapped to ellipses.
Preserving ratios of segments along the same line isn’t enough to keep circles, well, circular. The images of a circle’s diameters will still be bisected by the image of its center, but the transformation might not preserve their overall lengths.