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Ive seen rotation, translation, homothety, inversion, in geometry, and they have rules, on what is preserved, and what is not. For example, homothety preserves angles, but not lengths.

I've never seen a transformation that skews the image. It could be quite useful to transform a parallelogram into a rectangle, if skewing preserves the thing you want to prove.

Is there such a transformation, and what does it preserve?

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Perhaps you're looking for shear mappings and shear matrices? They preserve area and volume, among other things. –  Ben Jan 14 '13 at 1:30

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up vote 3 down vote accepted

The transformation $(x,y) \mapsto (x+ y \cos\alpha, y)$ "shears" a rectangle in the x-direction by an angle $\alpha$ to produce a parallelogram. It preserves area, but not angles or lengths.

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