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Why can any affine transformaton be constructed from a sequence of rotations, translations, and scalings?
Assuming that I have a set of points in a co-ordinate system (I know their coordinates), then I use a combination of transformations (Rotation, Scaling, Shearing and Translation) to get it to a new system (where again, I know the new co-ordinates), How do I find out the values of shearing, rotation, scaling and translation? Any method other than Iwasawa?
What I have tried:
The only thing I've realized is that if I have set of old coordinates and a set of new co-ordinates,
$$[Old] \times Transformation Matrix = [New]$$ $$Transformation Matrix = [Old] ^{-1}[New]$$ This gives me the cumulative Transformation matrix, how do I break it down to tell me what the shear, rotation, scale and translation was?
The translation, of course, will not be expressed in the matrix, but rather it is given by adding a fixed vector to all the old coordinates.
Now supposing your old and new coordinates share the same origin (after translation), you can compute the matrix $M$ for which transforms old coordinates to new: $[x_1\dots x_n]M=[y_1\dots y_n]$. I'm pretty sure the conditions you want make $M$ nonsingular, so the polar decomposition of $M$ will tell you what rotation and what stretching has occurred.
I don't know about shear explicitly: is it just the result of stretching going at different rates in different directions?