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I'm looking for the name of the geometric 2D-transformation that transforms 3 arbitrary non-aligned points into 3 other arbitrary non-aligned points. I know that it is a mix of scaling/rotating/translating/shearing, but what is the name of the transformation? thanks

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affine transformation? –  Heike Aug 15 '11 at 16:18

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I believe you are looking for Affine Transform.

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Technically that is not quite right: an affine transformation can introduce collinearity where there was none (in other words, I don't think most definitions of affine transformations require the transform to be invertible). –  Willie Wong Aug 15 '11 at 16:22
@willie: I don't see any invertible requirement. And I believe given any two sets of three points $\{A,B,C\}$ and $\{A',B',C'\}$, there is an affine transform mapping $A$ to $A'$ etc (not too sure though). –  Aryabhata Aug 15 '11 at 16:27
Yeah affine transform is what I was looking for. Thanks –  lezebulon Aug 15 '11 at 16:30
Well, maybe I am reading too much into it, but I take "non-aligned" to mean "non-collinear" in the original question, which would imply that the transformation is invertible. And yes, an affine transformation in the plane is given by a linear transformation + a translation, so that is a matrix + a point, or six degrees of freedom. Specifying the image of three points also takes six degrees of freedom. You are correct that given three arbitrary points and their images in the plane there can be specified one unique affine transformation. –  Willie Wong Aug 15 '11 at 16:35

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