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For example, I have set of points in 3D. Points lie on straight line. Transformed set of points lies on straight line too. How to check if transformation is affine?

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According to the fundamental theorem of affine geometry, it suffices to check that the transformation is a bijective collineation:

Theorem. Let $\mathbb A^{n}$ be an affine space over $\mathbb R$ with $n > 2$ and fix $a \in A$. Let $\phi :\mathbb A^{n}\to \mathbb A^{n}$ be a bijection which takes each three collinear points into collinear points. Then there exists a point $b\in \mathbb A^{n}$ and an invertible linear map $F$ such that $\phi(x) = F(x-a) + b$ for all $x \in\mathbb A^n$.

The proof can be found in Berger's Geometry 1 (Springer, 1987, pp. 52-56).

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Thank You. –  qutron Dec 22 '10 at 14:39
    
@Andrey Rekalo:Could you explain why we use affine transformations?.Why couldn't we use vector transformations?Is there anything relevant in using affine transformation other than the fact that affine spaces has no origin? –  justin Nov 18 at 12:53

an affine transformation between two vector spaces $$F:X\rightarrow Y$$ (one might define it more general) is defined as $$y = F(x) = Ax + y_0$$

where $A$ is a constant map (might be represented as matrix) and $y_0\in Y$ is a constant element.

So, to check if a transformation is affine you might try to proof that such $A$ and $y_0$ exist.
Sincerely

Robert

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Thanks) –  qutron Dec 22 '10 at 14:40
    
@Robert Filter:Could you explain why we use affine transformations?.Why couldn't we use vector transformations?Is there anything relevant in using affine transformation other than the fact that affine spaces has no origin? –  justin Nov 18 at 12:54

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