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?
How to check if transformation is affine?
2 Answers
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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0@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? – 2014-11-18
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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0@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? – 2014-11-18