# What are affine spaces for?

I'm studying affine spaces but I can't understand what they are for.

Could you explain them to me? Why are they important, and when are they used? Thanks a lot.

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What do you mean, "what are they for"? –  celtschk Aug 23 '12 at 8:29
@celtschk I mean: why do I study that? what can I do with affine spaces? what are the use of affine space? –  Surfer on the fall Aug 23 '12 at 8:31
Does the fact count that the very space we live in is, to a good approximation, an affine space? –  celtschk Aug 23 '12 at 8:35
@celtschk Very interesting.. can you explain (or give reference link) your last sentence? –  Surfer on the fall Aug 23 '12 at 8:43
Well, the Wikipedia article "Affine Space" begins with "In mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space." Since we live in an approximately Euclidean space, we also live in an approximately affine space. –  celtschk Aug 23 '12 at 8:58

A vector space is an abstraction of how geometrical vectors (in the plane, say) behave. You can form linear combinations of vectors. Not all vectors are equal; in particular, there is the zero vector which is special. And so on...

An affice space is an abstraction of how geometrical points (in the plane, say) behave. All points are equal. You can't add points. However, you can subtract points (giving a vector as the result). And you can form convex combinations of points. And so on...

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The euclidean space $E$ of high school geometry (2d or 3d) is an affine space, but one with extra structure: You can measure lengths and angles; among the angles there are distinguished ones, namely right angles, and among the ellipses there are distinguished ones, namely circles.

The affine structure of $E$ is what remains when you throw away compass, set-square and protractor. Parallelity is still recognized. The allowed tools left are the ruler and a device to determine the ratio of lengths on parallel lines.

A theorem of affine geometry in the plane is the following: Assume that the lines $a$ and $b$ intersect in a point $P$, that $A_1$, $A_2\in a$, $\ B_1$, $B_2\in b$, and that ${\rm vec}(PA_1)={\rm vec}(A_1A_2)$, $\,{\rm vec}(PB_1)={\rm vec}(B_1B_2)$. Then the lines $A_1\vee B_1$ and $A_2\vee B_2$ are parallel.

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So, like most abstractions, affine spaces may or may not be helpful to you, depending on how your brain works. But the particular affine space $R^3$ is very important. As celtschk pointed out, it's the space we live in. And, in particular, it's the space we compute in.