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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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  • $\begingroup$ What do you mean, "what are they for"? $\endgroup$
    – celtschk
    Aug 23, 2012 at 8:29
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    $\begingroup$ Does the fact count that the very space we live in is, to a good approximation, an affine space? $\endgroup$
    – celtschk
    Aug 23, 2012 at 8:35
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    $\begingroup$ @celtschk: that's so 20-th century. Now, people live in graphs, you know? $\endgroup$
    – PseudoNeo
    Aug 23, 2012 at 11:27
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    $\begingroup$ @PseudoNeo: No, it's 19th century; in the 20th century we lived on a curved manifold. ;-) (But then, I explicitly wrote "to a good approximation".) $\endgroup$
    – celtschk
    Aug 23, 2012 at 12:15
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    $\begingroup$ Prepare yourself for bigger shocks in life: If affine spaces intimidate you or do not give you a sense of what they model, there is a host of things to come: manifolds with negative curvature,A scheme which is not quasi-projective; a Lie group that is not a matrix group. Spaces that are not locally compact; normed linear spaces that are not separable. $\endgroup$ Nov 2, 2016 at 14:10

5 Answers 5

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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 look alike; they can vary in direction and in magnitude, and in particular there is the zero vector which is special. And so on...

An affine space is an abstraction of how geometrical points (in the plane, say) behave. All points look alike; there is no point which is special in any way. 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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Regarding "when would I use it" -- affine space is the natural setting for computer-aided design, computer-aided manufacturing, and other computer applications of geometry. People who develop software in this field all know that you have to carefully distinguish points and vectors (even though they might both be represented as triples of numbers), and avoid "illegal" operations like adding two points. The more mathematically inclined members of the community understand that they are working in an affine space.

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.

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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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  • $\begingroup$ Given three points $p_1, p_2, p_3 \in A$ with $A$ being the affine space, I can calculate the angle between the vectors $p_1 - p_2$ and $p_3 - p_2$, just like in a vector space, no? How does the lack of an origin in the affine space influence the ability to determine angles (apart from the angle relative to an origin of course)? $\endgroup$ Aug 9, 2020 at 20:48
  • $\begingroup$ ... assuming there is an inner product on the associated vector space, which I assume was the essential point. But within the subset of affine spaces whose associated vector space is an inner product space, angles should still be a well-defined notion (conversely: the notion of angles can be lost just by going from an inner product space such as $E$ to, say, a topological vector space, which is not affine)? $\endgroup$ Aug 9, 2020 at 21:48
  • $\begingroup$ @MrArsGravis: This is the whole point of the answer: You cannot calculate angles in an affine space since an inner product structure is not given. $\endgroup$ Aug 14, 2020 at 21:07
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Affine spaces are important because the space of solutions of a system of linear equations is an affine space, although it is a vector space if and only if the system is homogeneous.

Let $T:V \to W$ be a linear transformation between vector spaces $V$ and $W$. The preimage of any vector $w \in W$ is an affine subspace of $V$. If $w$ is nonzero, then the preimage does not contain $0$ so is not a vector subspace of $V$. However, the preimage of any $w \in W$ is an affine space modeled on the vector space $\ker T$, the kernel of $T$.

A special case of this example is that the space of solutions of the matrix equation $Ax = y$ (for fixed $y$) is an affine space modeled on the null space of the matrix $A$.

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The first space we are introduced in our lives are euclidean spaces, which are the classical beginning point of classical geometry. In these spaces, there is a natural movement between points that are translations, i.e., you can move in a natural way from a point $p$ to a point $q$ through the vector that joint them $\overrightarrow{pq}$.

In this way, vectors represent translations in the euclidean space. Therefore vector spaces are the natural generalization of translations of spaces, but which spaces? Here is where affine spaces are important, because they recover the concept of points which the "arrows" (vectors) of a vector space move.

In conclusion, an affine space is mathematical modelling of an space of points whose main feature is that there is a set of preferred movements (called translations) that permits to go from any point to other point in an unique way and that are modeled through the concept of vector space. Or in other words, affine spaces represent the points that the arrows of vector spaces move.

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