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I am having trouble understanding what an affine space is.

I am reading Metric Affine Geometry by Snapper and Troyer. On page 5, they say:

"The upshot is that, even in the affine plane, one can compare lengths of parallel lines segments. If $A \neq B$ are points of an affine plane $P$, it is perfectly alright to choose the line segment $AB$ as a "unit length" for the plane. However, this unit length has only limited power because it can only be used to measure the lengths of line segments which are parallel to $AB$. In order to compare the length of line segments which are not parallel, we need a measuring rod which will work for all directions at the same time. Such a measuring rod is provided by Euclidean geometry or by one of the non-Euclidean, metric geometries."

On pages 6 and 7, they say an affine space consists of

(1) $X$ is a nonempty set

(2) $k$ a division ring

(3) $V$ is an $n$-dimensional left vector space over $k$

(4) an action $(X,V,k)$ of the additive group of $V$ on $X$

Here's what I want to know: If I read the stuff above, I understand the individual propositions. But I still do not know what an affine space is. Like I can't visualize it or see any meaning in these propositions that tells me what I can do with this mathematical structure.

And Snapper and Troyer make it very confusing since they may constant close analogies between vector spaces and affine spaces that really don't distinguish the two well, at least to me.

Can someone (1) explain what an affine space is (2) provide one or more examples (3) explain how the above fits with your definition of an affine space

Thanks!

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  • $\begingroup$ Read "Linear Algebra" BY Larry Smith for a basic and a good understanding. $\endgroup$ – Hirak Dec 20 '14 at 5:38
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I write $P + v$ for the action of the vector $v$ on the point $P$.

An affine space is, in fact, probably the very first visualization you had of vector spaces: e.g. you thought of the plane full of points, and that vectors were arrows that went from one point to another.

The concept of affine space I know requires the action of $V$ on $X$ to be transitive and faithful: this means that, in an affine space, we can define subtraction: $P - Q$ is the unique vector $v$ such that $Q + v = P$. The pair $(Q, v)$ can be pictured as an arrow from $Q$ to $P$.

We can even define nearly arbitrary linear combinations of points: the restriction is that the coefficients have to sum to zero (thus giving us a vector) or sum to one (thus giving us a point).

e.g. if I write $P + \frac{1}{2}Q - \frac{3}{2} R$, I 'really' mean the vector

$$P + \frac{1}{2}Q - \frac{3}{2} R = (P - R) + \frac{1}{2} (Q - R)$$

or any other similar rearrangement into "legal" operations. (they would all give the same answer)

Similarly, if I write $\frac{1}{2}P + \frac{1}{3} Q + \frac{1}{6} R$, I mean the point

$$ P + \frac{1}{3} \left(Q - P \right) + \frac{1}{6} \left(R - P \right) $$

I believe I've seen some definitions of affine space that don't make reference to vectors at all: they are instead axiomatized in terms of the arithmetic of points.

The notion of vector space is, in fact, equivalent to the notion of a pair consisting of an affine space and a point on the space. A vector space already has the structure of an affine space; it just comes equipped with a distinguished point $0$. Conversely, given any affine space and a choice of a point $O$, we can complete its vector space structure by treating every other point $P$ as the vector $P - O$.


Beside from thinking like vectors, convex combinations (where the coefficients sum is one and are all positive) can be thought of as averaging between points. For example,

  • $\frac{1}{3} P + \frac{2}{3} Q$ is the point on the line segment $PQ$ that lies two thirds of the way from $P$ to $Q$

They're called convex combinations, because of the relation to convex sets: any convex combination of points in a convex set is again in the set. In fact, the convex hull of a set of points is precisely the set of all convex combinations of those points.

e.g. the set of all points inside the triangle $\Delta PQR$ are of the form $aP + bQ + cR$ where $a+b+c = 1$ and $a,b,c$ are all positive. (if we allow $a,b,c$ to be zero as well, then this set of points would include the triangle itself)

More general affine combinations (the coefficients sum to one) are similar:

  • $2P - Q$ is the point on the line $PQ$ that lies on the other side of $P$ from $Q$ that is twice as far from $Q$ as it is from $P$.

All convex combinations, actually, can be made out of binary ones: for example,

$$ \frac{1}{2} P + \frac{1}{3} Q + \frac{1}{6} R = \frac{5}{6} \left(\frac{3}{5} P + \frac{2}{5} Q \right) + \frac{1}{6} R $$

(how did I find this? Let $S$ be this point. I picked points on the plane for $P$, $Q$, and $R$ and found where the line $RS$ met the segment $PQ$: that point is the point that appears in the parentheses on the right hand side)

For linear combinations, you can just pretend the points are vectors. It doesn't matter which point you choose in the affine plane as the origin, you'd get the same result. e.g. my earlier example satisfies

$$P + \frac{1}{2}Q - \frac{3}{2} R = (P - O) + \frac{1}{2} (Q-O) - \frac{3}{2} (R - O)$$

no matter which point you choose for $O$.

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  • $\begingroup$ Is it possible to insert a drawing of what you described? Specifically the part with the fractions. I'm a visual thinker and it might help it gel better. $\endgroup$ – Stan Shunpike Dec 20 '14 at 8:08
  • $\begingroup$ @Stan: I've made an edit. (do you get pinged when I edit my post?) $\endgroup$ – Hurkyl Dec 20 '14 at 17:02
  • $\begingroup$ Don't think so. Thanks for telling me and for the added info. $\endgroup$ – Stan Shunpike Dec 20 '14 at 17:10

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