- #1

Lajka

- 68

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Hi,

I was reading a little about affine geometry, and something bothered me. Namely, in some books, there were some paragraphs that were written like "blabla, let's observe an affine plane for instance, and in the spirit of Descartes, we shall assign to each point in plane an ordered couple (x,y)".

Now, I don't get this, and it bothers me. I thought the whole point of an affine plane (for instance) was to get rid of the coordinate systems and the origin as some special point. Affine plane is just a bunch of 'points'.

If we identify a point in plane with an ordered couple (x,y), clearly we have assigned an 'origin' position to a point (0,0). Coordinates are just the numbers that tell us how "far away" are points from the "origin".

**So I don't think that it's okay to assign an ordered couple from [itex]R^2[/itex] to a point in an affine space just like that. What do you think?**

What I'm trying to establish here is to understand how exactly are we systematically bringing the numbers as a concept into the geometry. So the above feels like a shortcut to me, simply 'identifying' points with numbers.

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Here's where I got so far: If we take an affine space as it is, all we have in it are 'points', 'lines', 'line segments', and the ability to define parallelism and compare lengths of line segments (if they're congruent, or parallel in this case). We can introduce vectors in several ways:

as functions from the affine space onto itself, which map one point in the space to another one (addition is then uniquely defined as a composition of functions, which is pretty cool)

we take a vector space as some abstract concept (with all the rules for an abstract vector space over the field of real numbers), and we define a function f: AxA -> V, so that f(P,Q) = v "=" Q - P is a bijection for some fixed point P.

again, we take a vector space as an abstract concept, and we define a group action which is used by a vector space, as an Abelian group, to act on an affine space, namely "+":AxV -> A, or in other words, addition of a vector and a point to get another point. Space of points is something that's called torsor then, I think.

Just one question here: in the first case we are told what vectors exactly are, ordered couples of points. In the other two cases, I think we still don't know what vectors really are, we just have functions that connect them and our affine space. But regardless of that, we "identify" vectors as directed line segments in all three cases.

**Do I need to do add some additional steps to make this "identification' more rigorous?**

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Anyhow, my

*main*problem here, at the moment, is how to explain scalar multiplication of vectors. Sure, if I have [itex]v[/itex] from vector space V, I can easily construct [itex]2v[/itex], or any [itex]qv[/itex] as the directed line segment, where [itex]q \in Q[/itex]. Segment arithmetic I've seen in Hilbert's "Foundations of geometry" allows me to do this.

But irrational numbers are quite different beast. If V is a vector space, then it guarantees that vector [itex]\pi v[/itex] is also in space, for example. But I don't know how to construct directed line segment with that length.

**How to do this?**

**On the same note, how do you even determine the position of an irrational number on a line?**

Sure, you can easily construct [itex]\sqrt{2}[/itex] but I mean in general, for any irrational number.

In Hilbert's book, for instance, he takes some segment on a line do designate the unitary length, then he starts dividing it and dividing it and so on, which explains how to assign any rational number to any point, but I didn't see any explanation how to do this for irrational numbers.

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Now, if we surpass this problem, then I think I know what to do next. I choose an origin O, and I choose a basis {[itex]v_i[/itex]} in V, and then I can identify every point P as [itex]P = O + \sum \alpha _{i} v_{i} (i = 1, ..., n)[/itex].

With all this, I think I formally did two things: I established an isomorphism between an affine space [itex]A_0[/itex] and vector space V, and I established an isomorphism between vector space V and vector space [itex]R^n[/itex]. And that's how we finally get to assign numbers to any point in an affine space!

But I think that I should also say that these isomorphisms aren't "canonical" since I can easily arrange new ones with another point as an origin and with another basis. Also, to distinguish between vectors and points, we can use coordinates that have an additional element (0 if they represent vectors, 1 if they represent points).

Now, to introduce orthogonal basis, i clearly need to know which vectors, aka directed line segments, are perpendicular one to another.

**Am I supposed to define the "angle" the way Euclid did, or is there some other way?**(I can't define it via dot product, cause I'm yet to introduce one, and I need an orthogonal basis if I want to define the standard dot product in the first place).

And then,

**how can I make sure that all my vectors, aka directed line segments, have the same length, because I want an orthonormal basis?**

Euclidean arithmetic only allows us to compare the lengths of congruent line segments (which are parallel to each other), and my vectors are perpendicular to one another.

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I believe this suffices for an explanation of Cartesian and skew coordinate systems, but I'm still not sure how to systematically, in this faashion, introduce curvilinear coordinate systems, e.g. a polar coordinate system. That's something I have no idea how to do.

**so I would appreciate if someone could point me in the right direction.**

Thanks in advance for any help I can get.

Cheers.