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Definitions.

  • By a vector space, I simply mean an $\mathbb{R}$-module.

  • By an affine space, I mean a vector space $X$ (the "translation space") together with a set $P$ (of "points"), together with an additively-denoted monoid action of $X$ on $P,$ such that for all points $p,q \in P$, there exists a unique vector $x \in X$ such that $q = x+p$.

  • By an inner-product affine space, I mean an affine space $(X,P)$ together with an inner product structure on $X$.

  • By the vectorial dimension of an affine space $(X,P),$ I mean the dimension of $X$.

  • By a finite-dimensional affine space, I mean an affine space whose vectorial-dimension is finite.

  • By a Euclidean space, I mean a finite-dimensional inner-product affine space.

  • For each natural number $n$, write $\mathbb{E}_n$ for the unique $n$-dimensional Euclidean space, up to isomorphism.

Question. Do any books or articles develop basic Euclidean geometry (i.e. results about $\mathbb{E}_n$) from this perspective?

goblin GONE
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  • I haven't read the book in detail, but I think this is precisely the approach taken in Dieudonné's Linear Algebra and Geometry. – Will R Sep 05 '15 at 23:58
  • Also, this question is perhaps a duplicate of http://math.stackexchange.com/questions/953202/elementary-geometry-from-a-higher-perspective. – Will R Sep 06 '15 at 00:30
  • @WillR, this request is much, much, much more specific than that. – goblin GONE Sep 06 '15 at 00:31
  • Okay, I'll grant you that the essence of your question is different, at least in so far as specificity, and perhaps motivation, are concerned. But the answers are arguably the same, in that, if there exists such a book, it fits into the answers to the other question.. – Will R Sep 06 '15 at 00:34
  • @goblin: What are you looking for you when you say "basic Euclidean geometry"? You haven't been very specific about that. – Rob Arthan Sep 06 '15 at 00:36
  • @WillR, well none of those answers mention affine space at all. So its not at all clear how to find an answer to my question there, excepting that you've pointed me toward Dieudonne's text. – goblin GONE Sep 06 '15 at 00:36
  • @RobArthan, true, but I don't know enough Euclidean geometry to be able to say what should be covered in such a text. My feeling is that as long as it covers a few results that would have been of interest to the Ancient Greeks, but from a modern perspective, I'm willing to go along with the author. – goblin GONE Sep 06 '15 at 00:44
  • But you are not willing to go along with the author if the term "affine space" doesn't appear in the abstract of the book? – Rob Arthan Sep 06 '15 at 00:48
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    @RobArthan, more precisely, I am not willing to go along with the author if all their spaces have a distinguished point $0$ thought of as the "origin". – goblin GONE Sep 06 '15 at 00:49
  • @WillR, I've had a good look at it, and it seems that Dieudonne's text takes Euclidean spaces as having a distinguished basepoint. Obviously, this is a very bad convention. Its possible that later in the book, he introduces Euclidean spaces in the sense of my question, but calls them something else. – goblin GONE Sep 07 '15 at 00:47
  • I'm just going to throw this out there, but what about Audin's Geometry? From the intro: "The first idea is to give a rigorous exposition, based on the definition of an affine space via linear algebra, but not hesitating to be elementary and down-to-earth. I have tried both to explain that linear algebra is useful for elementary geometry (after all, this is where it comes from) and to show “genuine” geometry: triangles, spheres, polyhedra, angles at the circumference, inversions, parabolas..." Chapter 1: Affine Geometry – pjs36 Oct 07 '15 at 01:09
  • @pjs36, thanks! This looks like exactly what I was looking for. – goblin GONE Oct 07 '15 at 11:28
  • @pjs36, why not post this as an answer? – goblin GONE Oct 08 '15 at 01:52
  • @goblin I figured Dieudonné's book would be similar, and when it didn't quite fit the bill, I decided to test the waters with a comment first :) – pjs36 Oct 08 '15 at 12:51

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Perhaps you might like Audin's Geometry.

From the intro:

The first idea is to give a rigorous exposition, based on the definition of an affine space via linear algebra, but not hesitating to be elementary and down-to-earth. I have tried both to explain that linear algebra is useful for elementary geometry (after all, this is where it comes from) and to show “genuine” geometry: triangles, spheres, polyhedra, angles at the circumference, inversions, parabolas...

Chapter 1 is titled Affine Geometry, the next three chapters are about Euclidean geometry (generalities, in the plane, and in space), followed by projective geometry and then a few chapters on classical topics (conic sections, curves, surfaces).

pjs36
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