Talk:Affine space

message: A recent edit changed some signs in the informal description. The version before the edit makes sense to me, the current one doesn't. I'm hardly an expert; any thoughts on this? 84.73.177.141 (talk) 15:10, 9 June 2012 (UTC)
replies: text: Yes, you are right. I undo that wrong edit. Thank you. Boris Tsirelson (talk) 16:23, 9 June 2012 (UTC)
replies: text: Thank you, too! 84.73.177.141 (talk) 14:03, 10 June 2012 (UTC)
thread_title: Wrong signs in informal description?

message: In the article there is (at least) a strongly misleading definition of affine subspace. I think it would be clearer to state something like:
"An affine subspace (also called, in some contexts, a linear variety, a flat, or, over the real numbers, a linear manifold) B of A is a subset of A such that B with the inherited translation space is itself an affine space."
Reason for this suggestion is the question (since already people take this wrong) ocuring on this side: http://math.stackexchange.com/questions/1930925/how-do-i-prove-that-a-given-set-is-an-affine-space/
ChristianTS (talk) 01:15, 18 September 2016 (UTC)
replies: text: Indeed! Probably, introduced by User:D.Lazard on 26 Nov 2015. Boris Tsirelson (talk) 04:26, 18 September 2016 (UTC)
replies: text: OOOPS, yes I have introduced this wrong definition. The reason of my mistake is the following. I wanted to define subspaces without an a priori definition of the associated vector space. In fact, the given definition is incomplete: the "inherited translation space" is effectively the set of the differences of points of the subspace. But one has to add (and I'll do that) that the subspace is stable by these translations. D.Lazard (talk) 08:46, 18 September 2016 (UTC)
replies: text: In fact, for fixing my error, it suffices to fix one of the points in the set of differences, and this gives a definition of a subspace, which is easier to test. I have edited the article accordingly. D.Lazard (talk) 21:15, 18 September 2016 (UTC)
thread_title:  misleading/wrong definition of affine subspace 

message: The caption of the first illustration ends:  "The difference a − b of two of its elements lies in P 1 and constitutes a displacement vector."  Shouldn't this be: "...lies in P 2..." ? RobLandau (talk) 00:30, 17 September 2018 (UTC)
replies: text: No, it is correct as written.  --JBL (talk) 01:57, 17 September 2018 (UTC)
replies: text: I also agree with that comment, since the translation should be in the affine space.— Preceding unsigned comment added by 2a00:a040:196:33d3:b17c:c410:c7d7:b89 (talk) 07:26, 30 June 2021 (UTC)
replies: text:  No; the vector is translation vector parallel to the space but it does not belong to the space.  The confusion is common and natural because vectors can be drawn anywhere.  In the case of an affine subspace H of R2 or  R3, the relation "the arrow representing vector v can be drawn in such a way that it lies in the plane H" is the relation "v is parallel to H", not "v belongs to H".  (The relation "v belongs to H" would instead be that, if the arrow is drawn so that its tail is at the origin, then its head is pointing to a point in H.)  Here is an analogy: the displacement vector between the points (1, 2) and (2, 1) in the Euclidean plane is (1, −1).  The two points lie in the first quadrant, but the vector does not.  The situation is the same here.  --JBL (talk) 17:01, 30 June 2021 (UTC)
thread_title:  First diagram 

message: The following pair of sentences in the article's introduction seem not entirely helpful:
"A Euclidean space is an affine space over the reals, equipped with a metric, the Euclidean distance. Therefore, in Euclidean geometry, an affine property is a property that may be proved in affine spaces."
Note that I am new to the subject of affine spaces. With respect to this article I am a consumer of information rather than a provider. But my mathematical background is adequate, and that does allow me to give an opinion on whether particular material is helpful. It seems to me that the first sentence,
"A Euclidean space is an affine space over the reals, equipped with a metric, the Euclidean distance."
is trying to tell me the following: 
Euclidean space = a particular affine space + a particular metric.
That sentence delivers meaning in respect to affine spaces, but in a sort of backwards way. 
But then the purpose of the second sentence, 
"Therefore, in Euclidean geometry, an affine property is a property that may be proved in affine spaces."
is really not clear to me at all.
I'd appreciate any replies below. Dratman (talk) 00:03, 23 September 2018 (UTC)
replies: text: I agree with all your comments; the article would be improved by removing those two sentences. --JBL (talk) 02:34, 23 September 2018 (UTC)
replies: text: As for me, the first sentence is OK. The second one could be moved from the intro to some section and expanded a bit; or, alternatively, removed. Boris Tsirelson (talk) 10:57, 23 September 2018 (UTC)

text: The fact is that most modern definitions of Euclidean spaces do not start from synthetic geometry, but from linear algebra. That is, a Euclidean space is defined as an affine space, such that the associated vector space is a normed vector space of finite dimension over the reals. This is an important fact that deserves to appear in the lead, possibly preceded by "In most modern definitions of Euclidean spaces". Also, this article deserves to have a section on Euclidean spaces, at least for clarifying the phrase "the Euclidean space \mathbb R^n" that can be found in many places in Wikipedia and elsewhere. The second disputed sentence would be better placed in this hoped new section when it will exist. However, as "affine property" is a common phrase in mathematics, and, thus, should be defined in this article, it must be kept in the lead until someone (maybe myself) will write this new section. D.Lazard (talk) 14:19, 23 September 2018 (UTC)
replies: text: My impression has been that when mathematicians refer to 3-dimensional Euclidean space they are talking about a system of Cartesian coordinates of the form (x,y,z) or perhaps more properly (r1,r2,r3), that is, a metric space derived from three perpendicular basis vectors, often written using unit vector notation as e1, e2, e3. But the word "affine" is seldom mentioned. Your new article text speaks of "the associated vector space". I would have thought that "Euclidean space" referred directly to that vector space based on e1, e2, e3. Is that not correct -- or rather, not modern? Dratman (talk) 16:09, 23 September 2018 (UTC)
replies: text: There is no basis in the definition of a vector space. Similarly, there is no origin in the definition of the Euclidean 3D space (otherwise, where should it be, at the center of Earth, of the Galaxy,...?) Moreover, a Euclidean space contains points, not vectors, and it is not a vector space. So, affine spaces have been introduced for "forgetting the origin", exactly as vector spaces have been introduced for "forgetting the standard basis". It is a basic theorem that the set \mathbb R^n is an affine space with itself as associated vector space, and that the dot product defines a norm that makes it a Euclidean space. Conversely, the choice of an origin on a Euclidean space, and of an orthonormal basis on it associated vector space induces an isomorphism (of Euclidean spaces) between the given Euclidean space and \mathbb R^n viewed as a Euclidean space. This makes \mathbb R^n a prototype for all Euclidean spaces, and explain that, when there is no need of changing the origin and the basis, one works often with \mathbb R^n instead of with a general Euclidean space. I agree that this is too WP:TECHNICAL for Euclidean space, but this needs to appear somewhere in WP, and this article seems the best place. D.Lazard (talk) 16:45, 23 September 2018 (UTC)

text: What you just wrote is very interesting to me. I have not seen Euclidean space described in just that way before. Your rapid description is intriguing, but parts of it are not clear and complete enough to convey everything necessary for comprehension. It sounds as if part of what you are discussing belongs in Euclidean Space; nevertheless I think the present article might accommodate most of the ideas you are referring to. It is my impression that the concept of an affine space is not well understood by, say, a typical engineering major, so you might be performing a real service by adding to the article an edited version of the points you have made here. Do you recommend a reference work for this material? Dratman (talk) 20:51, 23 September 2018 (UTC)
replies: text: I have rewritten the article Affine space exactly for this reason. The basic reference is Marcel Berger's Geometry. As I have said, I'll add a section on Euclides spaces to this article, when I'll get some time for that. D.Lazard (talk) 21:06, 23 September 2018 (UTC)

text:  The sentences still don't belong in the lead; even less so when expanded.  I have moved them to a new short section in the body.  --JBL (talk) 23:12, 23 September 2018 (UTC)
thread_title:  Sentences I don't find helpful 

message: Some time ago, and editor has systematically changed "frame" into "coordinate system". I have reverted this since it is wrong, although the two terms are strongly related: a frame is a set of geometric data that allows defining a coordinate system. In the case of vector spaces, frames are commonly called bases. In the case of affine and Euclidean spaces, a frame consists of a point and a basis of the associated vector space. This heterogeneity makes difficult to call it a "basis". As a frame may be considered independently of any coordinates, it is nonsensical to call it a coordinate system. D.Lazard (talk) 17:34, 10 April 2019 (UTC)
replies: text: I would say your analysis is mistaken.  What we mean by a coordinate system in an affine space is precisely the coordinate system generated by a frame, and conversely a coordinate system determines a frame.  They are equivalent notions.  The name "coordinate system" has more meaning for most people and should therefore be preferred.  Zaslav (talk) 07:08, 26 September 2021 (UTC)
thread_title:  Frames and coordinates systems 

message: it looks pretty, but what does it have to do with affine subspaces? 2A01:CB0C:CD:D800:B93E:F0CE:F0E6:4260 (talk) 14:20, 6 September 2020 (UTC)
replies: text: There is no mention of tesselation in this article. D.Lazard (talk) 14:26, 6 September 2020 (UTC)

text: I see. "Tesselation" appears in the name of a figure. But, here, this is simply a figure in an affine space of dimension two. D.Lazard (talk) 14:37, 6 September 2020 (UTC)
thread_title:  tesselation 

message: This edit certainly cannot be right (x is not in the image of \vec{f}).  I don't have time at the moment to figure out whether the right thing is just to revert you or something else; please check. JBL (talk) 23:52, 18 September 2022 (UTC)
thread_title:  Definitely not right 

message: Here are two interesting papers which generalize the theory of affine spaces over a field by considering "affine space over a module".
In both of them by a ring is meant an associative unitary ring in which 3 is an invertible element.
This topic has been brought up previously by @Ashley Y. I'm not sure how to begin incorporating this information but it seems important. FionaLovesCats (talk) 00:20, 7 May 2023 (UTC)
replies: text: International Mathematical Forum is published by Hikari, a publisher on Beall's list, and was de-indexed by MathSciNet.  Even setting these facts aside,  neither paper has any citations in MathSciNet after more than 10 years; so this is a case of "two people wrote two papers about something once, and no one else has ever noticed that they did so" -- very far from "important".  --JBL (talk) 00:36, 7 May 2023 (UTC)
replies: text:  Per WP:TALK#REPLIED, please don't edit comments after they've been responded to; or if you do so, please use strike-through or some other formatting so that other editors can understand what was originally there.  (In particular, because you removed the word "important" from your post, my response is now made potentially confusing or unclear.)  [Meta-commentary, since tone is difficult: this comment is meant as friendly advice, in the spirit of helping a newer member of the community learn some of the many, many, many community norms that are not necessarily obvious.] --JBL (talk) 21:57, 7 May 2023 (UTC)

text: You might more generally enjoy the concept of a torsor. See Baez's blog post https://math.ucr.edu/home/baez/torsors.html –jacobolus (t) 01:44, 14 May 2023 (UTC)
thread_title:  Affine space over a ring/module