Talk:Affine space

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Wrong signs in informal description?

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)[reply]

Yes, you are right. I undo that wrong edit. Thank you. Boris Tsirelson (talk) 16:23, 9 June 2012 (UTC)[reply]

Thank you, too! 84.73.177.141 (talk) 14:03, 10 June 2012 (UTC)[reply]

The informal description should have an image or two to make it clearer than the vectors from the origins to points a and b are different for Alice and Bob and what Bob calls a is a-p for Alice. For her, the resultant of Bob's parallelogram addition is (a-p) + (b-p) and to get to the same final point, Alice has first to follow p from her origin to Bob's and then add the resultant of the parallelogram addition: p + (a-p) + (b-p). Starple (talk) 08:51, 24 October 2017 (UTC)[reply]

Adding Points

"In an affine space, there is no distinguished point that serves as an origin. Hence, no vector has a fixed origin and no vector can be uniquely associated to a point. In an affine space, there are instead displacement vectors, also called translation vectors or simply translations, between two points of the space.[1] Thus it makes sense to subtract two points of the space, giving a translation vector, but it does not make sense to add two points of the space. Likewise, it makes sense to add a vector to a point of an affine space, resulting in a new point translated from the starting point by that vector."

IF P1-P2=V and P3-P4=V, then why not have P1-P2=P3-P4=V <=> P1+P4=P3+P2=V? — Preceding unsigned comment added by 188.4.199.76 (talk) 15:39, 14 April 2016 (UTC)[reply]

Because there is no sense in which the equation you wrote is true. Sławomir
Biały 15:58, 14 April 2016 (UTC)[reply]

In fact, this is the sum of points that has no sense. But with the standard notation for barycenters, it is true that

P_{1}-P_{2}=P_{3}-P_{4}\Longleftrightarrow {\frac {1}{2}}P_{1}+{\frac {1}{2}}P_{4}={\frac {1}{2}}P_{2}+{\frac {1}{2}}P_{3}.

This is the well known property that a quadrilateral is a parallelogram if and only if its diagonals bisect each other (I have not found the "if" condition in Wikipedia). D.Lazard (talk) 17:07, 14 April 2016 (UTC)[reply]

Yes. However, even the equality

P_{1}+P_{4}=P_{2}+P_{3}

makes some sense if we admit points outside the given affine space. The latter is an affine hyperplane in a vector space, and this extension is canonical... Boris Tsirelson (talk) 17:43, 14 April 2016 (UTC)[reply]

misleading/wrong definition of affine subspace

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)[reply]

Indeed! Probably, introduced by User:D.Lazard on 26 Nov 2015. Boris Tsirelson (talk) 04:26, 18 September 2016 (UTC)[reply]

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)[reply]

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)[reply]

First diagram

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)[reply]

@RobLandau: No, it is correct as written. --JBL (talk) 01:57, 17 September 2018 (UTC)[reply]

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)[reply]

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)[reply]

Sentences I don't find helpful

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)[reply]

I agree with all your comments; the article would be improved by removing those two sentences. --JBL (talk) 02:34, 23 September 2018 (UTC)[reply]

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)[reply]

Thanks much for the comments in reply to my point. I have removed the second sentence and made what I hope is a clarifying change to the first. Again replies would be helpful. Dratman (talk) 11:37, 23 September 2018 (UTC)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

Frames and coordinates systems

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)[reply]

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)[reply]

tesselation

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)[reply]

There is no mention of tesselation in this article. D.Lazard (talk) 14:26, 6 September 2020 (UTC)[reply]

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)[reply]

Definitely not right

Moved from User talk:Hooman Mallahzadeh § Definitely not right

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)[reply]

Please discuss about JBL's idea. Tnx. Hooman Mallahzadeh (talk) 04:04, 19 September 2022 (UTC)[reply]

Affine space over a ring/module

Here are two interesting papers which generalize the theory of affine spaces over a field by considering "affine space over a module".

Ostrowski, T.; Dunajewski, K. (2009). "An Affine Space over a Module". International Mathematical Forum. 4 (30): 1457–1463.

Ostrowski, T.; Dunajewski, K. (2010). "Affine Systems of Coordinates in an Affine Space over a Module". International Mathematical Forum. 5 (28): 1385–1391.

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)[reply]

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)[reply]

@FionaLovesCats: 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) Thanks for restoring the original version. --JBL (talk) 00:19, 14 May 2023 (UTC)[reply]

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)[reply]

Temperature as an affine space

@JayBeeEll I don't understand. You said that "nobody cares about absolute zero" is a very obscure way of saying "you're right, temperatures do not form an affine space", but isn't it the exact opposite? This very article states that when the special role of the zero vector in a vector space is "forgotten", we get an affine space. PBZE (talk) 23:52, 3 April 2024 (UTC)[reply]

In an affine space, any displacement can be applied to any point. There is no way to apply a displacement of -5 degrees C to the point -270 degrees C. --JBL (talk) 00:22, 4 April 2024 (UTC)[reply]

Temperature jumps out to me as an example, in part, because it's the prototypical example for an interval variable used when teaching about statistics in the context of psychology. It is also the most accessible example out of any given ones, and the problems with adding two temperatures in a non-absolute scale are more likely to be useful for getting a sense of the space, even if that example is wrong because some parts of the space generated are not real. I think it still might be worth having the example, with caveats.

I would also note that you cannot add absolute temperatures either. IndigoManedWolf (talk) 01:48, 4 April 2024 (UTC)[reply]

Temperature is not an example of an affine space, since, in an affine space you can add differences of two points (difference of temperatures), and multiply them by scalars; the middle of two points should be also defined. As none of these operations are meaningfully defined for temperatures, temperatures do not form an affine space.

Moreover, even if you were right by saying that temperatures form an affine space, this could not be used as an example in Wikipedia, because of the lack of a WP:reliable source for this assertion. D.Lazard (talk) 09:07, 4 April 2024 (UTC)[reply]

You can say that 10°C + 20C° is equal to 30°C (note the distinction between degrees Celsius and Celsius degrees), and therefore, 10°C + 10C° + 20C° = 30°C.

Similarly, you can say the average of 10°C and 20°C is 15°C, because 10°C + (20°C - 10°C)/2 = 10°C + 10C°/2 = 15°C.

And in Google-searching "celsius" + "affine space", I did find some (albeit obscure and informal) sources from academics which describe units of temperature as affine spaces.

PBZE (talk) 11:00, 4 April 2024 (UTC)[reply]

I am maybe willing to accept that the displacement space of temperatures (in "degrees Celsius") is a vector space, but it does not have a free and transitive action on the space of actual temperatures of things (in "Celsius degrees"). The first link does not describe temperatures as an affine space: it describes them as being measured in an "affine scale". (As you note, it also does not meet WP:RS.) It is definitely true that there is some not-purely-superficial relationship between the idea you're describing and the idea of an affine space, but the relationship is also not "is an example of". --JBL (talk) 18:15, 4 April 2024 (UTC)[reply]

Even if the zero of a temperature scale isn't absolute zero, it's still set to some value of physical significance, whether that's the freezing point of water or of brine. So, how can it really be a space where the origin is "forgotten"? XOR'easter (talk) 16:23, 4 April 2024 (UTC)[reply]