Talk:Affine transformation

All affine transformations are NOT dilations. Reverted many changes to the article.

Many changes were made to this article which effectively treat the term affine transformation as though it were the same as dilation. However, this is wrong: a dilation is only one particular type of affine transformation. Rotations, shearing, etc are all examples of affine transformations that aren't dilations.

The notion that all affine transformations are dilations is not supported by the source given, which is Coxeter 1969 (which calls dilations "dilatations"). You can see quite clearly in section 13.3, which is called affinities, at the bottom of page 202, that Coxeter says quite clearly: "A dilatation is a special case of an affinity, which is any transformation (of the whole affine plane onto itself) preserving collinearity." He then goes onto mention other affinities that aren't dilations, such as shears and reflections.

This means that many statements in this article are incorrect, such as the one in the opening paragraph "For every pair of line segments AB and A'B' in the affine plane, there is a dilation mapping the first to the second." This is again not what the cited source (Coxeter 1969 again) says: the actual statement on the page cited is "two given segments, AB and A'B', on parallel lines, determine a unique dilatation AB -> A'B'." The important thing is that the two segments lie on parallel lines. If they don't, then there's no dilation that's going to rotate one line segment into the other. If the condition that both segments lie on parallel lines is dropped, then the corresponding statement is not true in general for affine transformations: there are infinitely many affine transformations that map a line segment to itself. The correct corresponding statement is given on page 203, which is that two triangles IXY and I'X'Y' uniquely determine an affine transformation.

Note also that wolfram has separate pages for dilation and affine transformation which define the two terms correctly.

Additionally, I find this new picture to be =extremely= confusing, and much more so than the old one. It's much more jargon-y, focuses specifically on dilations, and defines some unnecessary stuff in the blurb (like a central dilation vs a translation). I also didn't find it clear that the affine transformation was taking the old triangle to the new one; it looked like the transformation was supposed to be applied to all lines shown at once. However, I do agree the old blurb ought to be made more informative, so I've changed that.

In the middle of my typing this, I note that user Gene Smith has reverted the page as well. I'm going to go through his revert and Rgdboer's latest version and merge some things to try and come up with the optimal revision.

And lastly, although I believe these edits were done in good faith - please, make sure you understand the precise nature of the claims being made by the cited source before editing the page! Math pages are hard to do right on Wikipedia, and I felt the page on affine transformations was one of the better ones. If anything it needed to be made a bit less technical, perhaps.

Battaglia01 (talk) 22:30, 28 September 2012 (UTC)

What is and how to read...

In the section on the Mathematical Definition, I don't know the meaning of, or how to say, the symbol following the equals sign. I have looked at numerous lists of math symbols, including the unicode blocks for math and for "letterlike" symbols and can't find it anywhere. Since its part of a graphic, I can't select it to copy and paste in a search. Of course it may be there and I'm just not seeing it. Nevertheless, I think a common language transcription, as if the expression were being spoken, ought to be provided. (I think that ought to be the policy for all maths in wikipedia. I'm not just picking on this one equation. I know doing that will mar the beauty of the concise math displayed. But how can anyone be expected to follow the math if they can't even *read* it? But I know this is not the place for that discussion.) Could someone please name that symbol for me?

Baon (talk) 15:10, 3 October 2012 (UTC)

I looked at the source and discovered the symbol is called 'varphi'. The entry on phi discusses this as a font variety associated with older fonts, and suggests that a stroked phi ought to always be used in mathematics... but I do not feel comfortable making a change here because I do not know if there are historical reasons or traditions for using this "loopy" form of phi. Baon (talk) 15:57, 4 October 2012 (UTC)

You have misinterpreted the phi page. It says that the stroked form is "required" for mathematics; what that means is, some mathematics needs it. It does not mean that the loopy form is not used as well, and in fact, it is used, quite regularly. It is not in any way "deprecated". --Trovatore (talk) 21:59, 10 October 2012 (UTC)

Isn't preservation of collinearity enough?

The top paragraph states that an affine transformation is one which satisfies two conditions: It preserves (i) collinearity and (ii) ratios between distances between collinear points. I would have expected (i) alone to suffice, and (ii) to be a consequence of (i) like the property that parallel lines are mapped to parallel lines. If it does not, then it would be very interesting to see a counterexample.

Or is it perhaps the case that (i) suffices for invertible transformations, but not in general? That would also be interesting to have stated explicitly, if true. 130.239.234.45 (talk) 17:01, 16 October 2012 (UTC)

Incomprehensible Gobbledygook

This article is completely incomprehensible to a non-mathematician. Could we try beginning in English? The purpose of an encyclopedia, such as this one ...

I recently received top grades in college calculus. I am currently working in computer graphics as a consumer of the methods others write that implement this math. If this is not within my realm of comprehension, for whom is it written? MartinRinehart (talk) 18:18, 30 August 2013 (UTC)

The best editors for choosing comprehensible words are often those who are freshly learning and working in the field. People familiar with the more abtract terminology will often be blind to what it is that others find difficult. You sound like you are in the ideal position to try your hand at editing the article for better clarity while you become familiar with the subject area. The lead section is the place to have the most immediate comprehensibility. You might find that some editors disagree with your wording choices, but that is all part of the cooperative editing process – which in the end leads to higher-quality articles. — Quondum 21:45, 30 August 2013 (UTC)

I am not a mathematician, and I understand everything except the statement "that preserves the affine structure", in the first sentence of the introduction. Also, the sentence "This means that an affine transformation sends points to points, lines to lines, planes to planes, etc" uses technical jargon ("sends ... to ..."), which may sound not familiar to beginners. Although this terminology is effective, commonly used and easy to grasp, I can be substituted with even simpler terminology. See my edit, and please let me know if you like it. Paolo.dL (talk) 10:21, 31 August 2013 (UTC)

Wikipedia has no interest in educating people. The editors are solely interested in inflating their egos with the most impressive words possible. For a better explanation see: This article by Wolfram quoted above. — Preceding unsigned comment added by Arydberg (talk • contribs) 08:06, 23 April 2016 (UTC)

aligning two separate images

Please add information about how this applies to aligning two separate images so that they can be optimally superimposed, particularly map layers. And information about available tools.-71.174.188.32 (talk) 13:05, 22 October 2015 (UTC)

Translation example

For completeness, the "Image transformation" section needs an example of a matrix and diagram for "translation."—Anita5192 (talk) 19:56, 10 November 2017 (UTC)

There appears to be a constraint in that section that would make this impossible. The affine maps are all linear maps, represented by square matrices, but you need an augmented matrix (or projective map) in order to represent translations. So, I would guess that they are missing for a reason. --Bill Cherowitzo (talk) 20:38, 10 November 2017 (UTC)

The matrices are augmented matrices. They are 3 × 3 matrices as defined in "Augmented matrix" and the diagrams are two-dimensional.—Anita5192 (talk) 21:00, 10 November 2017 (UTC)

Oops. My mistake, sorry. Upon closer examination, all these examples have a null translation component. I could provide the examples, but I don't think that I could mimic the graphic. --Bill Cherowitzo (talk) 23:32, 10 November 2017 (UTC)

Intro

I may be being very stupid here, but is there not a mistake in the following phrase in the introduction?

   "...every affine transformation $f\colon X\to Y$ is of the form $x\mapsto Mx+b$..."

It seems to me that the mapping should be from y not x, and so should read:

   "...every affine transformation $f\colon X\to Y$ is of the form $y\mapsto Mx+b...$"

There is no mistake here. The transformation f maps the domain X to the range Y, and hence maps the vector x to Mx + b.—Anita5192 (talk) 16:41, 24 November 2017 (UTC)

It is much worse: There is a wealth of different mistakes! Due to sloppy notation, the point sets and the vector spaces are confused and the action of the vector space on the point set and the addition in the vector spaces are confused. There is no Mx, since x is a point and M a linear function. If M is a map on X the result is such that the addition with b is not defined. Quite bluntly: This is complete rubbish. Moreover, not only the current intro is wrong, also the mathematical definition is wrong. There is a reason why math books make this more elaborate! — Preceding unsigned comment added by 87.163.202.64 (talk) 21:08, 25 January 2020 (UTC)

I agree the intro is a mess and so are portions of the article. The problem as I see it is that some editors have confounded the concept of "affine mapping" (in Berger's sense) between affine spaces and an "affine transformation" on a given affine space. True, an affine transformation is a special case of an affine mapping, but this article is supposed to be about the transformation, not its generalization. I propose taking all the affine mapping stuff and putting it into its own section and rewriting the lead to deal with the transformation.--Bill Cherowitzo (talk) 18:49, 28 January 2020 (UTC)

I've finally gotten around to start doing this. It was a bit more involved than I had envisioned at first. This still needs some resectioning and a pass through checking on uniformity of notation. --Bill Cherowitzo (talk) 20:03, 1 February 2020 (UTC)

Thank you for implementing this change, Bill. I think it would be a good idea, at the end of the section on affine maps, to add a sentence remarking that an affine transformation is a special case of an affine map (where the two affine spaces are the same), and (if the explanation is not completely trivial) to explain this. Joel Brennan (talk) 23:51, 10 November 2020 (UTC)

Affine mapping by its action on simplex

I came across a stunning equation that allows to solve the inverse problem -- find affine mapping, when its action on vertices of a triangle are known. The post I read is this one https://math.stackexchange.com/a/3224534/673024 , but there are links to the original works where the equation appears for the first time. Is it a good idea to add something from there to this article? — Preceding unsigned comment added by 94.153.230.50 (talk) 09:34, 13 May 2019 (UTC)

Subgroups

I feel like this page could do a better job explaining exactly what subgroups are and aren't included. The intro section does say "Examples of affine transformations include translation, scaling, homothety, similarity, reflection, rotation, shear mapping, and compositions of them in any combination and sequence.", which is good, but then it gets confusing with "Viewing an affine space as the complement of a hyperplane at infinity of a projective space, the affine transformations are the projective transformations of that projective space that leave the hyperplane at infinity invariant, restricted to the complement of that hyperplane." which, while correct, is confusing since thinking about homogeneous transformation matrices, affine transformations are exactly the ones that don't involve projection (that is, that have a bottom row of [0, ..., 0, 1]). —Ben FrantzDale (talk) 13:13, 16 November 2021 (UTC)

I have reverted your edit because it is confusing, and unneeded. Just before the place of your edit, it is said that affine transformations preserve the dimension of any affine subspaces. Thus no projection can be an affine transformation, and this is not useful to add a further warning. Also, as the lead does not mention any matrix, mentioning a bottom row is very confusing. D.Lazard (talk) 17:28, 16 November 2021 (UTC)

Misleading / incorrect example

It's written

"The functions f(x) = mx + b (...) are precisely the affine transformations of the real line."

This is not in agreement with the definition given at the beginning of the article, as an "automorphism" and "a function which maps an affine space onto itself". At least if m = 0, the image of such a function is a single point. But even for m different from 0, in how does it map the real line onto itself? Yes, the image (range, for Texans) equals R, but that is also true for g(x) = x^3, which clearly isn't an affine function. So, does it mean that their graph is a different 1-dimensional affine subspace of R²? But then it is no more an automorphism which means that it should map the space onto itself. In either case its contradictory. — MFH:Talk 23:55, 1 December 2022 (UTC)

Indeed, the condition $m\neq 0$ was ommitted (now fixed).

In this example, $\mathbb {R}$ is viewed as an affine space over itself (see Affine space § Vector spaces as affine spaces), and "automorphism" is meant as "affine space automorphism"

"A function which maps an affine space onto itself" is only the beginning of the definition. The function $g(x)=x^{3}$ does not satisfy the last part of the definition after "while".

D.Lazard (talk) 12:25, 2 December 2022 (UTC)