Talk:Affine transformation

Archives

This page has archives. Sections older than 365 days may be automatically archived by when more than 10 sections are present.

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

Dilation?

[A late reply to now-archived § All affine transformations are NOT dilations]

As you say that dilations are not the only kind of affine transformations, I'd suggest to do the following:
a) mention dilations in this article and have a link
b) figure out whether the picture https://en.wikipedia.org/wiki/File:Central_dilation.svg makes any sense in this article. The picture is also weird, theres's no "triangle A1C1Z" in the picture, so what's the role of it. I'd suggest to remove this picture; the table of tranformations above it is probably enough.
Vlad Patryshev (talk) 12:24, 8 May 2025 (UTC)[reply]

This is usually useless to answer to a twelve-years-old post.

The only mention of "dilation" in the article is in the caption of the figure of § In the plane. I have replaced it by homothety.

Section § In the plane had many issues. I fixed some.

Sure that only two edges of the triangle A1C1Z are drawn, but this does not mean that it is not in the figure. As it is incompletely drawn, I pushed it after the two other triangles. D.Lazard (talk) 15:09, 8 May 2025 (UTC)[reply]