# Talk:Affine transformation

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Field: Geometry

## Rtimes

Help, Wikitex doesn't support \rtimes! How are we ever going to write semidirect products here? Phys 19:32, 13 August 2003

## Line Preserving

"In a geometric setting, affine transformations are precisely the functions that map straight lines to straight lines." Don't projective transformations also map straight lines to straight lines? 128.163.141.13 21:09, 22 June 2007 (UTC)

## Example

Why do you have to use such a strange and confusing example? What about transformations with 3 or even only 2 dimensions? IMHO they will be understood by more people. -- KL47 23:40, 3 Dec 2004 (UTC)

It's the affine transform that Rijndael uses for its S-box. See Rijndael S-box or AES. Samboy 08:04, 18 May 2005 (UTC)

## affine isn't necessarily invertible?

The article says "The set of all affine transformations forms a group under the operation of composition of functions." Don't you have to specify invertible affine transformations? For example, projection of $R^3$ to $R^2$ is affine, but not invertible. Or have I misunderstood something?

You're right; I've added the word invertible. Michael Hardy 01:09, 6 Feb 2005 (UTC)

## Yes, simpler example needed here

I'm thinking this needs one more example specific to 2D affine transformations, maybe with a link to iterated function system fractals (where they're very commonly used) and to chaos game. I'll work on a good simple example for a chaos-game article, too, with pseudocode (it only takes about half a dozen lines!) and a few external links.

As a segue out of the "Affine transformations of the plane" section, it would make a lot of sense to give the equations for the 2D case -- which anyone with a good solid high-school algebra background can understand:

X' = Ax + By + C
Y' = Dx + Ey + F


If you know the locations of three (noncollinear) points before and after the transformation, there's a unique solution for real numbers A, B, C, D, E, F -- the problem amounts to solving two sets of three equations with three unknowns each.

Here's [a page I wrote up] some time ago about planar affine transformations as related to mapping applications -- I work for MapInfo, which regularly uses affine transformations to 'warp' vector information onto background raster images. Approximate and ugly sometimes, but very very fast. More MapInfo-specific writeups are here and

Interesting that "Affine transformations of the plane" specifies a parallelogram as the basic unit defining an affine transformation. I always visualized everything in terms of triangles... but of course a parallelogram is just a triangle with a fourth point defined by the other three.

Suggestions would be appreciated for what would/would not be reasonable to add to this article -- I'm new at this game --dvgrn 10:36, 13 Apr 2005 (UTC)

## The "vector" B

I am 99% sure this is a point, not a vector.

B is the translation vector, it translates coordinates it is applied to. Also note that vectors can denote points, so this comment did not make much sense to me. --62.163.111.132 (talk) 21:12, 15 January 2008 (UTC)
Strictly speaking, a point in an affine space is different from a translation vector between two points. Points don't have a "zero" whereas vectors do. This distinction is often overlooked. —Ben FrantzDale (talk) 14:10, 7 February 2008 (UTC)

As far as I can tell, the Arabic interwiki link pointing here is for "Linear transformation." --Urdutext 23:53, 8 January 2007 (UTC)

## "Affine function"

Affine function redirects here. It is mentioned on Karush–Kuhn–Tucker_conditions#Sufficient_conditions in contrast with convex function. This page doesn't include the string "affine function". Could someone explain that meaning? —Ben FrantzDale (talk) 02:18, 29 May 2008 (UTC)

## Relationship to Computer Graphics and Projective Geometry?

Much of the world's computer graphics software represents affine transforms in n dimensional space by linear transforms in n+1 dimensions. The n+1 space is treated (i think) as a projective space. Is it worth mentioning here? I'm too green to know. 192.55.12.36 (talk) 15:34, 2 June 2008 (UTC)

## Affinely dependent

Similarly they are affinely dependent if the same is true and also

$\sum_{i=1}^n a_i = 1$

I think this is incorrect. I don't have a book, but here is a sample from lectures:

An affine dependence between points ... is a linear dependence ... (Thus, in an affine combination, the coefficients sum to 1, while in an affine dependence, they sum to 0.)

Beroal (talk) 12:48, 11 November 2008 (UTC)

## Linear Manifold vs. Affine Subspace

Many authors don't use linear manifold to denote an affine subspace. For example, in "Intro to Hilbert Spaces" Halmos uses linear manifold to mean a linear subspace of a Hilbert space (pg. 21). The distinction here is that a linear manifold is not necessarily closed, and thus not necessarily a subspace in the Hilbert space sense. I think Dudley uses this definition also in "Real Analysis and Probability".

I suggest removing the (at best controversial, at worst erroneous) definition of linear manifold in this article and instead making linear manifold redirect to the topological vector space article.

Bradweir (talk) 02:43, 22 August 2009 (UTC)

## Names of A and b?

What are the textbook names of A and b in: $x \mapsto A x+ b\,$? In Portuguese textbooks, they are called by "coeficiente angular" e "coeficiente linear", which could be translated as "angular coefficient" and "linear coefficient". In Excel, they are called "slope" and "intercept". Albmont (talk) 12:49, 14 January 2010 (UTC)

## Two symbols for xor

Is there a reason to use two different symbols for xor in this article? -Craig Pemberton 07:35, 3 February 2010 (UTC)

So '+' and (+) both are used to mean XOR? This was very confusing. The use of (+) with no explanation needs to be fixed. Don't possess the knowledge to do so myself —Preceding unsigned comment added by 72.177.191.195 (talk) 18:30, 20 February 2010 (UTC)

I'm not sure about anyone else, but it seems to me that (+) seems to be the more accepted symbol for XOR. Why not just use that? Burningstarfour (talk) 02:52, 27 April 2011 (UTC)

## The example

I don't think a 7x7 matrix under a particular definition of addition really serves as the best example to illustrate what an affine transformation is, when the geometric interpretation is the only one most people will ever see. Maybe if someone has the time, just draw up a simple example with a shear and reflection, followed by a translation, in R^2, to give an illustration that is approximately similar to the visual of the fern at the top of the page. Charibdis (talk) 00:52, 25 May 2010 (UTC)

As mentioned above, the particular 7x7 matrix given is the Rijndael S-box, used in the AES cryptosystem. However, your point is well-taken; a geometric example would, in my opinion, be helpful. If I have time in the next few days I'll write one up. --Burningstarfour (talk) 02:56, 27 April 2011 (UTC)

## Huh??

Is this really an arena of human knowledge that defies a simple, straightforward, plain-language explanation? Really? Look, I'm not a mathematician, but I nonetheless would like to know what an Affine transformation is, but I have no more idea now than before visiting this page. — Preceding unsigned comment added by 66.192.126.3 (talk) 17:39, 28 October 2011 (UTC)

You are right. It needs a simple explanation. It's something like "a transformation in which straight lines remain strait". That is, a linear transformation along with a translation. —Ben FrantzDale (talk) 17:42, 28 October 2011 (UTC)

Please cf. Berger's book on page 38

In fact, one can read: The conclusion is that, heuristically, $f\,$ consists of a translation and a linear map. But I agre with you: it is not well-written.

Another definition is: Given two affine spaces $\mathcal{A}$ and $\mathcal{B}$, over the same field, a function $f:\mathcal{A}\to\mathcal{B}$ is an affine map if and only if for every family $\{(a_i,\lambda_i)\}_{i\in I}\,$ of weighted points in $\mathcal{A}$ such that $\sum_{i\in I}\lambda_i=1\, ,$ we have

$f\bigl(\sum_{i\in I}\lambda_i a_i\bigr)=\sum_{i\in I}\lambda_i f(a_i)\, .$

In other words, $f\,$ preserves barycenters. Mgvongoeden (talk) 19:27, 28 October 2011 (UTC)

## Planar mapping

Today the section on affine mapping of the plane was revised. Reference to eigenvalues of matrix A were removed as this link involves the reader in linear algebra that is unnecessary and perhaps beyond comprehension. The case of the plane lends itself to exhaustive treatment because 2 x 2 real matrices are well-known so all types of affine transformation can be described.Rgdboer (talk) 21:56, 14 August 2012 (UTC)

## Preserve parallels

Affine transformations respect the relation of parallel lines. This fact appears in major references. The term dilation is commonly used. Today's edits revise the lead paragraph accordingly.Rgdboer (talk) 23:20, 20 August 2012 (UTC)

"The term dilation is commonly used." Really? I have never heard this term used this way. A rotation is a kind of dilation? Do you have a source that uses the term "dilation" like this? —Keenan Pepper 21:41, 28 September 2012 (UTC)

## 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)