Talk:Conformal linear transformation

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Meaning of words

“The basis matrix” is a nonstandard, unexplained phrase. “The basis” needs a referent to have any chance of making sense (whence cometh a basis? this list is allegedly a list of properties of a particular kind of matrix) but I suspect the problem goes deeper. Is this just a pile of WP:OR? Please find a decent source and use it to replace this incoherent list with something that uses standard terminology correctly. 100.36.106.199 (talk) 12:34, 20 January 2024 (UTC)

"Basis" is standard terminology in linear algebra. See Basis (linear algebra), which is linked on the page in the very sentence you complain about. A basis matrix is a matrix that is a basis. Please read up on the subject before proposing that something is wrong with the page. Aaronfranke (talk) 06:24, 5 February 2024 (UTC)

The phrase “a basis matrix is a matrix that is a basis” is meaningless gibberish. I am sorry that your grasp of linear algebra is so poor that you do not understand that a rectangular array of numbers and a set of elements of a vector space are different kinds of objects, but really until you recognize the incoherence of what you’ve written (e.g., by comparing it directly to what reliable sources say) you should not be editing this article (or any other article on mathematics). 100.36.106.199 (talk) 13:02, 6 February 2024 (UTC)

I have reported your behavior here: https://wiki.riteme.site/wiki/Wikipedia:Administrators%27_noticeboard/Edit_warring#User:100.36.106.199_reported_by_User:Aaronfranke_(Result:_) Aaronfranke (talk) 08:09, 8 February 2024 (UTC)

For what it's worth, I think this report was inappropriate, both of you were violating policy by revert warring instead of discussing, both of you were using inappropriately aggressive language (I do this also sometimes, so this is not intended too harshly) and the IP editor should be unbanned. The IP editor's concerns were/are valid even if their language was dismissive/insulting. –jacobolus (t) 20:11, 8 February 2024 (UTC)

I reported 100.36.106.199 because I knew the discussion was becoming heated and I could not reasonably be expected to give an unbiased response to someone calling my words "meaningless gibberish" and telling me to stop editing any articles on mathematics. By asking someone else to step in, I was avoiding escalation. My actions were obviously suboptimal, I could have been more cordial, I should not have edit warred beyond a single revert, and I apologize for using the word vandalism. But overall I don't think my actions were unreasonable. If I had waited to report until later, things may have gotten worse. Aaronfranke (talk) 05:46, 9 February 2024 (UTC)

If you ever run into local trouble resolving a dispute about a new or obscure mathematics-related article, you can always feel free to find more eyeballs at Wikiproject Mathematics talk, where more watchers will see it. For other ideas, see Wikipedia:Dispute resolution. Anyway, hopefully we can all stay amicable from here forward. –jacobolus (t) 06:42, 9 February 2024 (UTC)

I too am confused by the term "basis matrix", despite knowing linear algebra including bases, matrices, change of basis, etc. Does it mean that the columns of the matrix form a basis? But apparently squareness is also required. So the matrix is invertible? Mgnbar (talk) 12:31, 8 February 2024 (UTC)

Also I don't understand the other phrases under dispute, such as "The basis may be composed of rotation." Please clarify all of the content that 100.36.106.199 was complaining about. Mgnbar (talk) 14:00, 8 February 2024 (UTC)

Presumably he means that the transformation can be expressed as a composition of dilation, reflection and rotation. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:26, 8 February 2024 (UTC)

Yes, that is precisely it. I am confused as to how "may be composed of" is misunderstood but "can be expressed as a composition of" is clear, these seem like differences in wording that convey the same concept to me. Aaronfranke (talk) 05:30, 9 February 2024 (UTC)

The sentences now in the article fully specify the kind of decomposition being used: “as a product of an orthogonal transformation and a uniform scaling”, “as a product of n reflections”. Without specifying the full decomposition considered, it does not make sense to talk about whether things can or cannot be decomposed. For example, uniform scaling can be written as a composition of multiple non-uniform scalings—so conformal maps can be written as product in which some of the factors are non-uniform scalings. If you want to say that that’s not allowed, you need to explain clearly what the rules of the game are. 100.36.106.199 (talk) 12:21, 9 February 2024 (UTC)

That makes sense, my wording there was suboptimal. To clarify, I was trying to list operations that could cause the output to be non-conformal, and the required degrees of freedom after decomposition. (The rest of my reply is in the context of 2D) Scaling by (1, 2) then (2, 1) results in it being conformal, yes, but then I would decompose that back as a scale of (2, 2). When I write about high-level transform decomposition I am thinking of filling in one set of blanks of "rotation = ???, uniform scale = ???" which can be losslessly composed again to fully describe the original matrix iff the matrix is conformal. Similarly, a matrix with orthogonal vectors would have "rotation = ???, scale = (???, ???)" as its set of blanks that that need filling in, which can be converted back losslessly. There is only ever one set of scale numbers in this decomposition, so scaling by "(1, 2) then (2, 1)" becomes "(2, 2)" and never "(1, 2) then (2, 1)". Aaronfranke (talk) 17:35, 9 February 2024 (UTC)

This kind of thing (you have a bunch of implicit assumptions about what you’ve written, and understanding what those assumptions are is necessary to assign meaning to the words) is the reason that I slapped a bunch of “clarify” tags on things: the text by itself was completely unclear. Maybe in the future you can consider this as a realistic possibility — I would have been happy to workshop wording with you. (The best way to say this particular thing is just to say that nonuniform scalings are not themselves conformal. FWIW I don’t think this parenthetical description is completely right, either — it assumes that all the scalings share the same eigenbasis, or equivalently that they all commute with each other.) 100.36.106.199 (talk) 12:20, 10 February 2024 (UTC)

The phrase "a basis matrix" makes sense, but the phrase "the basis matrix" does not. Earlier, the phrase high-level transform decomposition has no obvious meaning. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:26, 8 February 2024 (UTC)

Sorry, but what is "a basis matrix" even? Do you mean a matrix that can be a member of a basis for the (n m)-dimensional vector space of m x n matrices? That would be any non-zero m x n matrix. So you must mean something else. Do you mean a change-of-basis matrix? Mgnbar (talk) 16:32, 8 February 2024 (UTC)

Presumably a transformation from a preferred basis to a different basis, but even if my guess is correct the text should spell it out. If he doesn't mean a change of basis matrix then I'm completely in the dark. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 19:26, 8 February 2024 (UTC)

I apologize for any confusion, please see my long reply below and let me know if that clarifies things. Aaronfranke (talk) 05:26, 9 February 2024 (UTC)

I apologize for any confusion, in the contexts where I work in, "high-level transform decomposition" is common terminology. The intended meaning is that operations like "rotation" and "scale" are high-level, as opposed to the raw numbers stored in a 2x2 matrix, 3x3 matrix, etc representing a linear transformation. Aaronfranke (talk) 05:26, 9 February 2024 (UTC)

The article has gotten a lot better in the last day, thanks everyone (especially jacobolus). The section Conformal linear transformation#Two dimensions is still written in a private language and it should be converted to something that can be understood by someone other than its author. Obviously the other huge problem is the serious lack of sources at present. 100.36.106.199 (talk) 02:53, 9 February 2024 (UTC)

100.36.106.199: Let's stay polite, please. I'm happy to keep going a bit on the part after the lead, and let's try to figure out together what topics this article should cover and how they should be organized. –jacobolus (t) 03:39, 9 February 2024 (UTC)

Apologies. This was meant as descriptive not normative: there’s a particular kind of failure to use non-standard language that the authors don’t notice (often because they have a detailed set of models, expectations, etc. that are not spelled out in the text) that is very common in mathematics writing (especially less experienced authors; but I’ve gotten as far as submitting papers to journals that had this problem a couple of times). 100.36.106.199 (talk) 12:05, 9 February 2024 (UTC)

Name of the concept

This concept is valuable and under-applied, but I don't think the name "conformal linear transformation" is a good one for it. The name seems relatively rare in practice and not really in currency (edit: after searching there doesn't seem to be any particularly common name for this). I don't think primarily focusing on these transformations being "conformal" is at all helpful, since the geometry of >2-dimensional conformal transformations (Möbius transformations) is entirely different, and the 2-dimensional case (where the concept of "conformal" originated) dramatically more different still. What about a name like Tristan Needham's "amplitwist" or similar? If I needed a descriptive phrase I would go for something like "origin preserving similarity transformation". A few sources seem to use the names "homogeneous similitude" or "homogeneous similarity transformation", which seem like okay names.

One of these transformations is most naturally represented using geometric algebra, as a versor (product of invertible vectors). Sandwich multiplying a vector $v$ by a versor $A$ like $AvA^{\dagger }$ where $A^{\dagger }$ is the reverse of $A$ applies the transformation. So hunting for sources discussing this may turn up other names for the concept. The group of transformations is called the versor group or Lipschitz group. –jacobolus (t) 17:12, 8 February 2024 (UTC)

I'll note that we have the article conformal group, which is also not in a good state. (Also the article Möbius group is limited to two dimensions.) I'm not convinced the present article needs to exist, but some clarity in adjacent articles seems more pressing. Tito Omburo (talk) 00:29, 9 February 2024 (UTC)

Neither conformal group nor Möbius group (which are as far as I can tell the same thing) is the same as these orthogonal-transformation-composed-with-dilation transformations. I think this concept is worth having a separate article about, but it should be clear and clearly sourced.

The Möbius transformation article is very problematic because it mixes up the geometric transformations generated by reflections and sphere inversions (the actual subject of the term "Möbius transformation") with the particular representation of 2-dimensional Möbius transformations as fractional linear transformations of the complex projective line. I'd like to eventually fix this but it's going to take significant work. –jacobolus (t) 00:38, 9 February 2024 (UTC)

First definition in the conformal group article defines the "conformal orthogonal group" to be as in this article, but there is a lack of clarity as to the subject there, which overlaps with this article and Möbius. Tito Omburo (talk) 01:18, 9 February 2024 (UTC)

I don't thin "conformal orthogonal group" should be covered at the article conformal group. It's a big mistake to mash two largely unrelated topics together. –jacobolus (t) 01:29, 9 February 2024 (UTC)

Fwiw, I think this article should be renamed to conformal orthogonal group and generalized with sources. That seems to me like a much more common name, and then conformal group can be more satisfactorily scoped. Tito Omburo (talk) 11:57, 9 February 2024 (UTC)

I apologize if I seem like I'm introducing new terminology. Let me take a step back and explain my purpose for starting this article. I am not a mathematician. I am a software developer who works on game engines. In the context of physics simulations in game engines, it is helpful for physics objects to be "non-distorted". This is often accomplished by requiring that an object may only be translated and rotated as in a rigid transformation. However, it is often allowed to scale physics objects uniformly, since this does not "distort" the object, spheres remain spheres, angles remain similar, distance ratios are preserved, etc.

I was looking for a term to describe this and I came across Conformal map, which is a greatly helpful concept with a nice name so I ran with it, but it is too general, as evidenced by the example image in the Conformal map article which showcases a conformal map that preserves angles but not distance ratios. I needed a term to describe a transformation that allows translation, rotation, and uniform scale, but not non-uniform scale, skew, shear, etc, so I decided to start this article. After looking into this closer I believe that I may have been using the term "linear transformation" inaccurately, as it seems that this does not allow translations? A better term to describe what I was trying to convey may be "Conformal geometric transformation", or perhaps somehow relating to Similarity (geometry).

The way that this page has evolved has moved beyond my original goal of describing this operation since it now explicitly disallows translation, and I would appreciate assistance with creating a separate page that describes the concept I have described above, including selecting an ideal title and mathematicians writing the precise correct terminology that my non-mathematician mind does not contain. At first I believed that it was one user's mistake reading my words, but given this discussion it is clear that I am incapable of explaining myself using the correct terminology at the level of Wikipedia's standards and I need help from a mathematician.

I really hope that I can clear up any confusion my words have caused, in my mind they are not confusing. Above it was mentioned that "a basis matrix" is confusing terminology. Let me clarify, I am referring to this thing described at 6:20 in this 3Blue1Brown video. In this video he describes a linear transformation basis and then says "it's common to package these coordinates into a 2x2 grid of numbers called a 2x2 matrix". So... this is a 2x2 matrix, which encodes a 2D basis as numbers, therefore I am calling this a basis matrix, it is a matrix that contains numbers encoding a basis, or a basis encoded as numbers in a matrix. See also Godot Basis "A 3×3 matrix used for representing 3D rotation and scale. Usually used as an orthogonal basis", Bevy Mat2 and Mat3, etc. I can understand if this is not the normal term in mathematics contexts (as I am coming at this from computer science) but am bewildered at how the concept was not conveyed by my words.

As a final note for this opening message, I am utterly dismayed that my words have been continually dismissed as nonsense for minor mistakes or grammatical errors, like "a basis" vs "the basis". I assure you the concepts I am attempting to describe are real things, I would appreciate not being attacked for minor mistakes. Hopefully this message has helped clarify things. I am assuming that most users here are smart people, please try to understand what the intentions behind the words are and that not everyone who uses advanced math has a PhD in mathematics. Lots of people get the bulk of our math knowledge from 3Blue1Brown or Freya Holmér instead of a university or research paper. Aaronfranke (talk) 05:20, 9 February 2024 (UTC)

If you want to allow translations, then (a) what you are looking for is a transformation of a Euclidean affine space, not a Euclidean vector space (that is, the objects you want to transform are "points" rather than "vectors"), and (b) the transformation you are looking for is called a similarity transformation or similitude, and these are described at Similarity. Arguably the idea should be split into a separate article, in the same way we have separate articles for Congruence (geometry) and Rigid transformation (and in greater generality, Motion (geometry)). –jacobolus (t) 06:44, 9 February 2024 (UTC)

Yes, makes sense to me. I apologize for not knowing the distinction between an affine space and a vector space. I assumed that points are vectors and therefore a space with points is a vector space. It is extremely common for math educational materials to mention the interchangeability of points and vectors. Aaronfranke (talk) 07:02, 9 February 2024 (UTC)

Let me also say: Aaronfranke, sorry for any insults and miscommunication you've been facing here. Miscommunication is frustrating on all sides, and we should all strive to be open-minded, leave pedantic nitpicking aside in talk pages, and interpret each-others actions and words in the most generous possible light. Thank you for contributing to Wikipedia; I hope this experience doesn't discouraging you from trying to make improvements (including creating new articles) going forward.

In general most of our articles about particular geometric transformations are disorganized, incomplete, parochial (in the sense of privileging the interpretation/terminology/outlook of one or another niche subfield rather than giving a clear overview), sometimes contradictory, sparsely illustrated, weakly sourced, and could use a lot of help. (Much like articles about many other technical topics on Wikipedia.) –jacobolus (t) 07:07, 9 February 2024 (UTC)

We also have splits like Orthogonal group (where this article is now headed) versus Orthogonal matrix (probably closer in spirit to AF’s original conception). One could resonably try to describe the concrete and abstract under one roof, instead. 100.36.106.199 (talk) 12:16, 9 February 2024 (UTC)

That particular split makes more sense. For the present topic, I'm not really convinced that "conformal orthogonal matrix" is an independent topic from "conformal orthogonal group", or whatever the ultimate destination is. Tito Omburo (talk) 13:58, 9 February 2024 (UTC)

Sometimes mathematicians don't make a clear distinction between e.g. 2-dimensional real coordinate space, the Euclidean plane, a 2-dimensional Euclidean vector space, a generic affine plane, a generic 2-dimensional vector space, the complex plane, the inversive plane (a.k.a. Möbius plane), etc. These are all conceptually different, but depending on the context it can be convenient to conflate two or more of them. Personally I don't like this because I think it leads people (especially students and newcomers, but sometimes also experts) to significant misconceptions. But if you start e.g. programming in Matlab you'll get used to routinely mixing and matching them. –jacobolus (t) 07:15, 9 February 2024 (UTC)

A note specifically about affine space and translations: the affine transformations on n-space can be represented by matrices (as you noted in an edit summary at some point?) but in this representation the n-space is embedded in an (n+1)-space (as the vectors having last entry 1) and the matrices in question are (n+1)-by-(n+1). So this is a thing, but it’s not compatible with the representations that were explicitly discussed before (in which the matrices on n-space were n-by-n). 100.36.106.199 (talk) 12:13, 9 February 2024 (UTC)

The phrase "the vectors having last entry 1" does not make sense to me. Does it mean that the last component of all vectors is set to 1? Why? As for representing affine transformations with matrices, yes you can write a 2x3 matrix for a 2D transform, a 3x4 matrix for a 3D transform, etc, encoding the translation as the last column. This is why I specifically wrote "basis matrix" (meaning a square n-by-n matrix only representing a rotation/scale/etc basis and no translation) and "basis of a matrix" (meaning the left-most n-by-n square inside of a matrix with translation, the part only representing a rotation/scale/etc basis and no translation) earlier. By "the vectors having last entry 1" I am guessing you are referring to padding out the matrix with the bottom row as (0, 0, 0, 1) to make it a square matrix again, but only the translation column has the last entry set to 1, not any other vectors. Aaronfranke (talk) 17:18, 9 February 2024 (UTC)

@Aaronfranke You might be interested to read a book about matheamtical tools for computer graphics. The typical representation used in 2D and 3D graphics is homogeneous coordinates, where an extra coordinate is tacked onto the end, but any scalar multiple of the same vector is considered to represent the same "projective" point. This allows arbitrary projective transformations (including affine transformations) to be represented by linear transformations of a space 1 dimension higher (for general projective transformations, you need a rule that you always normalize vectors to have a 1 in the last coordinate before you try to draw them). Then you can use linear transformations (matrix multiplication) to handle translations as well as rotation and scaling; that's where "encoding the translation as the last column" comes from. This is convenient because we have computer hardware that is very good at matrix multiplication: GPUs are have been optimized to do 4x4 matrix multiplications very very fast, expressly for the purpose of transforming homogeneous coordinates representing 3-dimensional space. –jacobolus (t) 17:49, 9 February 2024 (UTC)

This is explained at Transformation matrix#Affine transformations fwiw. 100.36.106.199 (talk) 12:07, 10 February 2024 (UTC)

I'm sorry that you have been "utterly dismayed" by other editors' confusion. Speaking only for myself, my confusion was not overstated. I literally could not understand your text (despite being a mathematician who, coincidentally, has written 3D graphics engines). Anyway, I'm glad that you have clarified your background and goals, and that the article is proceeding. Mgnbar (talk) 14:49, 10 February 2024 (UTC)

Please do not conflate distinct editors. Nothing in The phrase "a basis matrix" makes sense, but the phrase "the basis matrix" does not. Earlier, the phrase "high-level transform decomposition" has no obvious meaning. is remotely close to claiming that what you wrote is meaningless gibberish, and, in fact, I suggested a possible meaning. When an editor says something that you disagree with, and you want to be taken seriously, identify the claim or the editor, not an anonymous "they". -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:52, 12 February 2024 (UTC)

I don't know what you are referring to, I'm not conflating editors. User 100.36.106.199 said 'the phrase “a basis matrix is a matrix that is a basis” is meaningless gibberish'. When I responded and quoted "meaningless gibberish", I mentioned 100.36.106.199 earlier in the same sentence. It is 100.36.106.199 both times, I am not conflating anyone. Actually, I have not even used the word "they" to refer to a person once anywhere on this talk page. Aaronfranke (talk) 18:51, 12 February 2024 (UTC)

I am referring to s a final note for this opening message, I am utterly dismayed that my words have been continually dismissed as nonsense for minor mistakes or grammatical errors, like "a basis" vs "the basis".; I'm the one that mentioned the inappropriate use of the definite article, and I never described that as "meaningless gibberish." There is no reference to 100.36.106.199 in or prior to the sentence I quoted. I suspect that you're thinking of an earller discussion, prior to the fork. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 20:06, 12 February 2024 (UTC)

I wonder if we can step back, chalk this one up to frustration and mild miscommunication, and then leave it be. This point doesn't seem worth belaboring. It's clear Aaronfranke felt like they were under fire from multiple directions, but also seems like other editors didn't really intend any personal insult.

@Chatul (and everyone else here) do you think it would work to merge this topic into a new article about similarity transformations in general, as I proposed below? If so, how should be structured and what should it include? –jacobolus (t) 21:47, 12 February 2024 (UTC)

Do you see a need for separate 2D and general articles? -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 22:24, 13 February 2024 (UTC)

I think we could make an article called Similarity transformation, which could cover the general case, basically as an expansion of Similarity (geometry)#In Euclidean space to a dedicated article. Currently Similarity transformation is a disambig page; we could take the title over and put a hatnote from there to Matrix similarity. Then we could edirect the name Conformal linear transformation to a section of it. –jacobolus (t) 23:03, 13 February 2024 (UTC)

I see what you mean now, and I apologize, but you put it in a really confusing way. I did not say "meaningless gibberish" to you, only "nonsense". I did not use the word "they", I mentioned "have been continually dismissed" without a subject. Please do not quote my words if you are going to change my words, my natural assumption is that if you quote me, you are quoting me exactly. Aaronfranke (talk) 16:42, 13 February 2024 (UTC)

All of the text inside {{tq}} and {{tqq}} was cut-and-paste from your edits; it doesn't get more exact than that. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 22:24, 13 February 2024 (UTC)

A different possibility: what if we make a dedicated article about similarity transformations, separate from Similarity (geometry)#In Euclidean space, and then merge this topic into one section there? –jacobolus (t) 16:04, 9 February 2024 (UTC)

Surprising that we don't already, but this seems like the best solution thus far. Tito Omburo (talk) 17:32, 9 February 2024 (UTC)