Talk:Shear mapping

Old question
Shear preserves all parallel lines as being parallel -- what is the name for a transform does not do this, turning a square into a trapezoid?

Re: shear matrix
I propose that this article and shear matrix be merged. Though that article was created with only two edits, it is a valuable resource on this topic. Only today did I learn of it and respond to the alert given by another editor. On the other hand, that article uses column vectors, instead of the in-line row vector approach. Since linear algebra depends on facility with both, this jump between pages gives some necessary exercise. Futhermore, the focus on two-dimensions at the outset, allows this article to be more introductory. In that view, the "fixed subspace form" section belongs more with shear matrix. Rgdboer 20:24, 4 September 2007 (UTC)

Moved here from Shear (mathematics)
Today I made the move and repaired most links in WP. If you find a reference to the old name, you're home now. The issue of row vector versus column vector continues to stand in the way of a merger with shear matrix; the value of ambidexterity being noted.Rgdboer 02:37, 14 November 2007 (UTC)

Return to row vectors
Someone using computer 147 94 187 222 located on cote d'azure near Marseille edited this page on 14 October 2008. The edit changed the matrix applications to column vectors. The editor is unidentified and gave no reason for the change; furthermore, there was some loss of sense in the text with the change. Today row vectors have been re-instated.Rgdboer (talk) 01:10, 20 January 2009 (UTC)

Matrices swapped
In the case of a shear parallel to the x-axis, the matrix is $$\begin{pmatrix}1 & m\\0 & 1\end{pmatrix}$$ and a shear parallel to the y-axis therefore has the matrix $$\begin{pmatrix}1 & 0\\m & 1\end{pmatrix}$$. This has been corrected :) (Reference: http://store.aqa.org.uk/qual/pdf/AQA-MFP4-TEXTBOOK.PDF page 18) Jambo6c (talk)
 * Thank you for your interest Jambo6c, and for your well-intended correction. However, if you check that GCE Math(6360) reference you supplied, it introduces a matrix $$\begin{pmatrix}a & b \\ c & d \end{pmatrix}$$ by saying that the first column is the image of i and the second column is the image of j. These words show that the authors have in mind the column vector application of a matrix as in
 * $$\begin{pmatrix}a & b \\ c & d \end{pmatrix}\begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix}a \\ c \end{pmatrix}$$.
 * In this article the row vector application is being used, not the column vector application as in the cited text. Therefore the horizontal and vertical descriptions were in fact correct before, so I have restored them. Again, your participation is appreciated.
 * Rgdboer (talk) 01:30, 22 March 2009 (UTC)

Changing from row vectors to column vectors
Further to the instincts of 147.94.187.222 and above, I'm changing the article from using row vectors to using column vectors.


 * Our most mainline articles use Mx → x' rather than xTM → x'T, as at eg Transformation matrix, Linear map, Euclidean vector
 * Introductory textbooks also overwhelmingly introduce transformation matrices by their effect on column vectors, rather than row vectors.

A couple of reasons might be hypothesised for this:
 * It means the matrices L, M, N for successive transformations compose in the same order as is conventional for the corresponding group operators
 * It means the transformation Mx reads from left to right in the same order as an English sentence: verb first, then object.

For these reasons, it makes more sense for the exposition here to use column vectors, rather than the mathematically equivalent row vector forms. Jheald (talk) 08:19, 19 May 2009 (UTC)
 * The reasons given for preference of column vectors do not sway me. Especially the comment on composing successive transformations is inappropriate: the composition can run left to right when row vectors are used, but when column vectors are used as input for transformation the compositions of matrices run in the opposite of reading order. Matrices are computational objects, so the order of operations is important for understanding. The other reason referring to language analogy is equally unconvincing since Subject-Verb order can be used to justify the row vector presentation. As you saw before your edit, the subject of row versus column vectors has already been discussed. Editing before seeking consensus is not appreciated. Seeking consistency through all WP articles doesn't make sense either since readers/students need to be able to navigate both presentations. This topic also arises in the discussion of function composition where one is resigned to the backward script of successive composed functions.  Matrices can be liberated from this regression by simply putting in a row vector before starting with matrix multiplication. Furthermore, since matrix multiplication is non-commutative this concern is a significant issue. Appeal to introductory textbooks as reason for use of the column vector does not address the material issues raised in this comment. I await your response so we can resolve this important format question.Rgdboer (talk) 02:14, 20 May 2009 (UTC)


 * The subject of row versus column vectors has indeed been previously discussed. As far as I can see, every single person apart from yourself has preferred column vectors.


 * The approach of introductory textbooks is strongly relevant, as this is our entry-level article on the introductory idea of shear transformations. If we can harmonise our approach we should, as a way to  make the access to this article as simple we can and minimise unnecessary stumbling blocks for the people with the least mathematical experience.


 * Finally, as you point out, this order of matrix composition matches the order of function composition. It is more helpful to rookie users to emphasise these similarities, rather than to adopt an opposite convention just for the hell of it.


 * You say you have raised "material issues" to support the use of row vectors. I have to say, in what you have written I don't see any; so if we can adopt the same convention used in key articles like Transformation matrix, in my view it is far preferable to do so.  Jheald (talk) 09:05, 20 May 2009 (UTC)


 * Thank you for your response, especially noting the level of this article and its role in introductory linear algebra learning. My primary reason for prefering the row vector convention is that is conforms with group notation. I am mystified by your comment:
 * "It means the matrices L, M, N for successive transformations compose in the same order as is conventional for the corresponding group operators."
 * Thinking through my experience with group theory, I recall the idea of an "opposite group" that is reflected in the article antihomomorphism. So far there is no article right on point, the closest being one connected to category theory. Since transformation theory is expressed adroitly with matrices, and transformations naturally begin composing, educators do well to prepare early for the consequent matrix products. The column vector convention needlessly leads students into a notational trap. Escaping the trap after capture by antihomomorphism or realization of the opposite group is sometimes necessary, but avoidance of the trap in the first instance is preferable.Rgdboer (talk) 20:49, 22 May 2009 (UTC)

Can I suggest maybe we seek additional third-party input? and both have judgment I'd respect. Jheald (talk) 21:29, 22 May 2009 (UTC)
 * See the paragraph in row vector for a statement of the preference. There you will also find a reference to Raiz Usmani (1987) where the row vector input convention has been applied with positive effect: left-to-right reading composition transformations and more compact text with row vectors.Rgdboer (talk) 23:03, 25 May 2009 (UTC)


 * &rarr; from WP:3O: Column vectors are currently the standard presentation, and should be used except in articles dealing with the differences. Pedagogically, Mv lends itself more readily to the interpretation as operators and state vectors; as well, it avoids sidetracking this article with discussion of dual and covariant spaces and introducing a minus sign under rotation. The issue of order of operations (do L, then do M, then do N to v written as vLMN vs. NMLv) is one for the sources to deal with, not this encyclopedia. - 2/0 (cont.) 15:28, 10 June 2009 (UTC)

Why not define the topic of this article correctly???
What in this article is called the "elementary" version of the shear mapping ought to be called the false version, since it is not a correct definition.

What is called the "advanced" version is the correct definition.

Instead of including a wrong definition, why not break the article up in the following way: 1) the 2-dimensional case, and 2) the n-dimensional case (n >= 2). (Actually, the n-dimensional version is valid, but trivially, in the 1-dimensional case as well.

(What is false about what the article calls the elementary version is that in an actual shear transformation, there is nothing whatsoever saying that the fixed subspace need be parallel to an axis.)Daqu (talk) 10:24, 26 November 2009 (UTC)
 * Thank you Daqu for commenting on the definition. By way of explanation, consider the group theory of planar mappings including rotation. All the shear mappings in the plane, including ones with invariant line not parallel to an axis, are in a conjugacy class represented by the shear mappings of this article's definition, where the subgroup of rotations provides the conjugation. While your more general idea of a shear mapping is valid, the necessity of providing a definition that can be easily understood means that the presumption of an axis serving for the invariant line is used. The use of a rotation to orient the general shear vertically or horizontally does not materially affect the meaning given.Rgdboer (talk) 20:21, 1 December 2009 (UTC)


 * Nothing can justify a false definition.  (I can't believe I need to say that.)  The question of pedagogy in an article is *also* an important one, but that is no reason to write a false statement.
 * Pedagogy is, however, a good reason to give simple examples. If you feel that having a *coordinate axis* be an invariant subspace is a good simple example, then by all means, mention it -- as an example.
 * Pedagogy is always a reason to write as clearly as possible, but it never justifies a false statement.Daqu (talk) 16:46, 10 December 2009 (UTC)


 * Please bring forward a reference for the correct definition. You must have something in mind, do you have a source with the same idea? Rgdboer (talk) 02:06, 12 December 2009 (UTC)

An arrowhead is up the Mona Lisa's nose!
Does ML's nose really need to be picked in an article about math? —Preceding unsigned comment added by 112.200.186.149 (talk) 12:30, 15 April 2011 (UTC)
 * Agreed, the image was distracting. Replaced it with one from matrix (mathematics) where the mesh demonstration is used for other transformations.Rgdboer (talk) 23:07, 15 April 2011 (UTC)

Nonsense
The following was removed today:
 * This mapping describes the motion of the fluid in a Couette flow, and the displacement of particles in a solid material under shear strain (which is the deformation that occurs when a metal sheet is being cut by shears, hence the name).

Such deception is to be deplored in this project.Rgdboer (talk) 01:12, 28 January 2015 (UTC)

Italic Type or Oblique Type?
Regarding the following:
 * The italics text can be thought of as normal text under a shear.

I think it would be more accurate to describe oblique, not "italic", type as having been sheared, as true italic type uses unique glyphs not producible by shearing. See Italic_type. Oblique type is often used as a surrogate for italic when italic glyphs are unavailable. Iddr (talk) 09:54, 6 April 2016 (UTC)

Horizontal or vertical shear?
The current description in the article would suggest that this is a horizontal shear. But the image name and description call it vertical. Is there some ambiguity here? Otherwise I would request to rename the image (and my derivate of it). Watchduck (quack) 00:27, 8 July 2022 (UTC)

Proposed merge of Shear matrix into Shear mapping
This was suggested more than 15 years ago by, and I think it was correct: there is not a true separation in the study of the two objects, they are just two manifestations of a common thing. To the extent that there are matrix-specific things to say (e.g., concerning elementary matrices) I don't see why they can't fit into a single article (neither one is very long). JBL (talk) 18:39, 23 May 2023 (UTC)


 * Support: The matrix representation of a shear mapping is better understood in the context of the mapping it represents, and indeed neither article is too long. Felix QW (talk) 16:06, 24 May 2023 (UTC)
 * @JBL – I say go for it! –jacobolus (t) 23:23, 5 August 2023 (UTC)
 * I very much agree on this also - I had assumed that the matrix was already in the Shear Mapping category, (as the matrix is an expression of the mapping) and only came across the other by accident when searching the site. Blayzeing (talk) 16:40, 13 October 2023 (UTC)