Talk:Rigid transformation

This article was nominated for deletion on 26 May 2026. The result of the discussion was no consensus.

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On the creator's talkpage, a bot notified him that a previous submission of this article came from this website. ----moreno oso (talk) 18:26, 24 June 2010 (UTC)

Inconsistent definition

I rewrote the definition, as it was inconsistent with the following statement, also included in the introduction: "In general, any rigid transformation can be decomposed as a translation followed by a rotation". The correct definition, consistent with that statement (and consistent with the common idea that a rigid transformation represents the linear and/or angular displacement of a rigid body), is:

Definition 1: v2 = R v + t, subject to:

"R is orthogonal", and
det(R) = 1 (R is not a reflexion)

I can't exclude that, in the literature, somebody may define a rigid transformation simply as a distance-preserving transformation:

Definition 2: v2 = R v + t subject to:

"R is orthogonal" (thus, a rotation or a reflexion)

However, distance-preserving transformations include translation, rotation and reflection, as they are not subject to det(R) = 1. And this is not consistent with the above mentioned statement and idea. So, in my opinion it is quite questionable to use definition 2 for rigid transformations.

Paolo.dL (talk) 13:13, 31 January 2011 (UTC)

Merge to Euclidean group

It seems clear that this article is a (less developed) duplicate of Euclidean group. The only difference that I can discern in the meanings of the titles is that a group is more abstract: a group can be defined in terms of its action on itself (or a space), whereas a transformation may be defined as a group's action on a space. However, the Euclidean group is generally defined less abstractly as the group of isometries of a Euclidean space, which is exactly what this article is about. Hence, this article should therefore simply be a redirect to Euclidean group (after a possible merge, but I suspect that there will be no change to the destination article). —Quondum 15:04, 2 July 2018 (UTC)

Support. However, we must add to the target article that "Euclidean group" is the common term in pure mathematics, while "rigid transformation" and "rigid motion" are more common in physics, typically in mechanics. D.Lazard (talk) 15:30, 2 July 2018 (UTC)

D.Lazard Not really, the term "rigid motion" (or simply "motion") originates in synthetic Euclidean geometry as an intuitive way of defining congruence between figures, and which can be either defined rigorously in terms of congruence between segments and angles (Hilbert's axioms) or taken as a primitive concept and axiomatised (see Motion_(geometry)#Axioms_of_motion), see also ^[1]. The term "rigid motion" or "rigid transformation" (where transformation usually means any bijective map between the points of space or plane) is probably more common in a certain branch of Mathematics (synthetic geometry) than in Physics; also, the term "rigid body motion" is much more common than "rigid motion" to mean the motion of a rigid body.--Ale.rossi91 (talk) 14:46, 17 October 2020 (UTC)

Rigid transformation vs rigid motion is not conventional

Large overlap with Motion_(geometry)

This article also has a large overlap with Motion_(geometry). The three articles (this, Motion_(geometry) and Euclidean_plane_isometry) should all be merged together maybe (?). — Preceding unsigned comment added by Ale.rossi91 (talk • contribs) 17:39, 17 October 2020 (UTC)

I agree that there seems to be a clear overlap. These articles are also directly contradictory about terminology. This article says that a "rigid transformation" is any distance-preserving transformation and that "rigid motion" or "Euclidean motion" must be orientation-preserving, but the article Motion (geometry) defines a "motion" as any distance-preserving transformation, and says that "rigid", "proper", or "direct" implies orientation-preserving. Can we get a few highly cited books or survey papers to clear up this terminological confusion? @D.Lazard, I notice you made a recent edit, what do you think?

Beyond that, I think the definitions here seem too limiting, and the discussion incomplete, e.g. it's easy to talk about rigid transformations of Euclidean spaces that aren't the full plane such as a cylinder or flat torus, as well as curved spaces like the sphere or hyperbolic plane. –jacobolus (t) 17:54, 19 December 2023 (UTC)

This follows a discussion at user talk:D.Lazard#Euclidean group/rigid motion about a revert done in this article. In this discussion, I wrote This page and the related ones are a mess, with incoherent terminology, improper formulation and some kind of pedantry (why using systematically "proper rigid transformation" when "rigid motion" is correct, simpler, and closer to the intuition of the concept?). I have fixed the lead of rigid transformation, but the mess remains in the remaining of the article, and many of the related ones. My fix of the lead consists of using "rigid transformation" for possibly indirect isometries and "rigid motion" for preserving-orientation isometry (this is the terminology in Motion (geometry), and, as far as I know, the most common one, especially in kinematics. Also, I clarified the nature of the rigid motions in low dimension, and specified that the terms "screw motion" and "rototranslation" are used only in dimension three.

Another terminology issue occurs in Displacement (mathematics), where a displacement is defined as a translation vector. This may correct when rigid motions are applied to points, but seems strange when applied to solids. Maybe, I am misled by the French meaning of "Déplacement", which is "rigid motion", but this article uses (in the last sentence of the lead) "displacement" as a synonym of rigid motion.

In short a large editing and restructuring work is needed on this vital subject, but, for the moment, I am unable to propose a framework for this. D.Lazard (talk) 19:21, 19 December 2023 (UTC)

You've provided no evidence that the term "rigid motion" is always used as you imply here. It seems like you're letting your own guesses based on personal experience guide you here, rather than existing highly-cited sources or scholarly consensus. –jacobolus (t) 19:34, 19 December 2023 (UTC)

To follow-up, I should be more direct: this term is not used as you imply in the sources I can find. I get a lot of examples like:

1. We recall that the classification theorem (often attributed to Michel Chasles1), states that every rigid motion of the plane different from the identity is either a reflection in a line, a translation, defined as the product of the reflections in two parallel lines, a rotation, the product of the reflections in two intersecting lines2, or a glide reflection, a translation followed (or preceded) by a reflection in a line parallel to the direction of the translation.

2. The basic tool for the study and creation of repeated patterns is symmetry. Synonyms are rigid motion and isometIy, which are often used to emphasize the defining property of symmetry, which is that symmetry is a distance-preserving transformation of the plane onto itself. Familiar examples are rotation about a given point by a given angle, translation in a given direction by a given distance, and reflection in a given line.

3. Every rigid motion of the "plane" is of one of the following types: i) Rotations about a fixed point P; ii) Translations in the direction of a line l; iii) Reflection across a line l; iv) Glide-reflections along a line 1.

4. If a first triangle ABC is congruent to a second triangle A'B'C', then it is possible to move the first triangle and perhaps turn it over so that it will coincide with the second triangle. Such a motion is called an isometry. That is, an isometry is a rigid motion (or map or transformation) of the points in the plane. The defining property of an isometry is that it preserves distances.

5. A transformation of Euclidean n-space Eⁿ (2≤ n < ∞) which preserves a single nonzero length must be a Euclidean (rigid) motion of Eⁿ onto Eⁿ. (In this paper, "preserves a single length" means that all pairs of points in the domain whose distance is a particular specific value have the same distance between their images in the codomain.)

6. A similar analysis shows that each g in F [frieze groups] induces a rigid motion σ_g of C [a cylinder] onto itself, and the only self-mappings of C that can arise from maps in one of the frieze groups listed in Theorem 1 are [...] (1) the identity map; [...] (2) a rotation of angle π about the axis through (1, 0, 0); [...] (3) the reflection across the plane x2 = 0; [...] the reflection across the plane x3 = 0; [...] (5) a rotation about the vertical axis of the cyinder followed by a reflection across the plane x3 = 0 (a 'rotary reflection').

7. A rigid motion of a pattern is a symmetry of the pattern if the pattern remains invariant under the rigid motion. There are five symmetries possible for frieze patterns: translation, reflection in a horizontal mirror, reflection in a vertical mirror, rotation by 180 ̊ and glide reflection. The collection of symmetries of a pattern is the symmetry group of the pattern. There are exactly seven distinct symmetry groups of frieze patterns.

8. A mapping is called a motion, a rigid motion or an isometry if it leaves all distances invariant (and thus all angles, as well as the size and shape of an object).

etc. –jacobolus (t) 20:03, 19 December 2023 (UTC)

I agree that this whole topic across multiple articles needs significant reorganization/restructure, clarification, better sourcing, and so on, and could be significantly expanded. –jacobolus (t) 19:35, 19 December 2023 (UTC)

I have always understood "displacement" in English to just mean a translation. Maybe there are broader uses in some sources though. –jacobolus (t) 20:09, 19 December 2023 (UTC)

I agree that my changes are unsourced; but the previous version was unsourced either. Moreover, it was clearly inaccurate by saying that "rototranslation" is used in any dimension. For "rigid motion", I followed the version of motion (geometry). However, I am not sure that terminology is exactly the same in pure mathematics and in kinematics; similarly, it seems, from a quick Scholar Google search, that "rototranslation" is a term that is used only in computational geometry. In any case, if the most common terminology depend on the application area, this must be clear in all articles on this subject. D.Lazard (talk) 21:03, 19 December 2023 (UTC)

A search turns up a bunch of results for "3-dimensional rototranslation". The term "rototranslation" seems to be used in papers in visual perception, differential geometry, computer vision, geology, molecular physics, mathematical art, ... –jacobolus (t) 21:22, 19 December 2023 (UTC)

@D.Lazard "Displacement" or "finite screw displacement" is used in this sense at least sometimes, e.g. doi:10.1016/j.mechmachtheory.2005.04.004. I wonder if we should move displacement (mathematics) to displacement vector and make a new article at displacement (mathematics) with this broader use (or perhaps under another title). I think screw motion should be distinct from screw theory (currently redirects there), maybe obtained by moving screw axis and turning "screw axis" into a redirect.

I just combined glide reflection and glide plane, but while the lead section now seems okay, the rest of the combined article still needs a lot of work (take a look). –jacobolus (t) 06:46, 20 December 2023 (UTC)

I agree. The only reason for which I redirected Screw motion to Screw theory rather than to Screw axis is that the definition of a screw motion in the former article does not require to read the remainder of the article, which is not the case for the latter article. D.Lazard (talk) 09:01, 20 December 2023 (UTC)

Proposal: this should be merged to motion (geometry), and the first sentence should say something like

A motion, also known as a rigid motion, rigid transformation, congruence, or congruent transformation, is a geometric transformation which is an isometry, that is, a mapping from a metric space to itself which preserves distances between points. When the space is orientable, proper or direct motions are those which preserve orientation, while improper or indirect motions are those which reverse it.

In Euclidean space, motions are sometimes called Euclidean transformations. [...]

–jacobolus (t) 21:15, 19 December 2023 (UTC)

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