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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.
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 . 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
Rigid motion, rigid transformation and (Euclidean) isometry are usually treated a synonyms; a wikipedia article should only state established conventions and not establish new ones, even if they are consistent within the article. This may in fact cause confusion in people who have found this article looking for a specific information.
Also, the definition stated in the article is inconsistent with that of the reference (which is about the term "rigid transformation" not "rigid motion"), which states
Rigid transformations will consist simply of rotations and translations.
Furthermore, the fact that one of the main contributors of this page User:Prof McCarthy, is also the author of the only (self-published) reference which states this convention is rather unusual and raises an issue of neutrality.
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)