Talk:Euclidean group

Inversion with respect to a point in 3D

How come "inversion with respect to a point" is "not preserving orientation" in 3D? BTW what it means to preserve orientation in 3D? --TMa

Improvements

I've done some work on the ordering of sections, and other tweaks. It shouldn't be too hard to put this into approved 'concentric' style. Charles Matthews 15:35, 13 October 2006 (UTC) OK, that should be somewhat better now. The only point of real concern I have is this: does the article really need the non-closed subgroups enumerated? I would have thought the closed subgroups were enough. Charles Matthews 15:52, 13 October 2006 (UTC)

If the overview is restricted to closed subgroups this has to be mentioned, you cannot say the subgroups are all of type A, B, or C, when there is also a type D. However, to clarify the restriction you have to explain it, so you end up briefly explaining the additional kind anyway.--Patrick 22:08, 13 October 2006 (UTC)

I don't agree: if it is thought of as a topological group, why not just explain the closed subgroups? I don't see the need for any more than that. Charles Matthews 12:23, 14 October 2006 (UTC)

I was thinking of the algebraic concept of a group. The concept of a topological subgroup seems more complex.--Patrick 23:39, 14 October 2006 (UTC)

Notation special Euclidean group : SE vs E+

In this article the notation $E^{+}$ is used for the special Euclidean group, but SO for the special orthogonal group (the rotations). I would prefer SE for the special orthogonal group thus making the notation consistent.

--Benjamin.friedrich (talk) 22:25, 2 February 2008 (UTC)

I guess you mean I would prefer SE for the special Euclidian group thus making the notation consistent —Preceding unsigned comment added by 128.178.150.69 (talk) 17:10, 6 October 2008 (UTC)

The Euclidean group E(3) as a matrix Lie group

When actually working with the E(3), it is very useful to write the elements of E(3) as matrices; this is done using homogenous coordinates.

I am unsure where to include information about homogenous coordinates: Possible options are the articles on rigid bodies, rigid body motion and the Euclidean group.

Here is a start for a section on homogenous coordinates (source: A Mathematical Introduction to Robotic Manipulation by Richard M. Murray, Zexiang Li, S. Shankar Sastry)

Homogenous coordiantes and the E(3):

Both points $p=(p_{1},p_{2},p_{3})^{T}$ of three-dimensional space and vectors $v=(v_{1},v_{2},v_{3})^{T}$ of three-dimensional space are usually written as 3-vectors. Since points and vecors differ in their transformation behaviour under Euclidean motions, we use alternative notation and write them as 4-vectors

$p={\begin{pmatrix}p_{1}\\p_{2}\\p_{3}\\1\\\end{pmatrix}}\quad and\quad v={\begin{pmatrix}v_{1}\\v_{2}\\v_{3}\\0\\\end{pmatrix}}$.

This has the benefit that we can represent an Euclidean motion being the composition of a rotation with rotation matrix W and a translation with translation vector T as a 4x4-matrix G, which reads in block-matrix form

$G={\begin{pmatrix}W&V\\0&1\\\end{pmatrix}}.$

Transforamtion of points and vectors is now given by simple matrix multiplication. --Benjamin.friedrich (talk) 22:40, 2 February 2008 (UTC)

Rigid body motion: why not a curve in E(3)?

The article says "A rigid body motion is in effect the same as a curve in E+(3)" but then goes on to explain that a curve can't jump between the two cosets. This explanation would seem more relevant if rigid body motion were defined as a curve in E(3) rather than E+(3), starting from the identity. --Vaughan Pratt (talk) 05:44, 10 June 2009 (UTC)

I have added more words. Charles Matthews (talk) 13:02, 10 June 2009 (UTC)

Looks great. Thanks. --Vaughan Pratt (talk) 01:52, 27 June 2009 (UTC)

Degrees of freedom

Currently the article notes these degrees of freedom for E⁺(3):

• rotation about an axis – 5
• rotation about an axis combined with translation along that axis (screw operation) – 6

Selection of an axis amounts to selection of a point on the unit sphere and presents two degrees of freedom. Amount of rotation about that axis gives another degree. Thus the total should be three. For a screw displacement the total should be four as the length of translation along the axis affords one more degree.Rgdboer (talk) 22:00, 6 August 2014 (UTC)

Okay, the center of that sphere has freedom to move, so the article is right. At Plücker coordinates one reads that lines in space have four degrees of freedom.Rgdboer (talk) 21:28, 7 August 2014 (UTC)

Isometries of E(4)?

I'm curious where this section is sourced, Euclidean_group#Overview_of_isometries_in_up_to_three_dimensions and whether it can be extended for Isometries of E(4)? Tom Ruen (talk) 16:47, 4 October 2014 (UTC)

We didn't define euclidean group.

It is fundamentally impossible to read about a math subject without defining precisely, formally what it means, it one of the most important piece of any math article, along with motivation, applications, theorems, properties... It amazes me there is not a place where it says "definition" in this article... this needs to be promptly fixed. — Preceding unsigned comment added by Santropedro (talk • contribs) 01:46, 21 January 2017 (UTC)

Yes. By you. And please sign your comments. YohanN7 (talk) 10:27, 21 January 2017 (UTC)

Is the name correct?

Is "Euclidean transformation" a good name for this article?
The concepts of Euclidean geometry are invariant with respect to arbitrary similarities (transformations of Euclidean space that preserve angles). For example, in plane Euclidean geometry one can define an equilateral triangle, and say that one triangle is twice as big as another one, and tilted 10 degrees with respect to it. But one cannot define "a triangle with side 2" or "a triangle with a horizontal base" or "an obtuse angle that turns to the left".
Once one has a proper reference figure (like an ordered list of three non-collinear points), one can define all those concepts relative to it. However, still many theorems of Euclidean geometry do not depend on such a frame: they talk about about congruent or similar figures, and may or may not involve handedness.
That is, in Euclidean geometry one deals with two basic groups of transformations: isometries (transformations of the space that preserve distances) and similarities (that preserve only angles). Each of the two groups has a subgroup that preserves chirality.
The rigid body motions are only the isometries that preserve chirality, which seems unquestionable. However, the rigid body tras]nsformations" are defined in the current version of this article as being the same as general Euclidean isometries. This is at the very least confusing, and posibly incorrect (since some authors may justifiably define "rigid transformation" as the same as "rigid motion").
It seems safer to have separate articles on each of those concepts, with unambiguous names (so that other articles can link directly to the proper one), and maybe have Euclidean transformation be a small article that points out those four options, with links to the above articles.
All the best,--Jorge Stolfi (talk) 01:26, 4 May 2019 (UTC)

Your question is "Is "Euclidean transformation" a good name for this article?". There is a problem, as Wikipedia does not has any article with this name. Euclidean transformation is a redirect to Rigid transformation, to which Euclidean isometry redirects also. I understand from your general considerations that you find "Euclidean transformation" ambiguous and "Rigid transformation" confusing. I would agree transforming Euclidean transformation into a disambiguation page and to move Rigid transformation to Euclidean isometry. But both must be discussed in the right place, which is Talk:Rigid transformation, not here. D.Lazard (talk) 08:55, 4 May 2019 (UTC)

shouldn't this article spend significantly more focus on screw motions?

It seems weird that screw motions (i.e. general orientation-preserving rigid motions in 3-space) are only really mentioned in the one sentence "Chasles' theorem asserts that any element of E+(3) is a screw displacement." It seems to me like the description and representation of screw motions (including an explanation of Chasles's theorem, etc.) should be something like half of this article. No? (And likewise for rigid transformation, which is currently more or less a stub.) –jacobolus (t) 06:14, 27 March 2023 (UTC)