Talk:Euclidean group

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Needs more exposition rather than lists. References would also help. Geometry guy 20:36, 31 May 2007 (UTC)

1 Inversion with respect to a point in 3D
2 Improvements
3 Notation special Euclidean group : SE vs E+
4 The Euclidean group E(3) as a matrix Lie group

[edit] 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

[edit] 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)

[edit] 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)

[edit] 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

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