Talk:Rotation group

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In "properties" it is said that the standard inner product can be written purely in terms of lengths; this is (the so-called polar identity (?)) valid for any inner product space (in fact, for any symmetric bilinear form, I think).

Btw, I didn't find much on this on any of the pages I searched, just some quite well hidden remarks on quadratic form. If s.o. has an idea, it would be nice to have this added somewhere (eg in bilinear form). — MFH: Talk 21:32, 21 Jun 2005 (UTC)

I don't understand the question. Can you rephrase it? What is it that you want more information on? In a standard real vector space, the "purely in terms of lengths" is just some simple vector space math, trigonometry, nothing deeper than that? linas 22:55, 21 Jun 2005 (UTC)

The length of a vector is given by its norm, which is given by the scalar product (or dot product), which in turn is defined by a symmetric positive definite bilinear form. But reciprocally, a norm defines a quadratic form which has an associated bilinear form (which I knew as its "polar form", but this term does not seem common here).

So one could define the rotation group w.r.t. any other (positive?) symmetric bilinear form. It seems to me if we take the bilinear form of index (3,1) i.e. with matrix diag(1,1,1,-1) we get the Lorentz group SO(3,1) pertaining to special relativity. — MFH: Talk 17:44, 22 Jun 2005 (UTC)

Yes, all of those statements appear to be correct; I detect no question. Traditional, narrow formal usage has the rotation group being SO(n) only, although you will occasionally find the broader usage you refer to, e.g. SO(3,1) might be called the "group of hyperbolic rotations". I've seen other non-SO(N) things called roatations as well, although the usage is informal and speaker-listener-context-dependent. linas 00:29, 23 Jun 2005 (UTC)

[edit] Request for technical explanation

• The first meaning of "rotation group" should probably just be merged with symmetry group, or at least direct readers there for a fuller explanation. Otherwise, it's a bit confusing.
• It's at least a little sad that this article about geometrical relationships doesn't have any diagrams. It would be a lot more accessible if each distinct concept were illustrated with a diagram. For example, the basic concept of what a rotation group is could be illustrated by showing some rotations which are and some rotations which are not in the rotation group. These pictures could demonstrate which properties of the same are and are not preserved. An example of an improper rotation could also be diagrammed; the "properties", "axis of rotation" ,and "topology" sections are definitely ripe for obvious illustrations.
• It's not clear in the introduction whether "the rotation group" and "SO(3)" are the same thing.
• Even having studied matrices in college well enough to be able to find determinants and eigenvectors and whatnot, I find the "orthogonal matrices" quite dense, and in some places very difficult to understand. I was not familiar with the term standard basis, and trying to read that article did not help. But come to basis, and it turns out that this is essentially the notion of an XYZ coordinate system, which most readers are actually already familiar with. This connection should be explained. This will also help give them an example to understand what it's saying about orthonormality, at a glance. The discussion of matrices would definitely benefit from a concrete example showing a specific matrix representing a specific rotation (preferably with a picture showing which properties the numbers in the matrix correspond to).

-- Beland 16:27, 18 December 2005 (UTC)