Talk:Improper rotation

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 Field:  Geometry

rotation is a coordinate transformation[edit]

so points remain the same, just the coordinate system changes; what about actual rotation (points are mapped into other points, the coordinate system remains the same)? - Patrick 10:33 13 Jun 2003 (UTC)
That's just a question of how you interpret the application of the rotation to a set of coordinates (or an arbitrary vector) in the context of your physical problem. Steven G. Johnson.
The new version of AxelBoldt is clearer. - Patrick 21:17 15 Jun 2003 (UTC)

Query what is said here about applying a rotation to a mirror image. The proper rotations form a normal subgroup, so the usual idea that you conjugate to get the symmetry group of an 'image' isn't what's meant.

Charles Matthews 09:24, 5 Dec 2003 (UTC)

I know nothing about this subject, but "(pseudovectors are invariant under inversion)" sounds to me like it contradicts other aspects of the article. Aren't pseudovectors reversed by inversion? Just as pseudoscalars change sign?)

The rule is that a pseudo-X transforms like X under improper rotations, except that it is additionally multiplied by –1. Scalars are invariant under inversion, so pseudoscalars get multiplied by -1. Vectors get multiplied by -1 under inversion, so pseudovectors are invariant <-- This statement is wrong ! --Lantonov 08:05, 20 April 2007 (UTC). —Steven G. Johnson 00:38, May 10, 2004 (UTC)

If the rule is that 'a pseudo-X transforms like X under improper rotation, except that it is additionally multiplied by –1' then why doesn't the article follow this rule through: 'pseudoscalars transforms like scalars under improper rotation, except that they are additionally multiplied by –1'. So far so good. But then: 'pseudovectors transforms like vectors under improper rotation, except that they are additionally multiplied by –1'. It follows that vectors are not multiplied by –1 therefore they do not flip sign therefore they are invariant. In the article it says the opposite. I tried to correct this error but it was reversed by Steven Johnson reinstating the erroneous statement. --Lantonov 07:15, 20 April 2007 (UTC)

x to -x really isn't a reflection, though. Charles Matthews 15:52, 10 Sep 2004 (UTC)

x to -x is indeed an improper rotation (-identity has determinant -1 in 3d, which is the only dimensionality in which pseudovectors are defined). Since pseudovectors are multiplied by an additional factor of -1 under improper rotations, they are left invariant by inversion. *Consider r×p where r and p are vectors, when r and p flip sign.) I'm sorry this was difficult for Lantonov to understand, but it is quite standard and supported in the references. — Steven G. Johnson (talk) 17:23, 18 January 2011 (UTC)

Disagree about merging with coordinate rotation[edit]

I find this a rather self-contained article, explaining a certain topic. I don't see much overlap with coordinate rotation if any at all. Also, coordinate rotation is a big article. As such, I see no value in merging, and somebody might find it harder to understand what an improper rotation is if looking it up in the coordinate rotation article. Oleg Alexandrov 01:34, 13 August 2005 (UTC)

I agree that this article per se is not a problem. But a merger is another issue. I don't see the need to separate mentions of a kind of rotation from coordinate rotation. It is natural to discuss if the rotation is proper or improper when discussing a rotation in general. As you can see the article discuss briefly how to define proper and improper rotations and that's an overlap, certainly. coordinate rotation can benefit from having a general discussion as well. -- Taku 03:14, August 13, 2005 (UTC)
Maybe you are right. However, merging would be a lot of work. You could start by inserting a nice discussion at about proper vs improper rotations at coordinate rotation. If that looks good, maybe merging would be OK. But again, the coordinate rotation is very big. Having this as a standalone article would be nice for the sake of not having the reader go through all of coordinate rotation to understand what an improper rotation is. Oleg Alexandrov 19:32, 17 August 2005 (UTC)


What is a rotoinversion supposed to be exactly? Is it the same thing as an improper rotation, or is it just the antipodal map? Tkuvho (talk) 17:13, 18 January 2011 (UTC)

A rotoinversion is synonymous with an improper rotation. An inversion is an antipodal map, and is a special case of an improper rotation. (Of course, a lot of this terminology is really only used for 3d in my experience, and in some cases is only applicable there. e.g. inversions only have determinant -1 in odd-numbered dimensions.) — Steven G. Johnson (talk) 17:28, 18 January 2011 (UTC)
OK. Note that inversion is used in a different sense in projective geometry and inversive geometry. My concern is that the term "rotoinversion" seems to be used in the sense of "antipodal map" at rotation group. Tkuvho (talk) 17:33, 18 January 2011 (UTC)

direct subgroup[edit]

What is direct subgroup and where I may find definition of it? I cannot find this term in wikipedia or google. Jumpow (talk) 21:32, 4 February 2017 (UTC)

I found definition in article Coxeter notation. I think, we have to make link. Jumpow (talk) 20:08, 7 February 2017 (UTC)

I added a link. Tom Ruen (talk) 21:33, 7 February 2017 (UTC)