Talk:Improper rotation

message: :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)
replies: text: 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)

text: 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)
thread_title: ''rotation is a coordinate transformation''

message: 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)
replies: text: 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)
replies: text: 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)
thread_title:  rotoinversion 

message: 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)
replies: text: I added a link. Tom Ruen (talk) 21:33, 7 February 2017 (UTC)
thread_title:  direct subgroup 

message: In the article it says:"... which [an improper rotation] is the combination of a rotation about an axis and a reflection in a plane perpendicular to that axis."
Further down it says:"In 3D, equivalently [why is this word in the sentence?] it is the combination of a rotation and an inversion in a point on the axis."
These two definitions aren't congruent!
The first definition is equivalent to an inversion, the second one is equivalent to a simple reflection!
Then follows:"Rotoreflection and rotoinversion are the same if they differ in angle of rotation by 180°, and the point of inversion is in the plane of reflection."
But at the beginning is says:"In geometry, an improper rotation,[1] also called rotoreflection,[1] rotary reflection,[2] or rotoinversion[3] ..."
Again, these two statements are incongruent!
--Felix Tritschler (talk) 11:36, 3 April 2018 (UTC)
thread_title:  Improper Definition