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According to the classic book Clifford Algebra to Geometric Calculus (CAGC), a versor is a product of vectors, and those vectors need not be invertible. Some texts, such as Geometric Algebra for Computer Science (GACS) require a versor to be invertible. However, the non-invertible concept is more general, and more useful, since many results hold for these 'k-products' of vectors, rather than just for invertible ones. A case in point is the formula for the contraction over a geometric product of vectors. Unit-versors are exactly the pinors, even unit-versors are exactly the spinors, and rotors are exactly the special spinors (GAGC has an erroneous claim here, which GACS propagates). I was hoping to get the non-invertible definition through as the most useful definition. What do you think?
-- Kaba3 ( talk) 01:41, 8 April 2013 (UTC)
I heard this mentioned in a mechanics class today in reference to a unit vector. I've added what I've found online, but it needs an expert's eyes. Still, it looks like it's a real thing — BenFrantzDale 02:51, September 9, 2005 (UTC)
I think I remember when the article on versors was practically empty! I want to complement you on the outstanding job you are doing. I like the way you have noted hamilton's terminology!
This is just a draft objecting, I want to come back later and refine it.
In classical notation as your outstanding article points out there is something called the tensor of a versor.
If the versor is written in quadranomial form it is just
The above operation always equals one for a versor.
What hamilton called the common norm of a versor is could be written as
common norm = Nq = qKq
In other words the common norm is quaternion times its conjugate. The 'common norm' and the 'tensor' of a versor were two completely different operations. The common norm of a quaternion is equal to the square of the tensor of the same quaternion.
The common norm of a versor is also equal to one.
One approach I considered, but don't like very well was that since there is more than one 'norm' if you called the tensor of a quaternion the Euclidian-Hamilton norm, that would distinguish it from the 'common norm', as well as the corresponding norm operations in
Along these lines terms used by both Hamilton in his classical theory and in other branches of math could be made less ambigious by affixing the word Hamilton in front of them.
For example of this nomenclature consider the section that you link to the Euclidian norm which you might consider writing a formula identical to that of the 'hamiltonian tensor' of a quaternion.
By affixing the term hamiltonian to the word tensor I mean the an abreviation of what hamilton called a tensor.
Your problem is confounded by the fact that they keep changing the names of the operations on quaternions on the main page. Thus leading to some major ambiguity.
While the classical quaternion point of view is very well defined, and can be well documented, and its definitions of every word are very precise, the same is not true about modern thinking on the subject.
Sometimes they are calling the classical tensor, the modulus, and sometimes the absolute value.
Eucledian norm is an operation on real vector spaces and the vector space associated with H is three complex numbers C3. So it is an operation on a different space, and therefore not identical to any operation on H
On the other hand the word tensor has been taken over these days by a whole branch of math. So using the word tensor, for the euclidian norm would confuse your readers, in your introductory paragraph.
Hobojaks ( talk) 03:51, 21 July 2008 (UTC)
The whole idea of rethinking the quaternion as some type of normed vector space is a point of view that needs to be documented in terms of the versor concept.
However some of these view are very much non-classical views.
I can provide you with some good references to material by writers who held this view, of which one of the important early ones is Gibbs. Gibbs book vector analysis devotes considerable time to his view of what a versor is, but it is really really different from hamiltons. What makes it interesting is that it is written by a writer well versed in classical quaternion thinking.
In my view the most interesting candidate corresponding to the idea of a norm in a normed vector spaces when it comes to quaternions is a version of the lorentz invariant. In other words take the product of each individual element of quaternion with itself, keeping the i j and k. You get
Which as Rgdboer points out is the scalar part of the square of a quaternion.
The thing is that the whole idea of a normed vector space to my way of thinking is pretty much quaternion negitive nomenclature.
A classical quaternion has only one product, and since it is a quaternion and not a scaler, it is excluded from the definition of a normed vector space.
It seems to me that the motivation behind normed vector spaces is to incorperate the idea of an angle into them, but every quaternion already has a built in angle characteristic, so elevating one characteristic of a quaternion above the others and bestowing the name norm on it so that you can define an angle, or for that matter define what you mean by distance seems a bit contrived.
In my way of thinking, and of course this is just my opinion and probably could not be included on wikipedia, a quaternion has a number of quantities that could be thought of as norms.
(1)The distance idea to my way of thinking is the tensor of the vector of the quaternion. This is the true eucledian norm.
(2)The lorenz invariant, as I am pleased to discover, in other words the four dimensional distance in space time is the scaler of the product of a quaternion with itself.
(3)The tensor of a quaternion, looks a lot like the norm of a real four space, so I can understand why you guys coming from the non-classical point of view that advocates remaking a quaternion into a normed vector space like it so much.
Once you start to extend the classical quaternion idea, depending on how far you go, you can't really be sure any more that the common norm is still equal to the square of the tensor.
Hamilton proved this was true for a classical quaternion, however I was thinking that if you wanted to avoid quaternion negative nomenclature and felt that the term tensor was to obscure for most of your readers, may I suggest simply in the classical section of your article."Square root of the common norm".
In another approach I was thinking of including the sentence: The square root of the common norm of a quaternion is a positive definite tensor of rank zero. This characteristic of a quaternion written has been found so useful that in contexts where no other kinds of tensors are being discussed it is simply called "the tensor". The term Hamiltonian tensor is preferable when discussing this characteristic in a mixed context to distinguish it from other uses of the word tensor. A versor rotates any vector in its plane through its angle, but preserves its length. A positive definite zero ordered tensor preforms an act of tension either streaching or shrinking a vector. If you pick two arbitrary vectors, you can multiply the first by a quaternion consisting of the proper choice tensor and versor and the result will be the second.
These two operations are commutative. In symbols.
qβ = Tq.Vqβ = Vq.Tqβ
Hobojaks ( talk) 21:20, 25 July 2008 (UTC)
Woops, the text below somehow got moved around, sorry for that.
Similarly the real part of the square of an ordinary complex number is the modulus of the corresponding split-complex number. Rgdboer ( talk) 21:51, 24 July 2008 (UTC)
The term evil tensor for the useful quantity we are discussing would not only be consistent with quaternion negative nomenclature, but helpful for remembering things because it starts with the letter e, helping people to remember that it corresponds with the idea of what is called in quaternion negative nomenclature the euclidean norm of real space. If the first basis vector of quaternion space is thought of as an 'imaginary' vector, then you might explain to your non technical readers that tensors are terrifying numbers. It would help them to rememberTq. This might also be reflective of a commonly held view of students of multi-linear algebra when first encountering the tensor of a quaternion concept.
It would unify the concept basis vectors and square roots for the non-technical readers since it is consistent with calling the square roots of minus one purely imaginary numbers, and with the naming conventions that generate expressions like real number.
It would give an introduction to the article a little less technical tone, than using the convention of naming concepts after the people who invented them, as in Hamiltonian Tensor Norm, or Euclidean norm.
I find 'the norm' very confusing because this could also mean the Lorentz invariant or scaler of a quaternion. Perhaps affixing some term with a negative connotation like satanic scaler would be in keeping with the traditional naming conventions of multi-linear algebra? A more neutral term starting with the letter for Sq might be special realitivistic norm.—Preceding unsigned comment added by Hobojaks ( talk • contribs) 12:10, 27 July 2008 (UTC)
A great need exists to explain the basic properties of a quaternion in a way people with out a phd in math can understand.
Below is an attempt that I started working on, but I am not really sure that I like it very much or if it would be allowed on wikipedia.
The main difficulty is finding words that are at the same time non-technical and non-ambigious.
Also this text is incomplete, because I quit working on it for today at the point were it took up the subject of the angle of a versor. I notice that no one has commented on the technique. —Preceding unsigned comment added by Hobojaks ( talk • contribs) 02:10, 28 July 2008 (UTC)
Hi I cut out the non-technical introduction that I was working on here and paisted it into my talk page. I don't think it could ever be acceptable for wikipedia, because it contains some made up words. In other words unless I could find a book were someone was already calling tensors terrible numbers, I don't think it would be appropriate.
One of these days if I ever write my own book on quaternions I think it is a good idea, but not for here and I don't think that there is any way to fix it.
Here is an important conclusion that I have reached. There are a lot of words that quaternion theory have in common, and it makes writing about linear algebra and quaternions in the same article hard because words like vector and tensor have meaning in each.
My conclusion is that when possible it may be better to have a section about one subject or the other, so as to not make the language so confusing. An entire article written using one nomenclature, for example a whole article on the vectors of linear algebra or on the vectors of quaternion calculus can use the word vector with out confusion.
Hobojaks ( talk) 17:40, 5 August 2008 (UTC)
Is a versor just another word for a member of SO(3)? —Ben FrantzDale ( talk) 02:42, 4 December 2008 (UTC)
In geometric algebra, it appears that "versor" has a broader meaning than just rotation. One paper I'm reading says versors are multivectors that "have the property that they are a geometric product of vectors." Is that right? Should this page be expanded to reflect this notion? —Ben FrantzDale ( talk) 21:44, 24 January 2009 (UTC)
There seems to be something seriously misleading going on in the "Presentation on the sphere" section! I'm talking about the part I have copied below. It makes it look like the arcs on the sphere form a spherical triangle, and the text sounds that way too.
The problem with this is that the edge CA can't really be a great circle arc! The product q'q is the rotation that you get by first rotating by q and then rotating by q'. Doing that rotation to point A will bring it to point C, but there are many rotations that have that effect and only one of them takes A along the great circle it shares with C. In fact, unless A, B, and C are all on the same great circle, AC is sure to be a small circle arc. Just imagine what happens to point B when rotated by q'q: rotating it by q and then by q' is very different from the result of the rotation that would take A to C along the great circle.
I'm having a hard time figuring out what this section is actually trying to say. Lilwik ( talk) 04:09, 1 May 2012 (UTC)
Why is there is a very bold unbalanced closing parenthesis in the first formula of this article? It bothers me if this is an example of someone's real notation for something. Please tell me it was a mistake and clean it up or enlighten me about why it could possibly make sense. — Preceding unsigned comment added by 209.66.90.5 ( talk) 23:55, 10 May 2012 (UTC)
I created a separate article for the meanings of the term versor in geometry and physics. A disambiguation page already existed, and at the very beginning of the article we say this article is only about an "algebraic rotation representation". In geometry and physics a versor does not represent a rotation, but the direction of an axis or vector. Paolo.dL ( talk) 10:58, 16 August 2013 (UTC)
I have no opinion on merits of a separate article versor (physics) mentioned above, but [1] obscured the fact that versors as a rotational formalism do not need to belong to H. The comment “It is customary… to specify the context… For other meaning of versor, see disambiguation” is indeed hypocritical. There is no special article about hyperbolic versors. One article should explain evolution of the term from “a norm-1 element of H expressed as blah-blah” in Hamilton’s jargon to “such thing q that q v q−1 in certain associative algebra specifies a rotation of v” in modern algebra. I tried to achieve it in the lead. Paolo.dL discarded “an algebraic representation of rotations” and left only “in classical quaternion theory”. By the way, how abstract algebra is related to versors? They rely upon associative algebras (with some additional structure such as *-algebra or a quadratic norm form) over the field of scalars. An associative algebra is already an algebra of certain signature, hence we are out of the domain of abstract algebra. Such things as ideal (algebra) or simple (algebra) are abstract. Versors are not especially so. Incnis Mrsi ( talk) 20:57, 22 January 2014 (UTC)
… used to represent a rotation about a fixed axis.
The sentence
does not seem to make sense, for several reasons:
Because the intent of the statement is so obscure that I cannot even repair it, and a versor has already been described as a rotation, I am simply removing it. — Quondum 02:31, 23 January 2014 (UTC)
Each versor has the form … where a is the angle of rotation (in radians), and r is a unit vector with the same direction as the axis of rotation. The same rotation is often viewed as the motion of a point rotating about r along a directed arc of a circle with radius 1.
A versor is a point on the 3-sphere, it is represented by an equivalence class of arcs on the 2-sphere, namely those with the same axis and the same arc length. Rgdboer ( talk) 20:35, 23 January 2014 (UTC)
I spent considerable time attempting to understand what these directed line (great circle) segments in S2 can do for Spin(3) or SO(3). I concluded that they can represent the conjugacy, but not a simple composition of rotations (essentially due to arguments already presented by user:Lilwik above). There is my new picture that shows a conjugacy in SO(3). It is an interesting observation that a simple geometric interpretation fails for the composition of geometric rotations (unfortunately I haven’t a picture for a composition), but works for the group conjugacy; versors also act by conjugacy.
Indeed, we need an equivalence relation that shifts beginning of the segment to any point of S3. We wouldn’t be satisfied with your “free rotation on the great circle”: two great circles in the general position do not intersect. A versor q1 rotates 1 along certain great circle. You always can start from certain v0 : q1 v0 = 1 and then say that for the product q2 q1 of versors v2 := q2 1 is the endpoint (I use this superfluous notation to emphasize that we have a group action). So, the product is represented by the shortest great circle route from v0 to v2. The problem is that this route does not pass, in general position, through 1 and you have to shift it transversally (via right quaternion multiplication, using associativity) to put its beginning point to 1 again. Incnis Mrsi ( talk) 20:31, 25 January 2014 (UTC)
Yesterday I was reluctant to think on Lilwik’s arguments thoroughly; that’s why I didn’t understand Quondum’s counter-arguments, and also stuffed my mind with operations in SO(3) and their geometric interpretation. Now I was reluctant to read attentively Quondum’s text – because of this, inventing the same construction by my own mind (after Roger Penrose and after Quondum’s explication) was prescribed. What is now in the article is virtually my original research, but I hope it doesn’t deviate much from the road of Sir Roger. I evaluated the stuff is the article mistakenly: it wasn’t actually an “incompetent gibberish” (I am sorry, Paolo.dL). There was only one mistake: explaining the directed segments on S2 as SO(3) rotations. It does not add anything to understanding. Sorry for this poor text: I have to sleep in this time. Incnis Mrsi ( talk) 23:57, 25 January 2014 (UTC)
A comment to
“ | It is fairly straightforward to see that there is a 1-to-1 mapping from unit quaternions to S3, albeit not natural (there is no "best" point in S3 to map 1 to). | ” |
— Quondum, the subsection above |
No so simple, indeed. When you get S2 as a Riemannian manifold, fix the 0 point there and the orientation, then you obtain the complex projective line up to U(1) symmetry only. You can multiply points to complex numbers, and you know their complex absolute values, but you do not know their exact values as a + bi. To establish an isomorphism you have to fix the 1 point somewhere halfway between 0 and ∞.
With versors we have an analogous (but not identical) fable: when you get S3 as a Riemannian manifold, fix the 1 point in it and the orientation, then you have a Spin(3) structure. You instantly define the scalar part as the cosine of the distance from 1, hence have the plane (unit sphere) of unit vectors, and the arc structure described above, where the orientation specifies the direction of arcs, hence the order of multiplication of versors. But the entire picture is still SO(3)-invariant. Unlike the situation with CP1 you can multiply S3 points, but not a point to a unit quaternion (specified with 4 components), nor can you get/put quaternionic values. You have to fix a correctly oriented orthonormal basis i,j,k in the space of 3-vectors for an isomorphism between a geometric picture and numeric components. Incnis Mrsi ( talk) 10:13, 26 January 2014 (UTC)
I attempted to make a picture that shows how one-sided multiplication action is related to directed great arcs on S2 (because I partially used a stereographic projection, it is shown as the (yellow) plane). How is it related to the discussion? Look how the action v ↦ 1 + i + j + k/2v is arranged near the 3-vectors’ sphere. Inside the red roundabout it pulls elements down (towards negative scalar part). Outside the red roundabout it pulls elements up (towards positive scalar part). On the red circle itself (note the geometry in the current version is a crap) it effects a 60° rotation, conserving the entire circle. When we compose this left multiplication with the right multiplication to the inverse versor, then up/down movements cancel each other on the whole S3, but the rotation will remain and become a 120° rotation.
The picture is unfinished (the region around −1 is almost empty, and a net of the octahedron of ±i, ±j, ±k is not visible yet), there are visualization glitches and some positions are incorrect, but do you have an timely advice about adding or altering something?
Incnis Mrsi (
talk)
22:49, 26 January 2014 (UTC)
T.Y. Lam, in his paper Hamilton's Quaternions p. 22 says: "Any quaternion q ∈ H∗ can be written in the form ρ ⋅ (cos θ + u sin θ), where ρ = N(q)1/2, and θ ∈ [0, π is called the polar angle of q. (Hamilton, who had a penchant for inventing strange names, called cos θ + u sin θ the versor of q.)" He makes no reference to the idea of versor as being used as in this article. If the term versor has come to mean a quaternion of norm 1 (which is not the same thing at all), a clearer picture of the origin of this use of the word would be appropriate. The two linked online dictionaries define the term versor compatibly with Hamilton's use (in a sense as part of a quaternion), and not in the main sense as used in the article. As the distinction is subtle, I'll labour the point: Hamilton's use appears to have described an operation applied to any quaternion and its result, not to an arbitrary quaternion that may be the result of such an operation (just as it sounds strange to refer to an arbitrary real number x ∈ [0, π as a polar angle), nor to a quaternion representing a rotation of a vector. Under his usage it would be proper to say that p is the versor of some quaternion, but it would not be proper to say that p is a versor. — Quondum 17:23, 17 February 2014 (UTC)
This image was removed as unhelpful in its detail. Rgdboer ( talk) 02:01, 1 June 2017 (UTC)
Representation theory more advanced than a general reader might assimilate:
New subsection Representation of SO(3) was substituted. Rgdboer ( talk) 21:52, 4 June 2017 (UTC)
Otherwise versors aren't closed under multiplication. I'll make the change, and if anyone can find a source that disagrees, they can roll it back. — Preceding
unsigned comment added by
62.172.100.253 (
talk)
14:00, 16 August 2018 (UTC)
[edit]
I've thought about it, and the original definition does work. Why? Because if is increased by , then you can negate , and subtract from again. One could even argue that it's more "efficient" to define it that way, because there are fewer representations of the same versor. But if a versor has then there are still multiple representations under the old definition. I don't see any advantage in it.
[edit]
To be conservative, I've rolled back the edit. — Preceding unsigned comment added by 62.172.100.253 ( talk) 14:12, 16 August 2018 (UTC)
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According to the classic book Clifford Algebra to Geometric Calculus (CAGC), a versor is a product of vectors, and those vectors need not be invertible. Some texts, such as Geometric Algebra for Computer Science (GACS) require a versor to be invertible. However, the non-invertible concept is more general, and more useful, since many results hold for these 'k-products' of vectors, rather than just for invertible ones. A case in point is the formula for the contraction over a geometric product of vectors. Unit-versors are exactly the pinors, even unit-versors are exactly the spinors, and rotors are exactly the special spinors (GAGC has an erroneous claim here, which GACS propagates). I was hoping to get the non-invertible definition through as the most useful definition. What do you think?
-- Kaba3 ( talk) 01:41, 8 April 2013 (UTC)
I heard this mentioned in a mechanics class today in reference to a unit vector. I've added what I've found online, but it needs an expert's eyes. Still, it looks like it's a real thing — BenFrantzDale 02:51, September 9, 2005 (UTC)
I think I remember when the article on versors was practically empty! I want to complement you on the outstanding job you are doing. I like the way you have noted hamilton's terminology!
This is just a draft objecting, I want to come back later and refine it.
In classical notation as your outstanding article points out there is something called the tensor of a versor.
If the versor is written in quadranomial form it is just
The above operation always equals one for a versor.
What hamilton called the common norm of a versor is could be written as
common norm = Nq = qKq
In other words the common norm is quaternion times its conjugate. The 'common norm' and the 'tensor' of a versor were two completely different operations. The common norm of a quaternion is equal to the square of the tensor of the same quaternion.
The common norm of a versor is also equal to one.
One approach I considered, but don't like very well was that since there is more than one 'norm' if you called the tensor of a quaternion the Euclidian-Hamilton norm, that would distinguish it from the 'common norm', as well as the corresponding norm operations in
Along these lines terms used by both Hamilton in his classical theory and in other branches of math could be made less ambigious by affixing the word Hamilton in front of them.
For example of this nomenclature consider the section that you link to the Euclidian norm which you might consider writing a formula identical to that of the 'hamiltonian tensor' of a quaternion.
By affixing the term hamiltonian to the word tensor I mean the an abreviation of what hamilton called a tensor.
Your problem is confounded by the fact that they keep changing the names of the operations on quaternions on the main page. Thus leading to some major ambiguity.
While the classical quaternion point of view is very well defined, and can be well documented, and its definitions of every word are very precise, the same is not true about modern thinking on the subject.
Sometimes they are calling the classical tensor, the modulus, and sometimes the absolute value.
Eucledian norm is an operation on real vector spaces and the vector space associated with H is three complex numbers C3. So it is an operation on a different space, and therefore not identical to any operation on H
On the other hand the word tensor has been taken over these days by a whole branch of math. So using the word tensor, for the euclidian norm would confuse your readers, in your introductory paragraph.
Hobojaks ( talk) 03:51, 21 July 2008 (UTC)
The whole idea of rethinking the quaternion as some type of normed vector space is a point of view that needs to be documented in terms of the versor concept.
However some of these view are very much non-classical views.
I can provide you with some good references to material by writers who held this view, of which one of the important early ones is Gibbs. Gibbs book vector analysis devotes considerable time to his view of what a versor is, but it is really really different from hamiltons. What makes it interesting is that it is written by a writer well versed in classical quaternion thinking.
In my view the most interesting candidate corresponding to the idea of a norm in a normed vector spaces when it comes to quaternions is a version of the lorentz invariant. In other words take the product of each individual element of quaternion with itself, keeping the i j and k. You get
Which as Rgdboer points out is the scalar part of the square of a quaternion.
The thing is that the whole idea of a normed vector space to my way of thinking is pretty much quaternion negitive nomenclature.
A classical quaternion has only one product, and since it is a quaternion and not a scaler, it is excluded from the definition of a normed vector space.
It seems to me that the motivation behind normed vector spaces is to incorperate the idea of an angle into them, but every quaternion already has a built in angle characteristic, so elevating one characteristic of a quaternion above the others and bestowing the name norm on it so that you can define an angle, or for that matter define what you mean by distance seems a bit contrived.
In my way of thinking, and of course this is just my opinion and probably could not be included on wikipedia, a quaternion has a number of quantities that could be thought of as norms.
(1)The distance idea to my way of thinking is the tensor of the vector of the quaternion. This is the true eucledian norm.
(2)The lorenz invariant, as I am pleased to discover, in other words the four dimensional distance in space time is the scaler of the product of a quaternion with itself.
(3)The tensor of a quaternion, looks a lot like the norm of a real four space, so I can understand why you guys coming from the non-classical point of view that advocates remaking a quaternion into a normed vector space like it so much.
Once you start to extend the classical quaternion idea, depending on how far you go, you can't really be sure any more that the common norm is still equal to the square of the tensor.
Hamilton proved this was true for a classical quaternion, however I was thinking that if you wanted to avoid quaternion negative nomenclature and felt that the term tensor was to obscure for most of your readers, may I suggest simply in the classical section of your article."Square root of the common norm".
In another approach I was thinking of including the sentence: The square root of the common norm of a quaternion is a positive definite tensor of rank zero. This characteristic of a quaternion written has been found so useful that in contexts where no other kinds of tensors are being discussed it is simply called "the tensor". The term Hamiltonian tensor is preferable when discussing this characteristic in a mixed context to distinguish it from other uses of the word tensor. A versor rotates any vector in its plane through its angle, but preserves its length. A positive definite zero ordered tensor preforms an act of tension either streaching or shrinking a vector. If you pick two arbitrary vectors, you can multiply the first by a quaternion consisting of the proper choice tensor and versor and the result will be the second.
These two operations are commutative. In symbols.
qβ = Tq.Vqβ = Vq.Tqβ
Hobojaks ( talk) 21:20, 25 July 2008 (UTC)
Woops, the text below somehow got moved around, sorry for that.
Similarly the real part of the square of an ordinary complex number is the modulus of the corresponding split-complex number. Rgdboer ( talk) 21:51, 24 July 2008 (UTC)
The term evil tensor for the useful quantity we are discussing would not only be consistent with quaternion negative nomenclature, but helpful for remembering things because it starts with the letter e, helping people to remember that it corresponds with the idea of what is called in quaternion negative nomenclature the euclidean norm of real space. If the first basis vector of quaternion space is thought of as an 'imaginary' vector, then you might explain to your non technical readers that tensors are terrifying numbers. It would help them to rememberTq. This might also be reflective of a commonly held view of students of multi-linear algebra when first encountering the tensor of a quaternion concept.
It would unify the concept basis vectors and square roots for the non-technical readers since it is consistent with calling the square roots of minus one purely imaginary numbers, and with the naming conventions that generate expressions like real number.
It would give an introduction to the article a little less technical tone, than using the convention of naming concepts after the people who invented them, as in Hamiltonian Tensor Norm, or Euclidean norm.
I find 'the norm' very confusing because this could also mean the Lorentz invariant or scaler of a quaternion. Perhaps affixing some term with a negative connotation like satanic scaler would be in keeping with the traditional naming conventions of multi-linear algebra? A more neutral term starting with the letter for Sq might be special realitivistic norm.—Preceding unsigned comment added by Hobojaks ( talk • contribs) 12:10, 27 July 2008 (UTC)
A great need exists to explain the basic properties of a quaternion in a way people with out a phd in math can understand.
Below is an attempt that I started working on, but I am not really sure that I like it very much or if it would be allowed on wikipedia.
The main difficulty is finding words that are at the same time non-technical and non-ambigious.
Also this text is incomplete, because I quit working on it for today at the point were it took up the subject of the angle of a versor. I notice that no one has commented on the technique. —Preceding unsigned comment added by Hobojaks ( talk • contribs) 02:10, 28 July 2008 (UTC)
Hi I cut out the non-technical introduction that I was working on here and paisted it into my talk page. I don't think it could ever be acceptable for wikipedia, because it contains some made up words. In other words unless I could find a book were someone was already calling tensors terrible numbers, I don't think it would be appropriate.
One of these days if I ever write my own book on quaternions I think it is a good idea, but not for here and I don't think that there is any way to fix it.
Here is an important conclusion that I have reached. There are a lot of words that quaternion theory have in common, and it makes writing about linear algebra and quaternions in the same article hard because words like vector and tensor have meaning in each.
My conclusion is that when possible it may be better to have a section about one subject or the other, so as to not make the language so confusing. An entire article written using one nomenclature, for example a whole article on the vectors of linear algebra or on the vectors of quaternion calculus can use the word vector with out confusion.
Hobojaks ( talk) 17:40, 5 August 2008 (UTC)
Is a versor just another word for a member of SO(3)? —Ben FrantzDale ( talk) 02:42, 4 December 2008 (UTC)
In geometric algebra, it appears that "versor" has a broader meaning than just rotation. One paper I'm reading says versors are multivectors that "have the property that they are a geometric product of vectors." Is that right? Should this page be expanded to reflect this notion? —Ben FrantzDale ( talk) 21:44, 24 January 2009 (UTC)
There seems to be something seriously misleading going on in the "Presentation on the sphere" section! I'm talking about the part I have copied below. It makes it look like the arcs on the sphere form a spherical triangle, and the text sounds that way too.
The problem with this is that the edge CA can't really be a great circle arc! The product q'q is the rotation that you get by first rotating by q and then rotating by q'. Doing that rotation to point A will bring it to point C, but there are many rotations that have that effect and only one of them takes A along the great circle it shares with C. In fact, unless A, B, and C are all on the same great circle, AC is sure to be a small circle arc. Just imagine what happens to point B when rotated by q'q: rotating it by q and then by q' is very different from the result of the rotation that would take A to C along the great circle.
I'm having a hard time figuring out what this section is actually trying to say. Lilwik ( talk) 04:09, 1 May 2012 (UTC)
Why is there is a very bold unbalanced closing parenthesis in the first formula of this article? It bothers me if this is an example of someone's real notation for something. Please tell me it was a mistake and clean it up or enlighten me about why it could possibly make sense. — Preceding unsigned comment added by 209.66.90.5 ( talk) 23:55, 10 May 2012 (UTC)
I created a separate article for the meanings of the term versor in geometry and physics. A disambiguation page already existed, and at the very beginning of the article we say this article is only about an "algebraic rotation representation". In geometry and physics a versor does not represent a rotation, but the direction of an axis or vector. Paolo.dL ( talk) 10:58, 16 August 2013 (UTC)
I have no opinion on merits of a separate article versor (physics) mentioned above, but [1] obscured the fact that versors as a rotational formalism do not need to belong to H. The comment “It is customary… to specify the context… For other meaning of versor, see disambiguation” is indeed hypocritical. There is no special article about hyperbolic versors. One article should explain evolution of the term from “a norm-1 element of H expressed as blah-blah” in Hamilton’s jargon to “such thing q that q v q−1 in certain associative algebra specifies a rotation of v” in modern algebra. I tried to achieve it in the lead. Paolo.dL discarded “an algebraic representation of rotations” and left only “in classical quaternion theory”. By the way, how abstract algebra is related to versors? They rely upon associative algebras (with some additional structure such as *-algebra or a quadratic norm form) over the field of scalars. An associative algebra is already an algebra of certain signature, hence we are out of the domain of abstract algebra. Such things as ideal (algebra) or simple (algebra) are abstract. Versors are not especially so. Incnis Mrsi ( talk) 20:57, 22 January 2014 (UTC)
… used to represent a rotation about a fixed axis.
The sentence
does not seem to make sense, for several reasons:
Because the intent of the statement is so obscure that I cannot even repair it, and a versor has already been described as a rotation, I am simply removing it. — Quondum 02:31, 23 January 2014 (UTC)
Each versor has the form … where a is the angle of rotation (in radians), and r is a unit vector with the same direction as the axis of rotation. The same rotation is often viewed as the motion of a point rotating about r along a directed arc of a circle with radius 1.
A versor is a point on the 3-sphere, it is represented by an equivalence class of arcs on the 2-sphere, namely those with the same axis and the same arc length. Rgdboer ( talk) 20:35, 23 January 2014 (UTC)
I spent considerable time attempting to understand what these directed line (great circle) segments in S2 can do for Spin(3) or SO(3). I concluded that they can represent the conjugacy, but not a simple composition of rotations (essentially due to arguments already presented by user:Lilwik above). There is my new picture that shows a conjugacy in SO(3). It is an interesting observation that a simple geometric interpretation fails for the composition of geometric rotations (unfortunately I haven’t a picture for a composition), but works for the group conjugacy; versors also act by conjugacy.
Indeed, we need an equivalence relation that shifts beginning of the segment to any point of S3. We wouldn’t be satisfied with your “free rotation on the great circle”: two great circles in the general position do not intersect. A versor q1 rotates 1 along certain great circle. You always can start from certain v0 : q1 v0 = 1 and then say that for the product q2 q1 of versors v2 := q2 1 is the endpoint (I use this superfluous notation to emphasize that we have a group action). So, the product is represented by the shortest great circle route from v0 to v2. The problem is that this route does not pass, in general position, through 1 and you have to shift it transversally (via right quaternion multiplication, using associativity) to put its beginning point to 1 again. Incnis Mrsi ( talk) 20:31, 25 January 2014 (UTC)
Yesterday I was reluctant to think on Lilwik’s arguments thoroughly; that’s why I didn’t understand Quondum’s counter-arguments, and also stuffed my mind with operations in SO(3) and their geometric interpretation. Now I was reluctant to read attentively Quondum’s text – because of this, inventing the same construction by my own mind (after Roger Penrose and after Quondum’s explication) was prescribed. What is now in the article is virtually my original research, but I hope it doesn’t deviate much from the road of Sir Roger. I evaluated the stuff is the article mistakenly: it wasn’t actually an “incompetent gibberish” (I am sorry, Paolo.dL). There was only one mistake: explaining the directed segments on S2 as SO(3) rotations. It does not add anything to understanding. Sorry for this poor text: I have to sleep in this time. Incnis Mrsi ( talk) 23:57, 25 January 2014 (UTC)
A comment to
“ | It is fairly straightforward to see that there is a 1-to-1 mapping from unit quaternions to S3, albeit not natural (there is no "best" point in S3 to map 1 to). | ” |
— Quondum, the subsection above |
No so simple, indeed. When you get S2 as a Riemannian manifold, fix the 0 point there and the orientation, then you obtain the complex projective line up to U(1) symmetry only. You can multiply points to complex numbers, and you know their complex absolute values, but you do not know their exact values as a + bi. To establish an isomorphism you have to fix the 1 point somewhere halfway between 0 and ∞.
With versors we have an analogous (but not identical) fable: when you get S3 as a Riemannian manifold, fix the 1 point in it and the orientation, then you have a Spin(3) structure. You instantly define the scalar part as the cosine of the distance from 1, hence have the plane (unit sphere) of unit vectors, and the arc structure described above, where the orientation specifies the direction of arcs, hence the order of multiplication of versors. But the entire picture is still SO(3)-invariant. Unlike the situation with CP1 you can multiply S3 points, but not a point to a unit quaternion (specified with 4 components), nor can you get/put quaternionic values. You have to fix a correctly oriented orthonormal basis i,j,k in the space of 3-vectors for an isomorphism between a geometric picture and numeric components. Incnis Mrsi ( talk) 10:13, 26 January 2014 (UTC)
I attempted to make a picture that shows how one-sided multiplication action is related to directed great arcs on S2 (because I partially used a stereographic projection, it is shown as the (yellow) plane). How is it related to the discussion? Look how the action v ↦ 1 + i + j + k/2v is arranged near the 3-vectors’ sphere. Inside the red roundabout it pulls elements down (towards negative scalar part). Outside the red roundabout it pulls elements up (towards positive scalar part). On the red circle itself (note the geometry in the current version is a crap) it effects a 60° rotation, conserving the entire circle. When we compose this left multiplication with the right multiplication to the inverse versor, then up/down movements cancel each other on the whole S3, but the rotation will remain and become a 120° rotation.
The picture is unfinished (the region around −1 is almost empty, and a net of the octahedron of ±i, ±j, ±k is not visible yet), there are visualization glitches and some positions are incorrect, but do you have an timely advice about adding or altering something?
Incnis Mrsi (
talk)
22:49, 26 January 2014 (UTC)
T.Y. Lam, in his paper Hamilton's Quaternions p. 22 says: "Any quaternion q ∈ H∗ can be written in the form ρ ⋅ (cos θ + u sin θ), where ρ = N(q)1/2, and θ ∈ [0, π is called the polar angle of q. (Hamilton, who had a penchant for inventing strange names, called cos θ + u sin θ the versor of q.)" He makes no reference to the idea of versor as being used as in this article. If the term versor has come to mean a quaternion of norm 1 (which is not the same thing at all), a clearer picture of the origin of this use of the word would be appropriate. The two linked online dictionaries define the term versor compatibly with Hamilton's use (in a sense as part of a quaternion), and not in the main sense as used in the article. As the distinction is subtle, I'll labour the point: Hamilton's use appears to have described an operation applied to any quaternion and its result, not to an arbitrary quaternion that may be the result of such an operation (just as it sounds strange to refer to an arbitrary real number x ∈ [0, π as a polar angle), nor to a quaternion representing a rotation of a vector. Under his usage it would be proper to say that p is the versor of some quaternion, but it would not be proper to say that p is a versor. — Quondum 17:23, 17 February 2014 (UTC)
This image was removed as unhelpful in its detail. Rgdboer ( talk) 02:01, 1 June 2017 (UTC)
Representation theory more advanced than a general reader might assimilate:
New subsection Representation of SO(3) was substituted. Rgdboer ( talk) 21:52, 4 June 2017 (UTC)
Otherwise versors aren't closed under multiplication. I'll make the change, and if anyone can find a source that disagrees, they can roll it back. — Preceding
unsigned comment added by
62.172.100.253 (
talk)
14:00, 16 August 2018 (UTC)
[edit]
I've thought about it, and the original definition does work. Why? Because if is increased by , then you can negate , and subtract from again. One could even argue that it's more "efficient" to define it that way, because there are fewer representations of the same versor. But if a versor has then there are still multiple representations under the old definition. I don't see any advantage in it.
[edit]
To be conservative, I've rolled back the edit. — Preceding unsigned comment added by 62.172.100.253 ( talk) 14:12, 16 August 2018 (UTC)