SO, SU, etc

SO, SU, etc

Post by mar » Sun, 13 Jul 2003 06:38:45



Hi,

Every so often on this group one sees the terms (abbreviations?)
SU,SE, SO, etc.  E.g. so-and-so is in SO(3).

Could someone recommend a web site or book that explains these terms?
I realise it's probably a huge area so I perhaps a good 'breadth
first' overview is what I'm after. Thanks.

 
 
 

SO, SU, etc

Post by Sylvain Croussett » Sun, 13 Jul 2003 08:00:27



>Hi,

>Every so often on this group one sees the terms (abbreviations?)
>SU,SE, SO, etc.  E.g. so-and-so is in SO(3).

>Could someone recommend a web site or book that explains these terms?
>I realise it's probably a huge area so I perhaps a good 'breadth
>first' overview is what I'm after. Thanks.

They are matrix groups, so any book on group theory should do.  See
http://mathworld.wolfram.com/SpecialOrthogonalGroup.html
http://mathworld.wolfram.com/SpecialUnitaryGroup.html

 
 
 

SO, SU, etc

Post by Just d' FAQ » Sun, 13 Jul 2003 18:44:39




Quote:>Every so often on this group one sees the terms (abbreviations?)
>SU,SE, SO, etc.  E.g. so-and-so is in SO(3).

>Could someone recommend a web site or book that explains these terms?
>I realise it's probably a huge area so I perhaps a good 'breadth
>first' overview is what I'm after. Thanks.

The one you'll see most often is SO(3), the Special Orthogonal group
in 3 dimensions. That's just a fancy way of saying 3D rotations. Bet
you can guess what SO(2) is...

An Orthogonal matrix has columns which form an orthonormal frame; in
other words, they are unit magnitude vectors with each perpendicular
to all the others. This can be succinctly stated as M^T M = I. (Since
this equation is its own transpose, we might just as well have talked
about rows instead of columns.)

A Special matrix has determinant +1. This implies that when used as a
linear transform it does not change volumes, and it does not flip the
orientation from right-handed to left. The groups SO(n) combine both
restrictions. So whereas O(n) includes reflections, SO(n) does not.

To form a group, every matrix must have an inverse; no singularity is
allowed. With that restriction alone we have the General Linear group
in n dimensions, GL(n). Imposing the determinant restriction gives the
Special Linear group, SL(n). In computer graphics these two don't show
up very often. (At least, not explicitly.)

Recall that a group is a set of actions satisfying a few reasonable
properties.

  1) You can do two things in a row as if they were one.
  2) You can do nothing.
  3) Anything you can do, you can undo.
  4) Actions accumulate in order.

More formally, we have composition, an identity, inverses, and the
obscure but essential associativity. We are not guaranteed that it
does not matter if you do A first then B, versus B first then A. In
formal terms, groups need not have commutativity; those that do are
called Abelian, after the brilliant young Norwegian Niels Henrik Abel.

  <http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Abel.html>

(The "e" of "Abelian" is stressed and pronounced as in "eel".)

Euclidean motions preserve rigidity, and thus consist of rotations and
translations only. It's pretty easy to see we have a group. Since we
don't want reflections sneaking in, we call it SE(n).

Rarely in computer graphics, but often elsewhere, we need matrices of
complex numbers. Then instead of Orthogonal matrices we have Unitary
matrices, satisfying not transpose times matrix gives identity, but
*conjugate* transpose times matrix gives identity, M^H M = I. (That
superscript "H" stands for "Hermitian".) As it happens, the Special
Unitary group in 2 dimensions (with complex coordinates), SU(2), acts
almost exactly the same as SO(3), so makes an occasional appearance.

All of these groups have an infinite number of members, and vary
continuously in their action. Specifically, the map from matrices to
their inverse is a continuous function of its one argument; and the
map from two matrices to their product is a continuous function of
both its arguments. This is not at all surprising, yet it is extremely
useful. Formally, we say these are Lie groups. The ones we discussed
are most of the classic examples. The name is that of Norway's other
especially famous mathematician, Marius Sophus Lie (pronounced "Lee").

  <http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Lie.html>

 
 
 

SO, SU, etc

Post by mar » Mon, 14 Jul 2003 02:33:22



Quote:> The one you'll see most often is SO(3), the Special Orthogonal group
> in 3 dimensions. That's just a fancy way of saying 3D rotations. Bet
> you can guess what SO(2) is...

Yep, but I'm having trouble guessing what SO(1) is :)  I'm also
wondering whether the fact that a rotation in 2D can be done with
successive shears (which are, presumably, not special orthogonol) has
any relevence.

Is the number in parenthesis always a positive integer? Starting at 2
and going up in increments of 1? Is there an upper limit, for example
is there an SO(4), SO(5)...SO(oo)?

Are quaternions in SO(3), SO(4), or both?

Quote:

> An Orthogonal matrix has columns which form an orthonormal frame; in
> other words, they are unit magnitude vectors with each perpendicular
> to all the others. This can be succinctly stated as M^T M = I. (Since
> this equation is its own transpose, we might just as well have talked
> about rows instead of columns.)

> A Special matrix has determinant +1. This implies that when used as a
> linear transform it does not change volumes, and it does not flip the
> orientation from right-handed to left. The groups SO(n) combine both
> restrictions. So whereas O(n) includes reflections, SO(n) does not.

> To form a group, every matrix must have an inverse; no singularity is
> allowed. With that restriction alone we have the General Linear group
> in n dimensions, GL(n). Imposing the determinant restriction gives the
> Special Linear group, SL(n). In computer graphics these two don't show
> up very often. (At least, not explicitly.)

> Recall that a group is a set of actions satisfying a few reasonable
> properties.

>   1) You can do two things in a row as if they were one.
>   2) You can do nothing.
>   3) Anything you can do, you can undo.
>   4) Actions accumulate in order.

> More formally, we have composition, an identity, inverses, and the
> obscure but essential associativity. We are not guaranteed that it
> does not matter if you do A first then B, versus B first then A. In
> formal terms, groups need not have commutativity; those that do are
> called Abelian, after the brilliant young Norwegian Niels Henrik Abel.

>   <http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Abel.html>

> (The "e" of "Abelian" is stressed and pronounced as in "eel".)

> Euclidean motions preserve rigidity, and thus consist of rotations and
> translations only. It's pretty easy to see we have a group. Since we
> don't want reflections sneaking in, we call it SE(n).

> Rarely in computer graphics, but often elsewhere, we need matrices of
> complex numbers. Then instead of Orthogonal matrices we have Unitary
> matrices, satisfying not transpose times matrix gives identity, but
> *conjugate* transpose times matrix gives identity, M^H M = I. (That
> superscript "H" stands for "Hermitian".) As it happens, the Special
> Unitary group in 2 dimensions (with complex coordinates), SU(2), acts
> almost exactly the same as SO(3), so makes an occasional appearance.

> All of these groups have an infinite number of members, and vary
> continuously in their action. Specifically, the map from matrices to
> their inverse is a continuous function of its one argument; and the
> map from two matrices to their product is a continuous function of
> both its arguments. This is not at all surprising, yet it is extremely
> useful. Formally, we say these are Lie groups. The ones we discussed
> are most of the classic examples. The name is that of Norway's other
> especially famous mathematician, Marius Sophus Lie (pronounced "Lee").

>   <http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Lie.html>

I didn't understand the last paragraph, partly because I don't know
what's meant by "members", or what a matrix's "one argument" could be
referring to.

But you've lifted the fog nicely. I gather this stuff comes under the
heading of Group theory (or maybe Lie algebra?), so I could look
further if I wanted, though the explanation you've given is more than
adequate, and excellent as usual. Thanks again.

 
 
 

SO, SU, etc

Post by Just d' FAQ » Mon, 14 Jul 2003 03:58:29




Quote:>> The one you'll see most often is SO(3), the Special Orthogonal group
>> in 3 dimensions. That's just a fancy way of saying 3D rotations. Bet
>> you can guess what SO(2) is...

>Yep, but I'm having trouble guessing what SO(1) is :)  I'm also
>wondering whether the fact that a rotation in 2D can be done with
>successive shears (which are, presumably, not special orthogonol) has
>any relevence.

For the record, the group SO(1) has only one member, the identity
element. O(1) has two members, +I and -I. As you guessed, shears are
neither special orthogonal nor of any particular relevance to this
discussion.

Quote:>Is the number in parenthesis always a positive integer? Starting at 2
>and going up in increments of 1? Is there an upper limit, for example
>is there an SO(4), SO(5)...SO(oo)?

Yes, the number in parenthesis can be any positive integer, and even
infinity. However instead of SO(oo) we may write SO; but that gets
into some rather sophisticated theory. Other possibilities exist, but
are best left for the day when you need them.

Quote:>Are quaternions in SO(3), SO(4), or both?

The correct answer is, neither. However unit quaternions form both a
Lie group and a sphere, S^3. This group is "isomorphic to" (acts the
same as) SU(2). Since +q and -q give the same rotation, we can say
that SO(3) is the "quotient group" of S^3 modulo the subgroup whose
only members are +1 and -1. The group SO(4) is larger than SO(3) or
S^3 alone, and in fact is their "product group".

Examples.
Let p and q be unit quaternions, and let v be any quaternion. Then

  q v q*

is q acting as a 3D rotation of the (i,j,k) part of v. This is our
usual way of mapping +/-q, a member of S^3, to a 3D rotation, a member
of SO(3). The fancier

  p q v q*

allows the pair (p,q) to act as a 4D rotation of all of v. This is one
way to map (p,+/-q), a member of S^3 x SO(3), to a 4D rotation, a
member of SO(4).

Quote:>> All of these groups have an infinite number of members, and vary
>> continuously in their action. Specifically, the map from matrices to
>> their inverse is a continuous function of its one argument; and the
>> map from two matrices to their product is a continuous function of
>> both its arguments. [snip]

>I didn't understand the last paragraph, partly because I don't know
>what's meant by "members", or what a matrix's "one argument" could be
>referring to.

The members of SO(3) are the individual rotation matrices. A 15 degree
rotation around the axis (0,3/5,4/5) is one member, for example. And
it's not the matrix that has one argument, it's the map from a matrix
to its inverse. The argument is the matrix to be inverted. The inverse
of our 15 degree rotation is a -15 degree rotation; and as you slowly
vary either the angle or axis, the inverse slowly varies as well.

Quote:>But you've lifted the fog nicely. I gather this stuff comes under the
>heading of Group theory (or maybe Lie algebra?), so I could look
>further if I wanted, though the explanation you've given is more than
>adequate, and excellent as usual. Thanks again.

Careful there. Most of group theory is about other kinds of groups,
not these "continuous" groups. Those groups are finite or discrete,
and so many of the techniques require adaptation. On the other hand,
Lie *algebra*, while vitally related, is not the same as Lie *group*.

Every Lie group has a unique associated Lie algebra, but one Lie
algebra can be associated with many different groups. For example,
O(3), SO(3), and S^3 have identical (isomorphic) Lie algebras. A Lie
algebra is a vector space with a "bracket" product. For the Lie
algebra of S^3, the vectors are 3D vectors, and the bracket is the
cross product. Best save that theory for later.

 
 
 

1. consiglio su impaginazione

Ciao,
a questo indirizzo http://www.temnostudio.com/page1.jpg
ho messo una prova di un impaginato che andr a far parte di una rivista di
moda in ambito "nunziale"...
Vorrei avere un parere da coloro che si interessano di tipografia e stampa
riguardo la gestione degli spazi, colori, equilibrio etc. etc.

Grazie mille!
Peppe

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