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.