>> 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
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
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".
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).
>> 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
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.