[lpetrich]

Now for groups of rotations and reflections. For a n-dimensional space, the group is called the orthogonal group O(n), and for reflections only, the special orthogonal group SO(n).

Members of O(n) have determinant values +1 or -1, while SO(n) has +1 only. The +1 means pure rotations, while -1 means rotations and reflections, sometimes called improper rotations. One might ask why not a count of reflections? Two reflections are equivalent to one rotation, so the -1 case means an odd number of reflections and +1 an even number.

So O(n) divides up into {pure rotations: O(n), rotations + reflections}

It is easy to show that the quotient group of this decomposition is Z(2):
rot * rot = rot
rot * rfl = rfl
rfl * rot = rfl
rfl * rfl = rot


This group's elements M satisfy M.MT = MT.M = I where the T means transpose.

For 1 dimension, it's very easy: SO(1) = {1} and O(1) = {1,-1}

For 2 dimensions, the elements of SO(2) have the form {{cos(a), -sin(a)}, {sin(a), cos(a)}} for all a between 0 and 2pi. The extra elements of O(2) have the form {{cos(a), sin(a)}, {sin(a), -cos(a)}}.

For 3 dimensions, the elements of SO(3) are composed from a unit quaternion vector {q0,q1,q2,q3}, "unit" meaning
Σi=03 (qi)2 = 1

Those elements are Mij = ((q0)2 - Σi=13 (qi)2) δij + 2*qi*qj + 2*q0*εijkqk

For O(3), the additional ones are - (SO(3) elements).

Note that M is a sort of square of q, that both q and -q map onto the same M.


For SO(4), one must use a pair of quaternions, q1 and q2, with M being (linear in q1).(linear in q2). To get the O(4) extras, one has to multiply the SO(4) elements by something like diag({1,1,1,-1}), since -I is in SO(4).