One bit in the presentation of rotations is that they are often presented as exponentials. The common approach appears to be to first take approximate an infinitesmal rotation, then apply them in succession.
I'm not terribly keen on that approach, because it isn't clear how the approximation goes away and becomes an exact answer, and because it isn't necessary.
Again, since we are rotating about the Z axis, we don't need to include it in the calculation, so for brevity I return again to the 2 dimensional rotation maxtrix.
[ cos Θ sin Θ sin Θ cos Θ]=
[ 1 - Θ^2/2!... - Θ + Θ^3/3!...
Θ - Θ^3/3!... 1 - Θ^2/2!... ]= [ 1 0
0 1 ]
+ [ 0 -1
1 0 ] Θ
+ [-1 0
0 -1 ] Θ^2/2!
+ [ 0 1
-1 0 ] Θ^3/3!
+ ...
Consider the matrix which marks the first order term in the equation above. When we multiply it upon itself, the other matrices appear.
[ 0 -1 [ 0 -1 = [ -1 0
1 0 ] 1 0 ] 0 -1 ][-1 0 [ 0 -1 = [ 0 1
0 -1 ] 1 0 ] -1 0 ][ 0 1 [ 0 -1 = [ 1 0
-1 0 ] 1 0 ] 0 1 ]
Designating this matrix as A, we can rewrite the Taylor expansion as
= A^0 + A Θ + A^2 Θ^2/2! + A^3 Θ^3/3!...
= (ΘA)^0 + (ΘA)^1 + (ΘA)^2/2! + (ΘA)^3/3!...
= exp[ ΘA ]
This expression is exactly correct, and in turn leads back to the expression used for the infintesmal rotation
exp[ ΘA ]
= lim ( 1 + (ΘA)/n ) ^ n
November 9, 2005 5:03 PM
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Note that the matrices of the form:
[ x -y ]
[ y x ]
are a ring, isomorphic with the complex numbers, and your A is i, the imaginary number.