Wednesday, October 1, 2014
Simple Rotations are Matrix Exponential Functions
One can rewrite the formula for the rotation R in terms of an exponential function. First let us recall that for vectors in the plane the equation for this formula is,
R(θ) = I cos(θ) + G sin(θ).
We can rewrite any vector v in the entire space as the sum of a vector in the plane and another perpendicular to it. So v = Pv + (I - P)v where P is a matrix that projects v onto the ab-plane. If we calculate the square of G = baT - abT we get
G2 = (baT - abT)(baT - abT) = b0aT - b1bT - a1aT + a0bT = -(aaT + bbT)
The part in brackets at the end is a matrix that will project any vector onto the ab-plane so we can let P = -G2 and the rotation formula for any vector will be,
R(θ) = I - P + [I cos(θ) + G sin(θ)]P = I - P + P cos(θ) + G sin(θ)
since GP = G. If we go to the Wikipedia article on matrix exponentials we will find that the expression on the right is just eGθ. So we conclude that for simple rotations,
R(θ) = eGθ = I - P + P cos(θ) + G sin(θ).
If we note that,
eGθP = P cos(θ) + G sin(θ) then,
R(θ) = I - P +eGθP = I - (I - eGθ)P,
the formula in the last blog, which works fairly well but isn't as elegant as R(θ) = eGθ.