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 = ba^T - ab^T we get

G² = (ba^T - ab^T)(ba^T - ab^T) = b0a^T - b1b^T - a1a^T + a0b^T = -(aa^T + bb^T)

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 e^Gθ. So we conclude that for simple rotations,

R(θ) = e^Gθ =  I - P + P cos(θ) + G sin(θ).

If we note that,

e^GθP =  P cos(θ) + G sin(θ) then,

R(θ) = I - P +e^GθP = I - (I - e^Gθ)P,

the formula in the last blog, which works fairly well but isn't as elegant as R(θ) = e^Gθ.