Feature Column from the AMS

calculation2The Mathematical Study of Mollusk Shells

Derivatives of rotation matrices

Suppose R = [a_ij] is a 3x3rotation matrix. This means that R preserves lengthsand angles, and preserves orientation. Since R timesthe first basis vector

   1     a₁₁R [0] = [a₂₁] = first column of R, etc.,   0     a₃₁

the statement"R preserves lengths and angles" implies that the three columns ofR have unit length and are pairwise orthogonal. This is equivalent to the statement that R R^T = I: thatR times the transpose of R (transpose means rows andcolumns are interchanged) is the identity matrix I.

Suppose now R_a is a smooth curve of rotationmatrices, with R₀=I. Differentiating the equationR_aR_a^T = I with respectto a givesR'_aR_a^T + R_aR'_a^T = 0. Setting a=0 then givesR'₀ + R'₀^T = 0, i.e thematrix R'₀ is equal to minus its transpose. So

       0 -f -g      0 -1 0      0 0 -1      0 0  0R'₀ = [f  0 -h] = f[1  0 0] + g[0 0  0] + h[0 0 -1].       g  h  0      0  0 0      1 0  0      0 1  0

where f,g,h are any 3 real numbers. This is the form ofour matrix r = lim_a->0(R_a-I)/a.

The next step is to rotate coordinates so that

             0 -c  0    r = [c  0  0].         0  0  0

If the new coordinates are derived from the old by a rotation matrixS, then in the new coordinates the same infinitesimalrotation r will appear as SrS^-1.Starting with

       0 -f -g        [f  0 -h]        g  h  0

a rotation of coordinates by an angle A = arctan(-h/g) about the z-axis changes r to

 cosA -sinA 0    0 -f -g     cosA sinA 0       0 -f -q[sinA  cosA 0]  [f  0 -h]  [-sinA cosA 0]  =  [f  0  0].  0     0   1    g  h  0      0    0   1       q  0  0

where q = -gcosA+hsinA. A further rotation by an angleB = arctan(-q/f) about the x-axis changes thatmatrix to

 1   0     0     0 -f -q    1    0    0        0 -c  0[0 cosB -sinB]  [f  0  0]  [0  cosB sinB]  =  [c  0  0] 0 sinB  cosB    q  0  0    0 -sinB cosB       0  0  0

where c = fcosB - qsinB.