calculation2The Mathematical Study of Mollusk Shells



Derivatives of rotation matrices

Suppose R = [aij] 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 RT = I: thatR times the transpose of R (transpose means rows andcolumns are interchanged) is the identity matrix I.

Suppose now Ra is a smooth curve of rotationmatrices, with R0=I. Differentiating the equationRaRaT = I with respectto a givesR'aRaT + RaR'aT = 0. Setting a=0 then givesR'0 + R'0T = 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 = lima->0(Ra-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.