calculation2**The Mathematical Study of Mollusk Shells**

## Derivatives of rotation matrices

Suppose

`R = [a`_{ij}] is a

`3x3`rotation matrix. This means that

`R` preserves lengthsand angles, and preserves orientation. Since

`R` timesthe first basis vector

1 a_{11}R [0] = [a_{21}] = first column of R, etc., 0 a_{31}

the statement"

`R` preserves lengths and angles" implies that the three columns of

`R` have unit length and are pairwise orthogonal. This is equivalent to the statement that

`R R`^{T} = I: that

`R` 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 `R`_{0}=I. Differentiating the equation`R`_{a}R_{a}^{T} = I with respectto `a` gives`R'`_{a}R_{a}^{T} + R_{a}R'_{a}^{T} = 0. Setting `a=0` then gives`R'`_{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 matrix

`S`, 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 angle

`B = 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`.