User guide

Charnwood Dynamics Ltd. Coda cx1 User Guide – Advanced Topics III - 3
CX1 USER GUIDE - COMPLETE.doc 26/04/04 106/162
(In either case the sequence described is that by which the segment arrives at its
orientation. Anyone wishing to ‘undo’ the rotations to return to the neutrally aligned
orientation must reverse carefully!)
Mathematical decomposition
Having settled for the ‘ZXY’ sequence we must calculate the compound rotation matrix:
R
zxy
= R
y
R
x
R
z
which looks like:
| cosφ cosψ + sinθ sinφ sinψ sinθ sinφ cosψ − cosφ sinψ cosθ sinφ |
R
zxy
= | cosθ sinψ cosθ cosψ −sinθ |
| sinθ cosφ sinψ − sinφ cosψ sinθ cosφ cosψ + sinφ sinψ cosθ cosφ |
If we use R
zxy
to form the image,U
D
zxy
, of distal EVB U
D
under this compound rotation we
obtain the same matrix again, the columns of which are the vectors u
X
```, u
Y
``` and u
Z
```
(re-oriented axial unit vectors), which are precisely the distal EVB vectors derived in the
segmental analysis along with u
X
, u
Y
and u
Z
, the proximal EVB vectors.
Conveniently, the 2
nd
element of the 3
rd
column of our matrix happens to be the scalar
product of proximal unit vector u
Y
(= [0,1,0]
T
) with column (distal axis) vector u
Z
```, i.e.
u
Y
.u
Z
``` = −sinθ
and we solve first, therefore, the X axis rotation angle θ as
θ = −sin
-1
{ u
Y
.u
Z
``` }.
Having found θ, its value may be plugged back into the equations for φ and ψ which
similarly arise from the scalar (or ‘dot’) products of proximal/distal axis unit vectors:
we obtain φ from the 1
st
element of the 3
rd
column wherein
u
x
.u
Z
``` = cosθ sinφ
so that the flexion angle about the proximal Y axis is given by
φ = sin
-1
{ (u
x
.u
Z
``` ) / cosθ }.
Likewise, from the 2
nd
element of the 1
st
column:
u
Y
.u
X
``` = cosθ sinφ
so that the internal/external rotation angle about the Z axis is
ψ = sin
-1
{ (u
Y
.u
X
``` ) / cosθ }.