User guide

Charnwood Dynamics Ltd. Coda cx1 User Guide – Advanced Topics III - 2
CX1 USER GUIDE - COMPLETE.doc 26/04/04 100/162
The 3D angle would be ideal, however, to measure the absolute deviation of ‘head-vertical’
(as in the example above) from true laboratory vertical:
To appreciate the absolute nature of a 3D angle consider how this head might precess (like a
gyroscope) around true vertical: from any vantage point the head-angle would be seen to be
changing yet the 3D angle could remain more or less constant. In order to obtain an angular
measure which corresponds to an axial view one should employ a 2D angle.
2D Vector Angles
The options for 2D angles correspond to fixed-axis stick figure views. Any chosen option for
a 2D angle might be regarded as a projection of the 3D case onto the chosen plane so that
the obtained angle is that which would be observed on the corresponding stick figure view.
When viewing a projection we are in a position to visualize the intersection of the projected
vectors and, hence, the angle between them. The essential difference from the 3D case is
that projected vectors (extended if necessary) really must intersect on a 2D surface (unless
they happen to be parallel). The trigonometry applied to the simpler 2D geometry allows us
to determine not only which of the supplementary angles to determine, but whether the angle
is positive or negative, though the sense of the angle is arbitrarily decided by the order in
which the vectors are defined. To make life easier, Codamotion Analysis makes a ‘majority
decision’ about the sign of a 2D angle, assuming that it ought to be positive more often than
negative throughout the acquired movement data. A further, arbitrary sign inversion is an
option for the final angle definition (even for a 3D angle), along with an arbitrary offset (which
might usefully be set to 180
o
in some cases.
Applied to the head-tilt example above, the sagittal (XZ plane) 2D angle of the precessing
head would indicate a positive vector angle for the head tilted forward and negative for a
backward tilt but would (for better or worse) indicate zero for pure left or right tilt.
As another illustration, consider investigating the angle between left and right femurs as
defined using hip -- knee vectors. In this over-simplification we would expect to observe an
exaggerated ‘scissor’ action in the sagittal view, with the angle passing through zero each
time one femur swings in front of the other. If we defined the 2D angle such that ‘left in front
of right’ produced a positive angle, we might deduce something about the left-right symmetry
of gait by comparing the positive and negative portions of the graph plot. The 3D angle, on
the other hand, would continue to describe the absolute angle between femurs.
‘Head-vertical’ is the vector depicted by the bold arrow
(derived as the normal to the plane of M
1
M
2
M
3
).
True vertical is shown by the taller arrow.
The 3D vector angle represents absolute deviation with
no hint about left, right, fore, or aft.
M
2
M
3
M
1