PDA

View Full Version : Applications of anticrossproducts and cross divisions

pdleva23
01-22-2010, 03:55 AM
Some subscribers discovered anticrossproducts and cross divisions (de Leva,
2008) by reading my recent posting about "moment arms in 3-D", and they
thought they were designed to define moment arms. Not at all.

Although a cross division can be used to define moment arms, I designed the
first cross division for a practical purpose. I hate writing scalar
equations, so I was looking for a clean "coordinate-free" vector-algebraic
formula to compute the force applied by a "load-shaft" on a force plate for
its calibration (see fig. 2 in
http://dx.doi.org/doi:10.1016/j.jbiomech.2007.09.030). When I found it, I
realized it was the missing tessera in Gibbs and Heaviside's mosaic of
vector-algebraic operations (such as vector sum, dot product and cross
product), i.e. a cross division (by the way, the dot division is just
impossible!). Then I found other kinds of cross divisions and other
applications. For instance:

ANGULAR VELOCITY OF A POINT

An "orthogonal cross division" (the simplest kind of cross division; symbol:
"/o") can be used to define the angular velocity of a point with respect to
another point O:

- omega = v /o r

where v is the velocity of P along any trajectory, even a non-circular,
non-spherical and non-planar one, and r the directed distance from O to R
(see fig. 4 in http://dx.doi.org/doi:10.1016/j.jbiomech.2007.09.030).
Compare this vectorial formula with its well known scalar version, which is
valid only for circular or spherical motion (v perpendicular to r):

- |omega| = |v| / |r|

Can you see the two advantages of the cross division in this case?

1) It returns not only the magnitude, but also the direction of omega
2) It is valid even when v is not perpendicular to r (non-constrained 3-D
motion of P).

CENTER OF PRESSURE

The center of pressure (CoP) is traditionally computed from force-plate
readings, by solving a system of three scalar equations. A single cross
division does exactly the same, but it is easier to write. (Similarly, a
cross product is just a symbol representing three scalar expressions, but it
is easier to write.)

You can find other examples in my paper, together with the definition of the
elementary concept of anticrossproduct, which is the theoretical foundation
of my study.

REFERENCE

de Leva P., 2008. Anticrossproducts and cross divisions. Journal of
Biomechanics, 8, 1790-1800
(http://dx.doi.org/doi:10.1016/j.jbiomech.2007.09.030)

With my kindest regards,

Paolo de Leva
Laboratory of Locomotor Apparatus Bioengineering
Department of Human Movement and Sport Sciences
University of Rome, Foro Italico
Piazza Lauro de Bosis, 6
00135 Rome - Italy
http://www.lablab.eu