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

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