Dear Jonas and all,

Ton and I have had a brief exchange of notes on a sub-topic of the current
discussion - that of whether the decomposition of the dot product a.b
=ax.bx+ay.by+az.bz works in non-orthogonal axis system. I resorted to an
old textbook on mathematical physics which said "no" with some fairly heady
math to explain. Ton's come up with the same answer but in a much more
insightful manner:

Let a and b be vectors, and i,j,k be unit vectors along the coordinate axes
which may be non-orthogonal. Let ai, aj, ak be the scalar components along
each axis. Then:

a = ai.i + aj.j + ak.k
b = bi.i + bj.j + bk.k

Hence:

(a.b) = ai.bi.(i.i) + ai.bj.(i.j) + ai.bk.(i.k) +
aj.bi.(j.i) + aj.bj.(j.j) + aj.bk.(k.k) +
ak.bi.(k.i) + ak.bj.(k.j) + ak.bk.(k.k)

= ai.bi + aj.bj + ak.bk +
(aj.bi + bj.ai).(i.j) +
(ak.bi + bk.ai).(i.k) +
(ak.bj + bk.aj).(j.k)

In the JCS, axes 1 and 2 are orthogonal, and axis 2 and 3 are orthogonal,
so we lose the cross terms with (i.j) and (j.k). The (i.k) term
remains :-(. So it seems that you need to add this term if you wanted
to compute total power from JCS angular velocities and JCS moment
components.

We are both agreed that vector relationships must hold whatever the
co-ordinate system used (whether orthogonal or not) but the way in which
these are calculated from the basic components will depend on the
characteristics of the co-ordinate system.

Richard


Richard Baker

Gait Analysis Service Manager, Royal Children's Hospital
Flemington Road, Parkville, Victoria 3052
Tel: +613 9345 5354, Fax +613 9345 5447

Adjunct Associate Professor, Physiotherapy, La Trobe University
Honorary Senior Fellow, Mecahnical and Manufacturing Engineering, Melbourne
University

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