Re: Newton's 3rd law? was Re: inertial forces (definition)

[brian whatcott <inet@INTELLISYS.NET>]

At 08:42 10/26/99 -0500, you wrote:
> brian whatcott wrote:
>
> > "Principal normal to a space curve at a point:
> > The line perpendicular to
> > the space curve C at the point P
> > of C and lying in the osculating plane of C at P.
> > The positive direction
> > of the principal normal to C at P
> > is chosen so that the tangent, principal normal,
> > and binormal to C at P
> > have the same mutual orientation as the
> > positive x, y, and z axes."

> OK, Brian, if you're so damn smart, what is "osculating"?
>
> poj

Hehehe...the Victorian sense of "kissing" says it all for me.
Let me look it up anyway to find if some untoward sense creeps in.

Hmmm...starting with Oxford, I see I am too easily pleased:
COD insists that esculating of a curve or surface implies contact
 'of a higher order', to coincide in three or more points.

James & James  offer this, if I skip from reference to
reference in the customary manner:
"Order of contact of two curves.
 ...~ is n when the nth order differential coefficients,
from their equations, and all lower orders, are equal at the point
 of contact, but the (n + 1)th  order differential coefficients are
unequal....a measure of how close the curves lie together in the
 neighborhood of a point where they have a common tangent."

I take it that the case is simpler where you consider a plane
osculating with a curve; you then concern yourself only with the
coefficients of the curve.

An allied definition - much plainer, is the one for the order
of a curve:
" The greatest number of points (real or imaginary) in which any
straight line can cut it."

The mind naturally wanders towards more familiar territory - where
a fair curve through experimental points is sometimes needed.

   The Rechenduden, in translation as the Universal Encyclopedia
of Maths, hastens to explain there is exactly one rational integral
function which is of at most (n - 1)th degree which joins those n
points.

 And we rejoice (or possibly not) to read that both Lagrange and
Newton have a recipe for a suitable interpolation formula.

Nevertheless, we whisper behind our hands, that experimental data
would need to be very, very well established to warrant a cut
from a fair curve.   Better by far, the subconscious asserts, to
satisfy the urge to fair with a least squares fit of lower order -
perhaps even a straight line in doubtful cases.

Sincerely

(p.s) If stitching together a few dictionary entries qualifies
me as smart, why am I not rich?

brian whatcott <inet@intellisys.net>
Altus OK