Re: parametric: x = cos(t), y = sin(t)

In article <071220071512205042%plsperry@[EMAIL PROTECTED]
>,
 Paul Sperry <plsperry@[EMAIL PROTECTED]
> wrote:

> In article <see-3F8102.10530107122007@[EMAIL PROTECTED]
>, Barb Knox
> <see@[EMAIL PROTECTED]
> wrote:
> 
> > In article <061220071507321524%plsperry@[EMAIL PROTECTED]
>,
> >  Paul Sperry <plsperry@[EMAIL PROTECTED]
> wrote:
> > [SNIP]
> > 
> > > So, in my view, it is improper or, instead, meaningless to say "Let
t
> > > be angle between lines M and N" (although I'll confess to doing so
at
> > > every op****tunity). Also, it is awkward to properly state things
like
> > > "There are 2*Pi radians in a circle" - never mind the fact that
there
> > > are _no_ radians in a circle - or like "The sum of the angles of a
> > > triangle is Pi radians".
> > > 
> > > I guess I am asserting that there is no such thing as an angle (how
> > > many angles can dance on the head of a pin?). At least I have never
> > > seen a definition that I found satisfying. In any case, it doesn't
help
> > > to swap off one word - "angle" - for another - "inclination".
> > > 
> > > ...So there!
> > 
> > I guess I don't (yet) see the difficulty with the definition that an 
> > angle between 2 rays with a common origin is the length of the
interior 
> > arc produced by the rays' intersection with the unit circle centred on

> > that common origin.
> > 
> > This handles the cases of an angle having zero radians (a degenerate 
> > arc), a complete circle having 2 pi radians, and the sum of the angles

> > of a triangle being pi radians.
> > 
> > What am I missing?
> 
> Well, OK. The use of "interior angle", is a little vague.  There is
> also the problem of orientation - is the angle Pi/4 or 7Pi/4 (or
> -Pi/4)?
> 
> Defining an angle to be a length (whatever _that_ may be) seems counter
> intuitive to me but that's alright.
> 
> I wonder what one does in "real world" situations where there are units
> associated with lengths. ("How far is it from here to the grocery
> store?" "Oh, about 5.63.") I guess one must require unitless "lengths"
> or else two lines would form different angles depending on whether our
> unit circle has a radius of one foot or one mile.

That's not a problem, since the length of the arc is measured in terms 
of "units", where the unit circle has a radius of one "unit".  It can be 
feet, miles, light-years, or whatever -- you get the same arc length in 
terms of those units.  So the result is that the angle is dimensionless, 
being actually the ratio of 2 lengths (the denominator being 1 "unit").

> Anyway, I didn't (and don't) intend to be taken too seriously in my
> exegesis about angles. Thanks for taking the trouble to respond.

It was fun.

-- 
---------------------------
|  BBB                b    \     Barbara at LivingHistory stop co stop uk
|  B  B   aa     rrr  b     |
|  BBB   a  a   r     bbb   |    Quidquid latine dictum sit,
|  B  B  a  a   r     b  b  |    altum viditur.
|  BBB    aa a  r     bbb   |   
-----------------------------