Re: definition of parallel lines

["John S. Denker" <jsd@MONMOUTH.COM>]

Justin Parke wrote:
> At lunch today we were talking about maps and lines of longitude etc (9th grade earth science) and I commented on the interesting fact that two people could start at different points on the equator and head north, their directions being parallel to one another, and end up meeting at the north pole.  I gave this as an example of two parallel lines intersecting.  (Wanting to appear clever :)  )

Clever indeed.

<nit> If you want to be extra careful, note that
they are parallel at one point and not parallel
elsewhere</nit>

> Then a colleague said, "so how do you know the lines are parallel?"
>
> This is, IMO, a very good question.

A verrrrry good question.  Very deep.

Let me start by answering a slightly different question.
You need to have _straight_ lines before it becomes
interesting to ask whether they are parallel.

So, how do we know that the lines are straight?

Answer:  My favorite model for this is masking tape.
The tape is non-stretchy, as you can verify by trying
to stretch a piece.  Next, note that it has finite
width.  It is non-stretchy across its width, and also
across innumerable diagonals, so it can hold itself
straight, the way a truss holds itself rigid with
cross-bracing.

This notion of straightness, defined by cross-bracing,
has numerous good properties.  For one thing, if you
stick an initial piece of tape to a surface, it defines
a unique way of laying down the next piece, and then
the next.  And it is reversible:  You can retrace the
such a tape-path in the reverse direction and get the
same result.

I find it intriguing that in mathematics, lines are
defined to be straight and have zero width, but in
physics, if you want to make sure it is straight, it
needs to have nonzero width (so the cross-braces have
some leverage).  You can pass to the limit of infinitesimal
width, but not zero width.

For thousands of years, mathematicians have asked us
to draw straight lines, but have depended on physics
to actually draw them.  (Borrowing a sentiment from
Misner/Thorne/Wheeler -- can't find the exact reference.)

See M/T/W page 249 for a picture of cross-bracing.

===

Straightness can also be _defined_ by the proposition
that a straight "line" is the shortest path between
two points.  More precisely, it is an extremal path
(either shortest or longest).  You can show that this
notion is equivalent to the previous masking-tape
truss notion.

===========

Once we have straight lines, it is easy define what we
mean by parallel.  For each point Ai on line A, find
the _separation_, i.e. the distance from Ai to the
nearest point on line B.  If at some point Ai*,
the separation is constant to first order, the lines
are parallel at that point.  (Constant to first order
in distance along the line A.)