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Re: definition of parallel lines



The mathemaical definition of a "straight" line is one that is a geodesic in
the space containing the line. This means that a line is defined as
"straight" if it is the shortest possible line connecting two points. In
plane geometry, the two-dimensional "space" is "flat," "straight" has its
usual meaning, and there exists only one line through a specified point that
is parallel to a given line. This is the famous Parallel Postulate of
Euclidian Geometry. However, if one works in a "curved" space, such as the
surface of a globe, every geodesic (hence "straight") line is a great
circle. There are NO parallel lines in this space. Every pair of great
circles will intersect. If the space is curved outwards, like a saddle, then
there are infinitely many lines through a point that are parallel to a given
line. These two cases (no parallel lines, and infinitely many parallel
lines) are the basis of the non-Euclidean geometries. As to why two
longitude lines seem to be parallel at the equator, the Earth is
sufficiently large that it may be considered "locally flat" within small
areas. For larger areas, check a road map. Many "T"-shaped intersections in
country roads exist because of the necessity of adjusting survey grids for
the curvature of the Earth. The effect is most obvious in the midwest/Great
Plains.

Vickie Frohne



-----Original Message-----
From: John S. Denker [mailto:jsd@MONMOUTH.COM]
Sent: Thursday, September 05, 2002 3:24 PM
To: PHYS-L@lists.nau.edu
Subject: Re: definition of parallel lines


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.)
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