Name: Alex Nemser
My name is Alex Nemser and I live in Cambridge, MA. My question is
regarding non-Euclidean geometry. Here goes: If two people stand on two
parallel lines opposite each other but facing the same direction and they
start walking, will they ever meet? I was wondering because if they are
facing north and travelling along latitude lines, they will meet at the
pole but what if they are facing east or another direction or what if
they're not on any line at all... Maybe you can help, thanks!
Yes, certainly they will meet. Your intuition about walking up
lines of longitude (not latitude) is correct -- they will meet at the
poles. In this case you are talking about a elliptical non-Euclidean
geometry, that which describes plane figures on the surface of the
Earth. In this geometry two straight lines (great circles) will
intersect at exactly two locations, or poles, which are opposite one
another on the globe.
``Lines'' of latitude are not straight lines in this geometry,
however, but rather curved arcs. So you can make no conclusions about
people walking along them. They are not straight because a straight
line, or geodesic, is always defined as that path defining the
shortest distance between two points. On the surface of a sphere that
kind of line is always a great circle. To construct a great circle
cut the sphere with a plane that passes through the two points in
question and the center of the sphere. The great circle is the
intersection of this plane and the surface of the sphere. You can see
that all lines of longitude are great circles, but only the equator
among ``lines'' of latitude is.
To see that a great circle is always the shortest distance, imagine
rotating the globe such that the great circle route passes directly
over the top of the globe (the ``North'' Pole) and your destination is
exactly as far ``South'' of the ``North'' Pole as is your starting
point. If you take any non-great-circle route to your destination,
you must deviate east or west from the great circle, and then return.
Clearly the extra distance you walk east and back west, or vice versa,
makes the non-great-circle route longer.
If that doesn't convince you, get yourself an orange and mark two
points on it with a pen. Take a string and stretch it tightly between
the two points, and you will see that it always makes a great circle.
That is, if you draw a marker along the string and then cut the orange
with a knife along the marked line, your cut will go through the
center of the orange.
Well, the whole difference between Euclidean and other geometries is their
treatment of parallel lines. In Euclidean (plane) geometry, parallel lines
never meet, and they are always separated by the same distance. In
Riemannian (elliptical) geometry, there are no parallel lines, as any two
straight lines will always meet somewhere. This is similar to what happens
to longitude lines on the earth's surface. The longitude lines on the
earth are not parallel; they are sections describing the intersection of
the earth's surface with planes that all intersect at the earth's axis. In
Lobachevskyan (hyperbolic) geometry, non-intersecting straight lines can
exist, and they will not always be the same distance from each other.
There will be some points that are closest to each other, and the lines
will ever diverge away from those points.
I'm not sure this fully answers your question. If it doesn't, clarify and
Richard E. Barrans Jr.
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Update: June 2012