Re: Curved space

On 17 Aug 2006 10:14:33 -0700, "DougC" <prigator@xxxxxxx> wrote:


Robert Carnegie wrote:

However, one space that definitely is curved is the surface of the
Earth. Taken as an ideal globe (which it isn't), it's rather a good
model of non-Euclidean geometry, if you consider "straight" lines as
the great-circles of the globe. I think all of Euclid's axioms work on
the globe except for the one about finding parallel "lines"; that's the
point. Then you see that every triangle has internal angles greater
than 180 degrees...

There are no "great circles" on a globe. The curves appear only when a
straight line is depicted on the distorted flat map made by Mercator
projection. The distortion of lines and triangles gets greater the
farther from the equator until the poles are lost to infinity.
Navigators understand that and still prefer the flat maps and charts.
Not a mystery.


No quite. The intersection of a plane and the surface of a sphere
is a circle. If the plane passes through the center of the sphere,
the circle is a great circle. Yes, there are great circles on a
globe.

It is easily proved that an arc of a great circle is the shortest
distance between two points on the surface of a sphere.

The Mercator projection has the enormous navigational advantage of
preserving direction. "North" or "East Northeast" or "bearing of
268.2 degrees" are always the same direction on the map, no matter
where you are.

.