Re: Poll on *Really* Wide Angle Lenses

In article <nP2dncw3a4aDD2bfRVn-rA@xxxxxxxxxxx>, Nostrobino <not@xxxxxxxxxx> writes 

You need to qualify that slightly - it's still geometry if you do it on
the
surface of a sphere, and it's not contradictory there (because on a
sphere,
parallel lines really do meet),


Well, not really. The problem with "parallel lines on a sphere" is that it's
a contradiction in terms if *straight* parallel lines are assumed. You can
have parallel lines going around a sphere (e.g., lines of latitude on a
globe) and they never meet. But they aren't straight lines either, except
when viewed from one, and only one, plane, which of course must be different
for each parallel line.

Not really. To a mathematician, the postulates of Euclid are just assumptions which may or may not apply. If you accept the last postulate (it's either the 5th or 6th, can't remember) - parallel lines never meet, but remain the same distance apart even if produced to infinity - you get the geometry applicable to the surface of a sphere - Euclidean geometry

However, this is only convention. If you look at non-Euclidean geometry, you can change this postulate. Riemannian geometry assumes that parallel lines always meet (eventually). In 2-dimensional space, this corresponds to geometry on the surface of a sphere, where straight lines are ^always^ great circles (note lines of latitude are not straight in this geometry - except the equator of course). In this geometry, incidentally, the angle sum of a triangle always exceeds 180 degrees; the conventional 180 degree rule depends on the (discarded) last postulate of Euclid.

Conversely, Lobachevskian geometry assumes that parallel lines continuously diverge. In 2-dimensional space, this corresponds to geometry on a surface something like a horse saddle; that is, where the centre of curvature of the two orthogonal directions are on opposite sides of the plane. In this geometry, the angle sum of a triangle is always less than 180 degrees.

Both these systems have been extensively studied, in 2 dimensions and also in 3 (and higher) dimensions.

There is no particular reason why we choose to think in Euclidean geometry, especially as we actually live on the surface of a sphere. I guess it may be partly because it was put forward a couple of millennia earlier, and partly because on a small scale we seem to see ourselves on a plane surface.

David
--
David Littlewood
.

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