Re: Curved space

Do we have a material difference of opinion on great circles and arcs?
Surely we can agree that on a sphere - which the Earth is not, but is
nearly - a route across the surface of the sphere between two points
that is an arc of a cirle whose centre is the centre of the sphere -
or, /equivalently/, the intersection of the sphere and a plane
containing the centre of the sphere - is the shortest route on the
surface between those endpoints.

In three-dimensional Euclidean geometry, such an arc is a curve, of
course. But Euclidean geometry is merely the outworking of certain
axioms, hypothetical assumptions, about space, points, lines, straight
lines, and their relations (and very effective in, for instance,
architecture); it has been so successful that some people have trouble
accepting non-Euclidean geometry, but non-Euclidean geometry will be
merely the outworking of a different set of axioms. That still doesn't
clearly make sense, but it isn't an original idea to offer spherical
geometry as an example.
http://en.wikipedia.org/wiki/Spherical_geometry
We are calling the circles concentric with the sphere, and arcs of
those circles, "straight lines"; the surface of the sphere is our
two-dimensional space, and points are the ordinary points of the
sphere; we can cross-verify the altered axioms against known properties
of geometry on a Euclidean sphere; we can do the working in Euclidean
accounting - so the conclusions do make sense, but in a non-Euclidean
way.

As for maps... there are various projections of sphere to genuine plane
that conserve different properties of the spherical space. None is
completely faithful, or else we wouldn't see globes on sale in the
geography department. And the map is merely a representation of the
sphere - what matters is the real thing.

For that matter, in the real world, spatial distance is not the only
consideration. The airspace territorial claims of flyover nations have
to be taken into account. This leads to great headaches whenever
someone sets out on a record-breaking round-the-world trip.

.

Relevant Pages

  • Re: The sphere and hyperbolic geometry
    ... My geometry - the surface of the sphere, all points and all circles ... axiom, ...
    (sci.math)
  • Re: The sphere and hyperbolic geometry
    ... My geometry - the surface of the sphere, all points and all circles ... axiom, ...
    (sci.math)
  • Re: Generation of microstructure
    ... I have to generate the geometry of a micro structure using ... The size of the circles is defined ... Then, determine the largest amount of overlap, and "expand" ... each x, y, and z value of the initial sphere centers ...
    (comp.soft-sys.matlab)
  • a question about geometries
    ... Does the concept of geometry, which includes lines and points, require ... between any two points there exists what you're defining as a line, ... So, for example, if you try to define a geometry on a sphere ... sphere with all great circles as lines certainly does ...
    (sci.math)
  • Re: The sphere and hyperbolic geometry
    ... My geometry - the surface of the sphere, ... the equator, or one is North and the other South of the ... If we have two distinct diameters on the ...
    (sci.math)