Quote:
Originally Posted by Archy
"....one can have the origin at the centre of the Earth, and the Z-axis along the polar axis. A person standing at the North Pole is at position (0, 0, R), at the South Pole (0,0, -R) where R is the Earth's radius.
A coordinate transformation from (0,0,R) to (0,0, -R) is equivalent to an upside down reflection
If you are travelling in a plane at a constant velocity you are in an plane's inertial frame of reference not the Earth's frame of reference. Your position and velocity is relative to the plane's frame of reference.
If your upside down it means the plane has turned upside down.  "..
You have defined a special set of conditions, namely a frame of reference with the origin 0,0,0 at the earth.
Consider the universe: there are a very large number (some would say infinite) locations for the origin of a frame of reference. Those locations that are within the earth, are by comparison an almost infinitesimally small fraction. Choosing the centre of the earth as the origin, as you have done, is a very special case. In the general case my statement is true: there is no up or down.
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I'm using a localized geometry to describe the Earth nothing more.
Cosmologists use a localized geometry to describe the observable Universe and a global geometry to describe the entire Universe.
The geometry for the observable Universe is Euclidean. That's not to say that the space-time around every object in the Universe is flat.
Black Holes and Neutron stars are clear exceptions.
The local geometry of these objects needs to be treated separately.
It is perfectly valid to use a local geometry for any object.
Quote:
Originally Posted by NotPrinceHamlet
What I mean by euclidean space is just flat 3d space, with all 3 axes orthogonal to each other and all uniformly scaled - then even if the ant doesn't realise that it is upside down at the south pole, relative to the north pole, it is.
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Quote:
Originally Posted by NotPrinceHamlet
You have chosen a special type of scale : it is Cartesian.
But it is not the only way to view things. For instance, maps of the earth use various projections.
The Mercator projection has a non uniform scale for latitude.
The Cartesian system is OK for small scale things, but it is not a particularly useful method of determining the locus of the shortest path from here to New York. Spherical geometry and specially constructed polar (nonlinear) grids give the result in seconds.
In other circumstances, a referential system might use imaginary numbers and scales eg the scale might be the square root of -1: I've used such a referential system to solve certain problems to do with x-ray crystallography.
In other circumstances one could use a spherical referential system. A triangle has 180 degrees in a plane system, but can have 180 to 540 degrees if drawn with a pole as the apex and the base on the equator.
One could also use a conical referential system: Australia's geological maps use such a system.
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Remember this is a thread on General Relativity which involves the geometry of space not the geometry of bodies.
Regards
Steven