why latitudes are not parallel to each other but longitudes are?
Actually, latitudes are parallel to each other - the longitudinal lines are the ones that meet at the poles. It's an arbitrary coordinate convention, there is no deep significance to it; you could, for example, use two sets of "longitudinal" coordinates to describe points on a sphere - you'd need four different poles for that.
but i've heard a definition of 'parallel': if line A is perpendicular to both line B and line C, then line B and line C are parallel.
since the equator is parallel to every longitude, so longitudes are parallel?
I'd imagine that definition was only intended to apply to straight lines.
look at a globe, think about it.
That is a definition from PLANE geometry. The surface of the earth is not a plane it is a sphere.
Both Riemann & Bolyay-Lobacevski-Gauss geometries ivalidate the parallels' axiom of Euclid...
An overall view one can get here
For your question though, longitude lines were designed to slice the earth through the poles. It just makes sense to do it that way.
Both sets of lines are referenced to the same source: the Earth's rotational axis. But one is parallel, while the other is perpendicular. Latitude lines slice the Earth along planes perpendicular to the axis while longitude slice the Earth along planes that pass through the axis. Latitudes form parallel concentric rings. Longitudes form slices, like an orange.
BTW, longitudes are always "great circles", i.e. they pass through the centre of the Earth and are at Earth's maximum radius. Latitudes are not great circles, with one exception: the equator.
Not really,u see the certers of the circles lie (actually form) the polar axis...
sorry i've typed something wrong...
that means, longitudes are Not parallel but latitudes are?
and my definition of parallel (if line A is perp. to both line B and line C, then B and C are ll) is not correct?
It is correct only in an Euclidean geometry...
so why don't the geographers design parallel longitudes but non-parallel longitudes?
also, no matter the longitudes are parallel to each other or meet at the poles, they are parallel to the equator. Doesn't it sound odd?
This is starting to hurt my head - latitude lines are parallel, longitude lines are not. Longitude lines are perpendicular to the equator.
It was done that way simply because it makes sense to do it that way. Try figuring out time zones (for example) with parallel longitude lines....
Its fine, but what the others are saying (I'll say it another way) is that lines traced on the surface of the earth are no longer lines, they are curves. When drawing maps, we pretend the earth is flat (generally - most maps are Mercator projections), but it isn't.
The really troubling thing when looking at a planiglobe is that Greenleand seems larger in surface than Australia... Australia is bigger in a ratio of ~11/3...
Right. Which is what makes them concentric - they all have the same centre (well, in two dimensions anyway.) They are concentrioc circles that have been translated along the axis.
Are you imagining a globe while you are asking your questions? I think that will clear up your confusion very quickly. (see attached)
If you try to draw a globe with parallel longitudes, you will end up creating two extra poles - and they'll be on the equator.
Something else you should note:
"2 lines perpendicular to the same line are parallel"
"2 parallel lines do not meet"
these two statements are only true in flat (Euclidian) geometry
On curved surfaces, 2 perpendicular lines are not parallel, and parallel lines can meet.
Though the longitudes are perpendicular to the equator, they still cross - at the poles.
Curves of constant lattitude are not really lines in spherical geometry, because they are not the shortest distance between two points. On a sphere, great circles are the shortest distance between two points.
Therfore lines of longitude are true lines in spherical geometry - they are great circles, and represent the shortest distance between two points. Lines of lattitude are not true lines.
As nearly as I can tell, the definition of parallel lines in 2d is that they are infinite lines that do not intersect.
This corresponds with the Mathworld definition, except that Mathworld only gives a definition for Euclidean geometry.
Neither lines of lattitude or longitude are parallel lines - one fails the intersection test, the other fails the line test. Since any two great circles on a sphere will intersect, there are no true parallel lines on a sphere.
wikipedia also mentions that there are no parallel lines in elliptic geomtery
This is addressing the question of "why", that Dave and Russ have responded to :
Parallel latitudes, and great circle longitudes designate respectively, the polar and azimuthal angles. These are none other than the [itex]\theta[/itex] and [itex]\phi[/itex] of a spherical co-ordinate system. What's special about these co-ordinates is that they form what is known as an "orthonormal basis". This is a fancy math way of saying that the co-ordinates of any point are always perpendicular to each other. You can see this readily from Dave's attached picture. Any latitude intersects any longitude at right angles (it's not JUST the equator that's perpendicular to the longitudes). This system of using an orthonormal basis is very useful for doing mathematical calculations relating to the earth.
Notice that this orthonormality condition is not met by the second system described in Dave's picture; you will even find points where the vertical and horizontal "latitudes" and are parallel. Such things make for a giant mathematical mess, giving everyone involved a terrible headache. The same kind of problem arises when you try to use two sets of "longitudes".
What's additionally nice about the system used (and this has been covered by Dave and Russ), is that it designates a single axis to the earth. This is a good thing, because the Earth naturally has one axis - the axis of rotation. The reason this is a good thing is that it becomes easy to describe azimuthally varying quantities, such as local time.
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