The 500km question.

Originally posted by ares834
No matter how you orient the axis, when you're moving along a sphere you will also be moving along a Z axis. Yeah, it may be very minor amount, but it is still there.

so long as the curvature is consistent this is moot. You would have the same change in Z for each leg of the trip

Originally posted by Master Han
Would this "observer in the air" not have to move around with the person, though? So this axis would not be stationary/absolute.

No? The observer stays stationary high in the air, able to see for a thousand kilometers in each direction.

He watches the traveler set off north. 500km later he says "stop, go east" and watches the next leg of the journey. 500km later he says "stop, go south" and watches the next leg of the journey. 500km later he says "stop, go west" and watches the next leg of the journey.

Originally posted by Symmetric Chaos
Another way of looking at it would be to say that "north" is a direction you pick at the beginning and that at each point you make a 90 degree turn to your left. That can be significantly stranger.

That doesn't really fit with the definition of cardinal directions, from what I know. Geographically, it would be based on latitude/longitude, the latter of which aren't parallel, and mathematically, wouldn't you need to have a stationary axis, with stationary cardinal directions?

On the 2000km sphere the first trip takes you to the pole, the second takes you to the equator, the third, takes you back to the starting point, and the fourth takes you to the pole.

IIRC, you can't travel "east" from a north/south pole.

Originally posted by Symmetric Chaos
No? The observer stays stationary high in the air, able to see for a thousand kilometers in each direction.

He watches the traveler set off north. 500km later he says "stop, go east" and watches the next leg of the journey. 500km later he says "stop, go south" and watches the next leg of the journey. 500km later he says "stop, go west" and watches the next leg of the journey.

Except that said traveler isn't really going "east" relative to the observer. He's really going three dimensionally into the two dimensional "axis", since he's on a sphere, even if only slightly.

Originally posted by Oliver North
so long as the curvature is consistent this is moot. You would have the same change in Z for each leg of the trip

It still makes defining the cardinal directions from an external axis meaningless, since they're two dimensional and can't arbitrarily be modified to include moving a certain, even consistent, Z distance.

Originally posted by Master Han
Except that said traveler isn't really going "east" relative to the observer. He's really going three dimensionally into the two dimensional "axis", since he's on a sphere, even if only slightly.

this really only depends on how you define it, and in neither case does it matter if the sphere is consistent and the opposing sides are parallel.

If you defined 500km irrespective of Z, the actual distance traveled for all 4 legs would be slightly more than 500km, if you take Z into account, slightly less.

Originally posted by Master Han
That doesn't really fit with the definition of cardinal directions, from what I know. Geographically, it would be based on latitude/longitude, the latter of which aren't parallel, and mathematically, wouldn't you need to have a stationary axis, with stationary cardinal directions?

Sounds like a conspiracy of cartographers I'm happy to put imaginary lines on imaginary spheres and see what happens. It starts getting messy if you put imaginary lines on real spheres.

I suppose the realization I've come to is that my original reaction was wrong, latitude and longitude do matter because the way you define the directions controls the entire problem.

Originally posted by Master Han
IIRC, you can't travel "east" from a north/south pole.

That's why I redefined the directions for that example so that you know what do do no matter where you are.

I already made a joke about how you can't go east from the poles. It is an interesting bit of ambiguity that only exists if you're actually on Earth. It pops up in a couple of places. Try to imagine what happens if you ask someone to travel 30000km south. It can't be done, the directions are clear, but it can't be done.

Originally posted by Master Han
Except that said traveler isn't really going "east" relative to the observer. He's really going three dimensionally into the two dimensional "axis", since he's on a sphere, even if only slightly.

Then I can't draw a square on a piece of paper.

Originally posted by Master Han
It still makes defining the cardinal directions from an external axis meaningless, since they're two dimensional and can't arbitrarily be modified to include moving a certain, even consistent, Z distance.

they could easily be modified to include it... since the curvature is consistent, it would simply mean that whatever correction you have to make washes out.

you would either have:

(X or Y) + Z = 500km

or

(X or Y) = 500km + Z

Originally posted by Oliver North
this really only depends on how you define it, and in neither case does it matter if the sphere is consistent and the opposing sides are parallel.

If you defined 500km irrespective of Z, the actual distance traveled for all 4 legs would be slightly more than 500km, if you take Z into account, slightly less.

But that isn't part of the cardinal directions' definition. That you're consistently adding a Z direction to every leg doesn't change the fact that they can't be defined as moving north, east, south or west on a sphere from an external axis. So you'd have to resort to using geographical definitions, which, as you've pointed out, would involve a net displacement.

Originally posted by Master Han
But that isn't part of the cardinal directions' definition. That you're consistently adding a Z direction to every leg doesn't change the fact that they can't be defined as moving north, east, south or west on a sphere from an external axis. So you'd have to resort to using geographical definitions, which, as you've pointed out, would involve a net displacement.

no... you would just be using parallel lines of longitude

there are pragmatic reasons we define our geography in the way we do, in some thought experiment done on a theoretical sphere, such restrictions wouldn't exist

Originally posted by Oliver North
they could easily be modified to include it... since the curvature is consistent, it would simply mean that whatever correction you have to make washes out.

you would either have:

(X or Y) + Z = 500km

or

(X or Y) = 500km + Z

I suppose that the Z's will cancel out algebraically, but it's still a bit of a stretch to argue that you could modify the definition of "cardinal direction" so...but at this point, we're devolving more into semantics than anything else. Perhaps the OP should clarify what he means by "north"/"south"/"east"/"west".

Originally posted by Oliver North
so long as the curvature is consistent this is moot. You would have the same change in Z for each leg of the trip

Of course. However, the curvature (of that section of the planet) may be different when moving west compared to the curvature when moving east.

Originally posted by Symmetric Chaos
That's why I made the sphere huge!

If you object to that answer than you also have to object to the notion that if I draw a square on a piece of paper (1 inch away from, me 1 in left, 1 inch toward me, 1 in right) I have no come back to my starting point. The paper is unlikely to be perfectly flat for a variety of reasons but I still know what will happen if I follow those motions without resorting to spherical geometry.

(So it was partially a joke.)

Ah, didn't notice your sphere was trillions of kilometers.

😛

Originally posted by Oliver North
no... you would just be using parallel lines of longitude

...and not-parallel lines of latitude to define North and South.

Originally posted by ares834
Of course. However, the curvature (of that section of the planet) may be different when moving west compared to the curvature when moving east.

not on a perfect sphere, no

Originally posted by Master Han
...and not-parallel lines of latitude to define North and South.

? no, the point is we are now using parallel lines to define north and south... there is no reason why they would have to meet at the poles. This isn't a quality of a "sphere", it is something humans developed because it was helpful on earth.

Originally posted by Oliver North
not on a perfect sphere, no

Sure it would. I'm not talking about the curvature of the sphere here, but the curvature of that section.

Imagine you start at the equator, if you "slice" the planet there you will have a different curvature than if you took a "slice" at, say, the Arctic Circle.

Originally posted by ares834
Sure it would. I'm not talking about the curvature of the sphere here, but the curvature of that section.

Imagine you start at the equator, if you "slice" the planet there you will have a different curvature than if you took a "slice" at, say, the Arctic Circle.

because you are slicing it at different angles. If you sliced it such that the cut was as perpendicular to the surface as possible, it would be identical

Originally posted by Oliver North
? no, the point is we are now using parallel lines to define north and south... there is no reason why they would have to meet at the poles. This isn't a quality of a "sphere", it is something humans developed because it was helpful on earth.

So how would you define what north and south are, from a system that still resembles the geographical one in the sense that it's measured from the perspective of a point on the surface, and not from an absolute, external axis?

Perhaps I am not visualizing this correctly, but without having the latitudinal lines meet at the poles, it seems entirely arbitrary how you draw them, aside from, in your modification, making them parallel to one another and perpendicular to the longitudinal lines. Because if you make them all point straight "up" relative from an external observer, they're going to meet at the "north pole", right?

Originally posted by ares834
Sure it would. I'm not talking about the curvature of the sphere here, but the curvature of that section.

Imagine you start at the equator, if you "slice" the planet there you will have a different curvature than if you took a "slice" at, say, the Arctic Circle.

Sorry, but I don't think you're accurate here. A sphere is entirely symmetrical; the "poles" could literally be any point on its surface.

I am accurate. Yes, you could theoretically place the poles anywhere you want. However, the north and south poles have already been "placed". Why the hell would we use a different set of poles if we are moving north and south?

Originally posted by Master Han
So how would you define what north and south are, from a system that still resembles the geographical one in the sense that it's measured from the perspective of a point on the surface, and not from an absolute, external axis?

Perhaps I am not visualizing this correctly, but without having the latitudinal lines meet at the poles, it seems entirely arbitrary how you draw them, aside from, in your modification, making them parallel to one another and perpendicular to the longitudinal lines. Because if you make them all point straight "up" relative from an external observer, they're going to meet at the "north pole", right?

just imagine the same thing that we do with latitude done with longitude. There aren't east or west poles.

ares: Yeah, point conceded.

Oliver: again, unless if I'm visualizing this incorrectly, this would suggest that there actually would be an effective "east" and "west" pole, since you'd reach a point where you couldn't travel any more North...despite being only in the middle of the sphere from a vertical standpoint, which wouldn't make any sense. I'm basing this off of flipping the longitude lines on their side.