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This may go in the philosophy forum: feel free to move it if necessary but it seems more like a basic geometry question.




Basically the question goes something like this:


If you travel 500km north, 500 km east, 500 km south, and then 500km west, what is your net displacement?





There are several answers to this question which rely on more information needing to be given.



1. Are those nautical miles?
2. Do we assume the earth is a perfect sphere or do we assume the earth as it actually is: an oblate spheroid? If the latter, then how accurate does our answer have to be?
3. Do we assume perfectly even terrain?
4. Where does this take place on the earth (This would relate to #2).
5. Are we using longitude and lattitude? If so, where?



For the sake of this question, assume it is done on a perfect sphere (don't worry about the terrain or the oblateness).


What is your answer? Don't scroll down to my answer because I don't want to taint/poison your reasoning. This question is debated and argued but geeks...and I think my answer is wrong.
























0 (zero)

On a perfect sphere? Wouldn't you wind back up where you started?

lines of latitude aren't parallel, therefore, moving north or south will displace you to some degree depending on whether you are above or below the equator, and how close to the pole. I'm sure someone could figure out some type of algorithm to describe it, though I'm fairly certain you would end up on the same line of longitude.

We are defining the coordinates on this sphere as they are on Earth, right? Two poles, north and south, whereas no such West or East poles exist?

EDIT: nvm

As long as you're dealing with an isotropic geometry you should end up where you started.

Originally posted by Oliver North
lines of latitude aren't parallel

I thought lines of longitude aren't parallel. But doesn't that have to do with the Earth's not being a perfect sphere? EDIT: no, it doesn't.

Assuming you start on the equator you would end up being east of your starting point. That's because the Earth has a larger "circumference" at the equator then it would 500 km north of it.

Originally posted by Master Han
I thought lines of longitude aren't parallel. But doesn't that have to do with the Earth's not being a perfect sphere?

ya, I mixed that up, and no, it has to do with the Earth having a north and south pole where longitudinal lines meet. It has more to do with naval navigation afaik.

^so then, are we defining the cardinal directions relative to the poles ala geography, or externally from the perspective of a math graph's use of cardinal directions? Because it seems that this thought experiment is being done on a hypothetical sphere, and not the Earth.

http://schoolworkhelper.net/wp-content/uploads/2010/10/latitudelongitude.jpg

ok, so it is pretty obvious when you look at it like this, but the image on the right shows it in the most extreme form.

So imagine you are starting at the prime meridian where it intersects the line of latitude ~in France. moving north would be essentially straight up. Moving east would follow the line of latitude to the right for 500km, and moving south would follow a line of longitude 500km down. However, because the lines of longitude are not parallel, you would be greater than 500km away from the prime meridian, though you would be on the same line of latitude, because lines of latitude are parallel.

Originally posted by Master Han
^so then, are we defining the cardinal directions relative to the poles ala geography, or externally from the perspective of a math graph's use of cardinal directions? Because it seems that this thought experiment is being done on a hypothetical sphere, and not the Earth.

which is why I asked for that clarification

Originally posted by dadudemon
1. Are those nautical miles?

No, they're kilometers.

Originally posted by dadudemon
5. Are we using longitude and lattitude? If so, where?

Using latitude and longitude isn't important to the question as far as I can tell.

Originally posted by dadudemon
For the sake of this question, assume it is done on a perfect sphere (don't worry about the terrain or the oblateness).

Assume a circumference of 2000km for simplicity and imagine that you begin at a point on the "equator" (this is just a tool for visualization).

500km north take you to the North Pole.
500km east no longer has a meaningful application
Divide by zero error.
Error...
Error......

Rebooting.

Assuming a circumference of several trillion km and imagine that you begin at a point on the "equator" (this is just a tool for visualization).
500km north takes you 500km along the Y axis.
500km east takes you -500km along the X axis.
500km south takes you -500km along the Y axis.
500km west takes you 500km along the X axis.

500-500=0
-500+500=0

Looks like no displacement to me.

Originally posted by Symmetric Chaos
Assuming a circumference of several trillion km and imagine that you begin at a point on the "equator" (this is just a tool for visualization).
500km north takes you 500km along the Y axis.
500km east takes you -500km along the X axis.
500km south takes you -500km along the Y axis.
500km west takes you 500km along the X axis.

500-500=0
-500+500=0


From what perspective is this axis drawn...? If it's above "you", the axis will have to move around...if it's away from the sphere, then traveling 500 km west won't actually move you exactly 500 km west in said axis, since you're traveling in three dimensions, and partially away from the plane.

Correct me if I'm wrong, of course.

Originally posted by Symmetric Chaos
Using latitude and longitude isn't important to the question.

that seems like the only relevant part of the question imho...

Originally posted by Oliver North
that seems like the only relevant part of the question imho...

But if you're not defining it by longitude/latitude, I don't see what other type of cardinal direction you could use. An external axis wouldn't really work, since moving "east" in that definition would involve moving off the sphere, right?

Originally posted by Master Han
From what perspective is this axis drawn...?

From the perspective of an observer in the air drawing two arbitrary perpendicular lines. At trillions of kilometers in circumference the planet is flat on a scale of 500km increments.

Originally posted by Symmetric Chaos

Assuming a circumference of several trillion km and imagine that you begin at a point on the "equator" (this is just a tool for visualization).
500km north takes you 500km along the Y axis.
500km east takes you -500km along the X axis.
500km south takes you -500km along the Y axis.
500km west takes you 500km along the X axis.

500-500=0
-500+500=0

Looks like no displacement to me.

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.

Edit: I see someone already stated this.

Originally posted by Symmetric Chaos
From the perspective of an observer in the air drawing two arbitrary perpendicular lines. At trillions of kilometers in circumference the planet is flat on a scale of 500km increments.

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

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.

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.)



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.

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. That's the classic way of getting a crazy, unexpected answer, I believe.

Originally posted by Master Han
But if you're not defining it by longitude/latitude, I don't see what other type of cardinal direction you could use. An external axis wouldn't really work, since moving "east" in that definition would involve moving off the sphere, right?

if you are defining what moving east means, then it doesn't have to. Sym is describing it as movement along an X axis arbitrarily drawn on the sphere. So long as the opposing sides of the shape made through this travel are parallel, there will be no displacement, even if the X/Y coordinates are mapped over a curved surface (so long as the curvature is consistent, etc).

This is why I think it is the only relevant question. If we use north/south/east/west in terms of how it is defined on Earth, the answer is the one I gave, if you map it in terms of north being an increase in the Y dimension and east in the X (and vice versa for south/west), the answer is the one Sym gave.

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?



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.

Originally posted by ares834
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?

the theoretical sphere isn't necessarily earth.

in fact, we don't know if it rotates. With no axis of rotation, anything other than arbitrary poles wouldn't make sense.

Originally posted by Oliver North
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

I'm slicing the planet along the lines of latitude which is how you move when going west or east.

Originally posted by ares834
I'm slicing the planet along the lines of latitude which is how you move when going west or east.

yes, and if you are making slices straight through the earth, as if slicing bread, you are cutting the surface at a different angle each time...

cmon guys...

Originally posted by Oliver North
the theoretical sphere isn't necessarily earth.

in fact, we don't know if it rotates. With no axis of rotation, anything other than arbitrary poles wouldn't make sense.

I'm not sure what you are saying here? Without arbitrary north/south poles, then how could one move north or south?

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.

Because as these new, hypothetical lines would form parallel rings, you can't rotate them, and they would eventually decrease in size down to zero, much like how longitudinal lines do at the poles.

Originally posted by Oliver North
yes, and if you are making slices straight through the earth, as if slicing bread, you are cutting the surface at a different angle each time...

cmon guys...

Ok?

What's your point here? Because, yes, that is exactly what I'm doing.

Because that's ultimately what is relevant when moving east and west.

Originally posted by Master Han
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.

sure? it would be a line, like the prime meridian or the international date line, but yes, if you placed those lines you could say there was a limit to how far west you could go...

that isn't theoretically necessary though, as you could just say one could travel west indefinitely around the sphere. the point is the lines don't have to meet.

Originally posted by ares834
Ok?

What's your point here? Because, yes, that is exactly what I'm doing.

the reason they have different curvature would be because the cuts go through the surface at different angles, not because spheres have different curvature near their poles.

Sorry, I did an edit there.

Yes, I understand the cuts are at different angles hence why the curvatures are different. But if you cut the sphere from an east/west direction then you are cutting the sphere up at different angles. And, well, that is what is relevant when moving in an east or west direction.

Originally posted by Oliver North
sure? it would be a line, like the prime meridian or the international date line, but yes, if you placed those lines you could say there was a limit to how far west you could go...

that isn't theoretically necessary though, as you could just say one could travel west indefinitely around the sphere. the point is the lines don't have to meet.

But if you start from the left or east-most side of the sphere, by your coordinate system, you would not be able to move north or south at all. It works on paper, but it hardly matches with what one would think about if ordered to travel "north". And nowhere is this definition of "north" used, even in pure mathematics. WADR...you just make it up.

ugh, ok, this is even more complicated than I thought...

so, imagine some sphere with no poles or anything like that. So long as you drew two sets of parallel lines that met at 90 degree corners, you could draw a square at any place on its surface.

However, once you extrapolate those lines around the surface to create lines similar to those of latitude, this is no longer true, and the only places squares could be drawn would be at locations along the "equator" (N/S or E/W), so long as 250km were above and below it (or to the right or left of it).

basically, you can't divide the surface of a sphere into squares... which I did actually learn in primary school and feel like an idiot for forgetting.

Essentially, the answer comes back to, how do we define the lines that the person travelling must take, and if we define things in terms of lines that run in some systemic pattern across the surface of the sphere, at best, there are only going to be some locations where someone could make the trip in the OP and end up where they started. However, a person could start at any location and make that trip so long as the N/S/E/W system they followed was either based more on an X/Y chart or drawn specifically so they fell into one of the "zones" where such squares were produced.

Originally posted by Oliver North
ugh, ok, this is even more complicated than I thought...

so, imagine some sphere with no poles or anything like that. So long as you drew two sets of parallel lines that met at 90 degree corners, you could draw a square at any place on its surface.

However, once you extrapolate those lines around the surface to create lines similar to those of latitude, this is no longer true, and the only places squares could be drawn would be at locations along the "equator" (N/S or E/W), so long as 250km were above and below it (or to the right or left of it).

basically, you can't divide the surface of a sphere into squares... which I did actually learn in primary school and feel like an idiot for forgetting.

Essentially, the answer comes back to, how do we define the lines that the person travelling must take, and if we define things in terms of lines that run in some systemic pattern across the surface of the sphere, at best, there are only going to be some locations where someone could make the trip in the OP and end up where they started. However, a person could start at any location and make that trip so long as the N/S/E/W system they followed was either based more on an X/Y chart or drawn specifically so they fell into one of the "zones" where such squares were produced.

To clarify, yes, we are using the Earth in the question but we are disregarding oblateness and terrain because it is a thought expirement, not a NASA question.



Based on Sym's answer and your answer, I think those are the best answers. Any other answer would be less correct. So, my original answer has to be wrong.


The only way my answer could be right is if the answer was posed without using the cardinal directions...but then that would end up being, "walk 500km in a straight line, take a perfect right, walk 500 km, take a perfect right, walk 500 km, take a perfect right, walk 500 km." Obviously, the question was not worded like that so I was clearly wrong.

Since longitudinal lines get progressively smaller as you move towards the north or south pole, wouldn't there be specific starting locations where the net displacement actually would be zero?

A spherical- or an ellipsoidal geometry would be anisotropic in case you're wondering.

I'm not going to bother with the ellipsoidal expression because that will end up with an elliptical integral.

So let's assume that the earth is spherical out of laziness.

We have the spherical coordinates.

http://i.imgur.com/XUeDgOP.png

I'll calculate distance from an arbitrary point on the sphere (x₁, y₁, z&#8321 to another (x₂, y₂, z&#8322.

http://i.imgur.com/96IaFvu.png

So let's begin. We'll create two orthogonal vectors denoting the change in direction of north and east respectively.

http://i.imgur.com/yTEv7l5.png

http://i.imgur.com/EMryFHb.png

In the next step we want to project the movement from (x₁, y₁, z&#8321 to (x₂, y₂, z&#8322 on the surface of the earth, but to do that we'll first have to normalize the vectors.

http://i.imgur.com/uUjw9LZ.png

http://i.imgur.com/vUSYshr.png

And finally we'll project the movement onto the curve drawn by the normalized vectors.

http://i.imgur.com/rH5EHKz.png

Apply this formula four times and voila, and you'll get your displacement. Obviously it depends on where on the earth you are.

Edit. I kind of ****ed up your thread with the size of the images. That wasn't intentional.

Originally posted by dadudemon
That's what I thought but my former professor said that since cardinal directions were used, they are most certainly important/relevant and that makes Oliver North's approach to the problem more correct than mine (which had me assuming 0 displacement).

Yeah, I quickly came to the conclusion that I don't know enough about how spherical coordinates are defined in cartography to answer the question as posed.

A little late, but I missed this:

Originally posted by Symmetric Chaos
Then I can't draw a square on a piece of paper.

Technically, you can't...given that this is a thought experiment, we're taking this hyper-literally.

Originally posted by Astner
And finally we'll project the movement onto the curve drawn by the normalized vectors.

http://i.imgur.com/rH5EHKz.png

Apply this formula four times and voila, and you'll get your displacement. Obviously it depends on where on the earth you are.

Edit. I kind of ****ed up your thread with the size of the images. That wasn't intentional.

That ends up making the net change in displacement, on the sphere's surface, 0.

Since cardinal directions are involved, that changes how the problem works, entirely.




If you have the time...I have a request: can you work out how this problem is answered if we assume our starting position is 1500km south of the north pole?


Since cardinal directions are used, the "Fattest" meridians will occur at the equator (meaning, the most distance we can possibly obtain between meridains is by traveling east to west or west to east on the equatorial parallel).


You should end up with a net displacement of something slightly west of your starting position. I do not have the patience to work that out and you're much better at this kind of stuff than I am.



And feel free to ruin any of my threads, at any time, by dropping you knowledge on them. Your type of input is always welcome in my threads.


Edit - If you want to get really pedantic/accurate, pretend the earth's oblateness is taken into consideration. If you want to take it to the most extreme pedantry, then also include the terrain changes in the earth and start at the Washington Monument (I chose that monument for obvious phallic reasons. ).

Why did you use r in place of rho? Is that a Mathematica, thing?

Originally posted by Master Han
Technically, you can't...given that this is a thought experiment, we're taking this hyper-literally.

Good answer.

I concede to the fact that I'm insufficiently familiar with the subject from at least two different angles.

Originally posted by dadudemon
That ends up making the net change in displacement, on the sphere's surface, 0.
No it doesn't, that's only the case if φ₁ = -φ₂.

Originally posted by Astner
No it doesn't, that's only the case if φ₁ = -φ₂.
I'm pretty sure it does if you do that 4 times and use your previous answers to get to the next step. Should be 0. stop at four iterations.

Plug it in, sequentially, with 500 being used for your vectors during each change.

Originally posted by Master Han
Since longitudinal lines get progressively smaller as you move towards the north or south pole, wouldn't there be specific starting locations where the net displacement actually would be zero?

potentially if you started 250km south and west of the point where the prime meridian and equator meet. Or I guess any meridian would work, the route would just need to be bisected by the equator.

How can the point where you start possibly have an effect on the final answer? Its a sphere, I can rotate it any way I want without changing anything.

Originally posted by Symmetric Chaos
How can the point where you start possibly have an effect on the final answer? Its a sphere, I can rotate it any way I want without changing anything.


You can't rotate it at will once the "problem" actually starts, at which point having a starting point at the "north pole" is certainly going to be different from having one near the sphere's bottom. If we use standard geographic definitions of NSEW, for example, starting at the North Pole would mean that you couldn't travel North, West or East at all, so it clearly matters where relative to the (fixed) sphere you are.

(correct me if I'm wrong, of course)

Originally posted by Master Han
You can't rotate it at will once the "problem" actually starts, at which point having a starting point at the "north pole" is certainly going to be different from having one near the sphere's bottom. If we use standard geographic definitions of NSEW, for example, starting at the North Pole would mean that you couldn't travel North, West or East at all.

(correct me if I'm wrong, of course)

Astner said: "Apply this formula four times and voila, and you'll get your displacement. Obviously it depends on where on the earth you are."

Why and how would my displacement vary depending on where I started?

Originally posted by Symmetric Chaos
Astner said: "Apply this formula four times and voila, and you'll get your displacement. Obviously it depends on where on the earth you are."

Why and how would my displacement vary depending on where I started?

Because latitudinal lines aren't parallel, and longitudinal lines grow more jam packed at the poles, where an equivalent angular displacement equals a far more massive arc-displacement. Again, I think that's right. It's the same fundamental reason why you can't travel north, west or east from the north pole.

To be honest, I don't exactly understand Astner's math, so I'm just assuming that he's using geographical definitions of the cardinal directions.

Originally posted by Symmetric Chaos
How can the point where you start possibly have an effect on the final answer? Its a sphere, I can rotate it any way I want without changing anything.
Not quite, the closer you get to one of the poles the less distance you'll have to cross to walk around it.

In fact I drew you a picture.

Two trajectories staring in the bottom right corner moving north then west then south then east. With one being equator-symmetric hence leaving no displacement, and the other not being equator-symmetric hence leaving one.

http://i.imgur.com/0KEj6LW.png

Cool! Thanks, Astner.

I think something that is also interesting to consider is what 500 km east/west is at the pole (or north at the north pole and south at the south pole). Because it really doesn't make much sense there. I'd say at the pole it becomes undefined, cause you could really make an argument for going in any direction from there.

Originally posted by dadudemon
Yes, exactly. Any "z" axis movement will be negated. Likewise, so will the y and x axes. That means if you're moving across the surface of a sphere and you account for your vectors in such a system, your net displacement, on that sphere, should be zero, if you moved as described, four times.
No, because the trigonometric functions of φ₁ makes the geometry anisotropic. You're not moving in straight lines.

Originally posted by dadudemon
The distortions only become involved once you mix in meridians and parallels. If you're doing that, then, yes, your distortions will show up.
Let me rephrase it for you then. North is longitude, and east is latitude.

Originally posted by Astner
No, because the trigonometric functions of φ₁ makes the geometry anisotropic. You're not moving in straight lines.

Then let me rephrase it for you, too: you'll be moving 500km in an arc in each vector, not in a straight line. As a matter of impossibility, you wouldn't be able to just cut straight through this hypothetical sphere...you must have some awesome tunneling equipment in China (or wherever you are).


Originally posted by Astner
Let me rephrase it for you then. North is longitude, and east is latitude.

Let me rephrase that for you, then: North-South Longitudes are meridians. East-West Latitudes are parallels.