The 500km question.

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.

Originally posted by Oliver North
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?

Yes, that is one of the correct answers. This is how one of my professors answered the question (we had fun arguing various stuff in that class):

"Since the earth is a sphere and due North and due South are traveled on meridians, but due East and West are traveled on parallels, this has an effect on the actual distance traveled on land when using these direct directions.

In the end, because of the way meridians converge as you travel north in the northern hemisphere, you would actually end up slightly west of your starting point. The distances in this problem are actually very small compared to the size of the earth, but the point is still valid. If you drive on country roads, which are often divided into “mile sections” in Oklahoma, you have probably observed what my dad always called “section corrections”. These are places where the north/south roads don’t meet in a straight line, and are necessary when you divide a spherical surface into squares!"

So, in that regard, my old professor was right. The meridians get "closer together" as you travel north.

But wouldn't that be wrong since you also travel south? Meaning, you're negating the "easterly effect" of traveling north by a "westerly effect" of traveling south?

I just thought about it: nope. You'd end up traveling in what looks like a parallelogram stretched on the surface of a sphere that "leans" to the right" if you viewed the traveled path from space. So, yes, you would end up farther west than your starting position. Someone worked out that math and it turned out to be 4.3 km west from the starting position, or something.

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

It would certainly make much easier to visualize if we used isotropic reference. I think involving the cardinal directions is what is throwing me off.

Originally posted by Symmetric Chaos
No, they're kilometers.

lol, pwned.

I was thinking 1/60 arc. Didn't realize that km is not the measure used for that...damn. I messed up, badly. 🙁

Originally posted by Symmetric Chaos
Using latitude and longitude isn't important to the question as far as I can tell.

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).

Originally posted by Symmetric Chaos
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.

To avoid the "0" thing, that's why the earth was/is used in that problem.

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.

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

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

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.

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

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.

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.