The 500km question.

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.

Spoiler:
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.

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.