Interesting question. (1 Viewer)

[seanieg89]

Thought up this question:

A man stands at some point P on the surface of a spherical planet of radius 1000km. He walks 1km north, 1km east, 1km south, and 1km west and ends up exactly where he started. What are the possible locations of P? (Use spherical coordinates to state your answer.)

Have solved it conceptually but haven't had paper yet to do the calculations. You guys have a try.

 

[Carrotsticks]

For those who are confused and thinking 'ummm isn't this obvious?' , squares on a spherical plane can have an angle sum more than 360 degrees.

Interesting problem, will try.

 

[seanieg89]

(His journey is not a 'square' technically, as the E-W segments are not geodesics on the sphere, they are lines of constant latitude. So in this case the angles at the 'vertices' ARE pi/2. But yes, the fact that we are working on a sphere is why the question is nontrivial.)

I am fairly confident that the equation you end up with has no simple closed form solutions, so its value is purely numerical. But arriving at the equation and being able to qualitatively describe its solutions is an interesting exercise in itself.

 

[Carrotsticks]

Wait, can this problem be solved using just basic knowledge of Spherical co-ordinates, or will it require more advanced knowledge?

Here is my current understanding of the situation, can you confirm? I don't want to be attempting a question off the wrong diagram.

Note: I probably should have named the angles alpha and beta to avoid confusion. Let P be (1000, theta, phi) such that after moving North, delta theta = alpha and after moving East, delta phi = beta.

 

[seremify007]

Dang I'm on a high school student maths forum and sitting here with absolutely NFI.

 

[bleakarcher]

This is university maths. ^

 

[Carrotsticks]

Dang I'm on a high school student maths forum and sitting here with absolutely NFI.

Spherical co-ordinates and geometry is usually taught at a tertiary level, hence this thread being in 'Extracurricular Topics' =p

 

[seanieg89]

Essentially carrotsticks, as long as you are aware that these lines correspond to the lines of constant latitude/longitude. (Don't entirely get your notation but its not terribly important.)

 

[seanieg89]

(And the equation describing the solutions can be found using only high school geometry, its just a little tricky to visualise naturally. I don't think it is possibe to find a solution to this equation of a nice form...unless I am mistaken and wolfram alpha is also lying to me. Although it is not hard to describe the solution set geometrically.)

 

[seanieg89]

(A variant with exact solutions can be found by removing the final westward leg of his journey and posing the same question I think.)

 

[largarithmic]

I'm pretty sure spherical coordinates are not required... cylindrical coordinates make the problem much nicer, but they're not required to solve the problem "conceptually". Anyway you can do it also like, by using spherical coordinates but in terms of like, cones etc. Not too hard, no integration of the coordinates required.

We note that whether or not the point is a solution depends only on its latitude, since the problem is invariant/symmetrical upon rotation about the planets axis - this follows from the definition of north/south/east/west being the directions up/down along the axis for N/S, and perpendicular to the axis for E/W. Now note because of this as well, if you move north 1km, move any amount E/W, then move down 1km, you will always arrive back at the original latitude/z-coordinate. Thats coz moving E/W is equivalent to rotating the planet, and latitude is invariant under that. so all you require is that the longitude before/after moving 1km east/west is the same. Now note that, the change in longitude when moving 1km east/west is directly proportional to the radius of the circle defined by all points of that same latitude (from definition of radians/angles), and the radius of that circle is given by , where r is the radius of the whole planet and z is the z-coordinate corresponding to that latitude (when taken in cylindrical coordinates). From this its obvious after moving 1km north, the absolute value of the z-coordinate has to be the same; i.e. it has to move 1km north to the reflection of that latitude about the equator. I.e. all such points lie on the circle of latitude thats distance 0.5 km from the equator; if you take the radius of the planet to be 1000km, that gives all points of latitude 1/2000 or -1/2000 radians or +/- 9pi/100 degrees.

 

[seanieg89]

(True, by spherical coordinates I more meant in terms of circles of latitude, didnt want to give away the form of the solution though.)

Re you response, the circle of latitude that is 0.5km north of the equator wont be a solution, only the one 0.5km south. But the main point of the question is that this is not the only solution, I won't say more as you are certainly clever enough to figure it out from here.

 

[largarithmic]

(True, by spherical coordinates I more meant in terms of circles of latitude, didnt want to give away the form of the solution though.)

Re you response, the circle of latitude that is 0.5km north of the equator wont be a solution, only the one 0.5km south. But the main point of the question is that this is not the only solution, I won't say more as you are certainly clever enough to figure it out from here.

oh right yeah, there are many many many solutions (infinite number) as you get closer and closer to the north pole

 

[Rezen]

Just a clarification on the question: What happens if you start off less then one kilometer below the north pole? does the man continue walking south after he passes the north pole or does he just stop since there is no more N/W/E and then goes S for 1km (How would one even determine what 'South' is from the north pole if all directions are south)?

 

[seanieg89]

The solutions cluster near the line 1km south of the north pole and near the south pole.

I intended the question to mean that P could not have been within 1km from the north pole, for then the man would not have been able to walk 1km north.