Another math problem! (Not HW)

[Julian]

This isn't HW, it's just a "bonus" question that my Physics Lab TA came up with, supposedly this isn't a trick question. Also, he encouraged us to ask others, but in before e-mailing him, etc, anyway.

On a 2D x-y plane, starting at the origin, you want to reach the point (1,1) with a continuous path. The total length/distance of the path needs to be of length 1.

How the hell would I go about doing this? When I try to take the logical approach, the "shortest line" is of length root 2, which is greater than 1.

From what I heard him talking about, it sounds like there's some path which gives distance=1, and you use some kind of proof to prove that it's continuous, or something, idk.

Anyone got any ideas?

[Khamsin]

What topics have you been studying in class lately?

[Jefe]

Is that the actual question?

Spoiler: show

http://i35.tinypic.com/zvo4m0.jpg

You want to get to that point, but limited to a length of 1?

[ilduce]

The two points would have to be on different planes. The Euclidean distance between two points of the plane with Cartesian coordinates (x1,y1) and (x2,y2) is
d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}. That's never going to change, the only conceivable way to get a distance of 1 between 0,0 and 1,1 is to manipulate the plane or separate the points on different planes. I sense trickery in this problem.

[Kaylia]

Punch your TA (but make sure you didn't misquote his question first). Common sense dictacte that a 2D xy plane refer to the normal eucledien distance.


If you want to be a smart ass, say that you used the discrete metric. Basically, distance between 2 points is 0 if it's the same point, otherwise it's 1. It would also be continuous with the definition of continuity you use in topology, but it's a bit more complicate to demonstrate. The definition of continuity ask you to cover the whole space, and since the whole space is limited to "0" and "1", it will works here.


You use distance function like this one in topology, but it's not something you can use randomly without mentioning it first. Beside, xy kinda imply a 2 dimensional spaces, and discret metrics doesn't use 2 dimension (can probably project your 2 dimensional space on a 0 dimensional space), but w/e, you can only give a retarded answer to a retarded question.


[edit]
Woozie or another mathematician can probably nitpick what I just said, but I think I have the essential right..I hope.

[Korietsu]

I'm going to go ahead and suggest line integration >_>.

[Ramor]

say you wormholed, teleported, waypoints, tele-holla'ed etc... your point the correct distance.

if he denies this claim, draw two points on a piece of graph paper, and then fold it so that the distance between the origin and (1,1) is 1

[Eliseos]

You could try using arc-length (why you would want to do so on a straight line is stupid). Parameterize sqrt 2 to be a distance of 1, then you've gone exactly a distance of one in the shortest path. The question is absolutely retarded though, if you are remembering it correctly.

[Kaylia]

I'm going to go ahead and suggest line integration >_>.

"The total length/distance of the path needs to be of length 1."

Nromally, you're not talking about length/distance when you do line integral (at least, not if (1,1) is a position). It would be an incorrect use of the language imo.

If you were talking about calculating arc length, you would find sqrt(2), unless it's not an eucledian space.

[SephYuyX]

Is that the actual question?

Spoiler: show

http://i35.tinypic.com/zvo4m0.jpg

You want to get to that point, but limited to a length of 1?

Isnt this correct? 1,1 from 0 is 1.

[Jefe]

The question ain't clear that's why I asked again. Anyway, my first thought was to fold the plane and viola.

[Kaylia]

You could try using arc-length (why you would want to do so on a straight line is stupid). Parameterize sqrt 2 to be a distance of 1, then you've gone exactly a distance of one in the shortest path. The question is absolutely retarded though, if you are remembering it correctly.

You could normalize your distance function instead. It's pretty much the same thing as you said, but a lot easier to write
d(x,y) = ||x,y||/sqrt2


Althought, I still think discrete metric and the implied continuity would be the most elegant answer to this question. If it's a trap question, you would definitively shut him up, and if he was serious, that's probably where the question originate from.

[Marootsoobutsu]

Make your starting point 0,1; 1,0; 2,1; or 1,2. >.>

Edit: Beated

[Woozie]

I think the question is asking to find a function such that function passes through the points (0,0) and (1,1), and the arc length of the segment of the path going from (0,0) to (1,1) is exactly 1. No changing the geometry of the plane or redefining the metric space. He PM'd me this problem a few days ago. I worked on it for hours and couldn't figure it out. Tried it again yesterday and still couldn't get it.

Edit: And from the way I understood it, the function had to be continuous. Otherwise I could easily make up some function that works.

Edit2: No, he's not talking about the distance between those two points on the plane, and no, you can't change the starting point (if I'm understanding this correctly).

[Nyima]

Well that's simple, Just come up with a new unit and have this new unit = sqrt2! Boom length of 1 "new unit"

[Silviya]

Does "continuous path" imply he wants a continuous function that passes through both (0,0) and (1,1) such that the length of the segment connecting the two points is 1?

If not just draw an arc or something with a length of 1 connecting the two.

If so then I'm thinking the function for a spiral could probably be manipulated to fit.

[Julian]

I think the question is asking to find a function such that function passes through the points (0,0) and (1,1), and the arc length of the segment of the path going from (0,0) to (1,1) is exactly 1. No changing the geometry of the plane or redefining the metric space. He PM'd me this problem a few days ago. I worked on it for hours and couldn't figure it out. Tried it again yesterday and still couldn't get it.

This.

All he did in class was draw an x,y plane, put a point at 0,0 and another at 1,1, and said to make a path that connects the two, is continuous, and has length of 1.

This is just kinda a bonus question, if anyone gets it right they don't have to do the next lab and get an auto 20/20 on it <.<

[Zalius]

stups, you just draw a line from the origin to the dot.

[SephYuyX]

This.

All he did in class was draw an x,y plane, put a point at 0,0 and another at 1,1, and said to make a path that connects the two, is continuous, and has length of 1.

This is just kinda a bonus question, if anyone gets it right they don't have to do the next lab and get an auto 20/20 on it <.<

Put a piece of paper on his desk, draw the plane, put the dots on, and then draw a line between them. You win.
You made a path, its continuous, and it's a length of one. No mention of having to make a formula for it.

[Kaylia]

I think the question is asking to find a function such that function passes through the points (0,0) and (1,1), and the arc length of the segment of the path going from (0,0) to (1,1) is exactly 1. No changing the geometry of the plane or redefining the metric space. He PM'd me this problem a few days ago. I worked on it for hours and couldn't figure it out. Tried it again yesterday and still couldn't get it.

Edit: And from the way I understood it, the function had to be continuous. Otherwise I could easily make up some function that works.

Edit2: No, he's not talking about the distance between those two points on the plane, and no, you can't change the starting point (if I'm understanding this correctly).

That's just impossible then. Arc length will always be longer than a straigth line, and the straigth line is obviously sqrt(2) with normal metric.

At least, as long you have triangle inequality, you won't be able to find such function, and sincerely, I don't know how you can break triangle inequality without changing the metric completely. That's why I proposed a new metric earlier, because I don't see how it's possible otherwise.

If you can find a solution, I would really like to hear it though. There is always weird mathematics that I don't know about.