Geodesics: Donuts are Delicious

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spectacu-awesome
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Geodesics: Donuts are Delicious

Postby spectacu-awesome » Fri Nov 02, 2007 11:23 am UTC

Could someone please help me with a problem?
Given two points on the surface of a torus, A(theta_1, phi_1) and B(theta_2, phi_2), what is the distance over the surface of the torus between the two points, and what is that path?

Put another way, an ant on the surface of a donut wants to travel from sprinkle A to sprinkle B. How far must the ant travel, and what is the shortest path?

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MartianInvader
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Re: Geodesics: Donuts are Delicious

Postby MartianInvader » Fri Nov 02, 2007 1:55 pm UTC

Well, it depends what sort of metric you have on the torus, but I'll assume it's the standard one you get from identifying the sides of a square with sidelength 1.

Try drawing 9 copies of this sqaure in a 3x3 pattern, and put your first point in the middle square. Now you can put 9 copies of your second point, one in each sqaure. Drawing the 9 line segments from your first point to the 9 other points gives you 9 candidates for shortest path. After thinking about it a bit, you should be able to figure out under which conditions you use which path.
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Re: Geodesics: Donuts are Delicious

Postby gmalivuk » Fri Nov 02, 2007 2:14 pm UTC

MartianInvader wrote:Well, it depends what sort of metric you have on the torus, but I'll assume it's the standard one you get from identifying the sides of a square with sidelength 1.

I would assume in general that, unless stated otherwise, you're talking about a torus in the real world, i.e. sitting in Euclidean 3-space. Is the inherited metric in this case the same as you'd get identifying the sides of a square (or variously-proportioned rectangles)?
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Re: Geodesics: Donuts are Delicious

Postby rhino » Fri Nov 02, 2007 3:16 pm UTC

gmalivuk wrote:I would assume in general that, unless stated otherwise, you're talking about a torus in the real world, i.e. sitting in Euclidean 3-space. Is the inherited metric in this case the same as you'd get identifying the sides of a square (or variously-proportioned rectangles)?


You're right to ask that question- in fact it's impossible to embed a locally Euclidean torus into R3 no matter how you distort it. You can be absolutely sure that a normal-shaped torus does not have a Euclidean metric!

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Re: Geodesics: Donuts are Delicious

Postby HenryS » Fri Nov 02, 2007 7:39 pm UTC

gmalivuk wrote:
MartianInvader wrote:Well, it depends what sort of metric you have on the torus, but I'll assume it's the standard one you get from identifying the sides of a square with sidelength 1.

I would assume in general that, unless stated otherwise, you're talking about a torus in the real world, i.e. sitting in Euclidean 3-space.
This would make the question much, much harder, so I'd assume like MartianInvader...

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Re: Geodesics: Donuts are Delicious

Postby gmalivuk » Sat Nov 03, 2007 12:34 am UTC

HenryS wrote:
gmalivuk wrote:
MartianInvader wrote:Well, it depends what sort of metric you have on the torus, but I'll assume it's the standard one you get from identifying the sides of a square with sidelength 1.

I would assume in general that, unless stated otherwise, you're talking about a torus in the real world, i.e. sitting in Euclidean 3-space.
This would make the question much, much harder, so I'd assume like MartianInvader...

It would indeed. But at what level is this question being asked?

(Also, general practice when someone asks what looks like an obvious homework question, especially when they don't admit it as such and haven't even deigned to come to the forum in months, is to overly complicate things. See, for example, the "mass of a feather" thread over in the Science forum.)
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Re: Geodesics: Donuts are Delicious

Postby dosboot » Sat Nov 03, 2007 3:37 am UTC

Would the shortest path still have to be one of those 9 segments, even though their arc lengths are not what we expect? That would reduce the problem to calculations.

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Re: Geodesics: Donuts are Delicious

Postby HenryS » Sat Nov 03, 2007 4:12 am UTC

gmalivuk wrote:(Also, general practice when someone asks what looks like an obvious homework question, especially when they don't admit it as such and haven't even deigned to come to the forum in months, is to overly complicate things. See, for example, the "mass of a feather" thread over in the Science forum.)
Fair enough. Well in that case, depending on the metric, the answer can be anything in (0,\infty).

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Re: Geodesics: Donuts are Delicious

Postby spectacu-awesome » Sat Nov 03, 2007 12:21 pm UTC

gmalivuk wrote:
HenryS wrote:
gmalivuk wrote:
MartianInvader wrote:Well, it depends what sort of metric you have on the torus, but I'll assume it's the standard one you get from identifying the sides of a square with sidelength 1.

I would assume in general that, unless stated otherwise, you're talking about a torus in the real world, i.e. sitting in Euclidean 3-space.
This would make the question much, much harder, so I'd assume like MartianInvader...

It would indeed. But at what level is this question being asked?

(Also, general practice when someone asks what looks like an obvious homework question, especially when they don't admit it as such and haven't even deigned to come to the forum in months, is to overly complicate things. See, for example, the "mass of a feather" thread over in the Science forum.)


yeah, sorry I've been neglecting the forums... :oops:

this is a problem of my own devising, and in true me-style, it has turned out RIDICULOUSLY complicated.

finding A distance is easy (parametri-fy (you know what I mean (OH GOD, how many more parentheses can I nest?(You, know, before I lose track of how many I have to use to close (including this, it's 5)))) the torus, then use the distance of a curve formula), finding the SHORTEST distance isn't. Wikipedia tells me to use "standard techniques of calculus and differential equations" to minimise this distance.
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Re: Geodesics: Donuts are Delicious

Postby LoopQuantumGravity » Sat Nov 03, 2007 2:48 pm UTC

You should use the "geodesic equation" in general to do this kind of thing. You also need to be familiar with the calculus of variations.

I found this with google, but you can probably easily find others by searching for "geodesic equation."
http://oregonstate.edu/~drayt/MTH437/hw/geodesic.pdf

It shows how to calculate the shortest distance between points on a sphere using the geodesic equation. The gab is the metric tensor, which you can calculate for a torus given the equations that generate it.
http://en.wikipedia.org/wiki/Metric_tensor (their G is the matrix gab from before).

If you really want, you can derive the geodesic equation from wikipedia's distance formula and the calculus of variations.

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Re: Geodesics: Donuts are Delicious

Postby mudge » Sun Nov 04, 2007 5:42 am UTC

if we're resorting to google, why aren't we googling "torus geodesic"?
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