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?

To get you going, have some appreciation in advance!

## Geodesics: Donuts are Delicious

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- spectacu-awesome
**Posts:**14**Joined:**Tue Mar 06, 2007 11:28 am UTC**Location:**the big C down under-
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### Geodesics: Donuts are Delicious

Graphing rational functions is a pain in the asymptote

We're number -[e^(pi*i)]

We're number -[e^(pi*i)]

- MartianInvader
**Posts:**782**Joined:**Sat Oct 27, 2007 5:51 pm UTC

### Re: Geodesics: Donuts are Delicious

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.

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.

Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!

- gmalivuk
- GNU Terry Pratchett
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### Re: Geodesics: Donuts are Delicious

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)?

### Re: Geodesics: Donuts are Delicious

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 R

^{3}no matter how you distort it. You can be absolutely sure that a normal-shaped torus does not have a Euclidean metric!

### Re: Geodesics: Donuts are Delicious

This would make the question much, much harder, so I'd assume like MartianInvader...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.

- gmalivuk
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### Re: Geodesics: Donuts are Delicious

HenryS wrote:This would make the question much, much harder, so I'd assume like MartianInvader...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.

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.)

### Re: Geodesics: Donuts are Delicious

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.

### Re: Geodesics: Donuts are Delicious

Fair enough. Well in that case, depending on the metric, the answer can be anything in (0,\infty).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.)

- spectacu-awesome
**Posts:**14**Joined:**Tue Mar 06, 2007 11:28 am UTC**Location:**the big C down under-
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### Re: Geodesics: Donuts are Delicious

gmalivuk wrote:HenryS wrote:This would make the question much, much harder, so I'd assume like MartianInvader...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.

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...

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.

Graphing rational functions is a pain in the asymptote

We're number -[e^(pi*i)]

We're number -[e^(pi*i)]

- LoopQuantumGravity
**Posts:**416**Joined:**Mon Oct 01, 2007 4:19 am UTC

### Re: Geodesics: Donuts are Delicious

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 g

http://en.wikipedia.org/wiki/Metric_tensor (their G is the matrix g

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

(I haven't slept and am sick, so I apologize in advance for any crazy errors that may or may not be here.)

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 g

_{ab}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 g

_{ab}from before).If you really want, you can derive the geodesic equation from wikipedia's distance formula and the calculus of variations.

(I haven't slept and am sick, so I apologize in advance for any crazy errors that may or may not be here.)

I study theoretical physics & strings, and am a recipient of the prestigious Jayne Cobb Hero of Canton award.

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For the people who are still alive!

And the science gets done and you make a neat gun

For the people who are still alive!

### Re: Geodesics: Donuts are Delicious

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

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