(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

This isn't actually homework, but just related to something I am working on.

Basically I need to find the distance between two points along a sphere. Not the linear distance, but the distance along the surface.

This would be akin to "arc length" except in three dimensions.

2. Relevant equations

2D arc length: [itex]s = r \alpha[/itex]

3D:

[tex]ds = dr \hat{r} + r d\theta \hat{\theta} + r sin(\theta) d\phi \hat{\phi}[/tex]

3. The attempt at a solution

I'm sure I already have an equation and a method of solving this by changing the coordinates to Cartesian, and then taking the dot product to find the angle and then using the 2D arc length equation to get the distance along the surface of the sphere between the two points.

However, I am writing this because I am interested in using the second equation I gave above to find the distance. I'm not sure if I'm doing it right but it looks like I could just integrate so I have "s" instead of "ds" and the unit vectors are constant so they come outside of the integrals:

[tex]s = \hat{r}\int_{r}\int_{\theta}\int_{\phi}dr + \hat{\theta}\int_{r}\int_{\theta}\int_{\phi}r d\theta + \hat{\phi}\int_{r}\int_{\theta}\int_{\phi}r sin(\theta) d\phi[/tex]

The first term would become {r-hat}*(r2-r1) right?

But what I'm not sure of are the other terms because just looking at the second term by itself you can split it up in this way:

[tex]\hat{\theta}\int_{r}r \int_{\theta}d\theta[/tex]

The thing that concerns me is that there is no "dr", so how do I go about integrating this? Similarly with the third term.

Can I not find the distance in this way? I don't see why I shouldn't be able to...

perhaps the "r" is a constant in this case...

I hope this question isn't too dumb, and if it is please just bare with me and try to help me understand. Any advice is appreciated.

Thanks.

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# Homework Help: Surface integration

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