# Length of sinusoid on a sphere

1. Apr 7, 2017

### Vrbic

1. The problem statement, all variables and given/known data
Originally the statement:
Find a length of two points on sphere. It was easy.
$\int \sqrt{g_{\phi\phi}}d\phi$
I hope you agree :-) But I have idea, how to find a length of path which is NOT a part of arc (circle). For example sinusoid. Is possible to find length of sinusoid on the sphere and how?

2. Relevant equations
$ds^2=g_{rr}dr^2+g_{\theta\theta}d\theta^2+g_{\phi\phi}d\phi^2$

3. The attempt at a solution
My attempt hit the snag very early :-)
A took $\theta=\pi/2+\sin{\phi}$
$d\theta=\cos{\phi}d\phi$
$ds^2=0+r^2d\theta^2+r^2\sin^2{\theta}d\phi^2$
$ds=r\sqrt{\cos^2{\phi}+\sin^2{(\pi/2+\sin{\phi})}}d\phi$
And now I don't know. I'm not sure if my procedure is so naive, and it exists better, or such problem doesn't have an analytical solution.

2. Apr 7, 2017

### Buffu

I don't agree until you tell me what does those symbols mean.

What does length mean here ? arc length or something else ?

3. Apr 7, 2017

### Vrbic

$r,\theta,\phi$ are spherical coordinates and $\theta=\pi/2$ is equator. $g_{ij}$ is metric tensor in these coordinates. By length I mean arc length (I hope it is same number when you take a ruler and measure sinusoid on a ball).

4. Apr 7, 2017

### Staff: Mentor

@Vrbic, please post questions involving integrals and tensors in the Calculus & Beyond section. These concepts are well beyond the Precalculus level.

5. Apr 7, 2017

### Buffu

@Vrbic Do you really require calculus here ? I think this question is perfectly feasible without calculus.

6. Apr 7, 2017

### Vrbic

Yes, I would like calculations (maybe both:-) ). I believe it is a training for work in general relativity. No?

7. Apr 7, 2017

### Ray Vickson

Calculus is definitely needed. Even the problem of the length of a sinusoid in a plane involves the (non-elementary) elliptic function.

8. Apr 7, 2017

### Buffu

Not that problem. I was talking about,

9. Apr 8, 2017

### Vrbic

Ok, elliptic function are needed for final solution, it seems not trivial. But how to get to them?
1) Is all right my procedure for finding length between two points on a sphere?
2) How to find length of path between two points connected by sinusoid (or sinusoid along all equator?)