How to Find the Length of a Circle on a Unit Sphere?

[whattttt]

Can anyone help with finding the length of a circle (theta) =pi/2 (latitude 90') on the unit sphere. I know it is related to the equation
L=
integral from 0 to T of
Sqrt(g_ij (c't,c't))
The formula is on the wikipedia page called Riemannian manifold so you can get a better idea what it looks like. Any help greatly appreciated. Thanks

 

[lavinia]

Can anyone help with finding the length of a circle (theta) =pi/2 (latitude 90') on the unit sphere. I know it is related to the equation
L=
integral from 0 to T of
Sqrt(g_ij (c't,c't))
The formula is on the wikipedia page called Riemannian manifold so you can get a better idea what it looks like. Any help greatly appreciated. Thanks

parametrize the sphere and write down the circle with the parameters. The do the integral.
easier might be to do a little geometry and figure out the radius of the circle.

 

[whattttt]

I assume the sphere is
x= cos(theta)sin(alpha)
Y= sin(theta)sin(alpha)
Z= cos(alpha)


For a circle theta = pi/2 can you please point me in the right direction how to implement the formula.
I guess g_ij is worked out from the sphere but am not sure how to do the rest. Thanks for any help

Is the final answer pi, as if it is I think I know how it works

 

[lavinia]

I assume the sphere is
x= cos(theta)sin(alpha)
Y= sin(theta)sin(alpha)
Z= cos(alpha)For a circle theta = pi/2 can you please point me in the right direction how to implement the formula.
I guess g_ij is worked out from the sphere but am not sure how to do the rest. Thanks for any help

Is the final answer pi, as if it is I think I know how it works

Write down the equation for the circle in your trigonometric coordinates. What is it?

 

[whattttt]

It works out to be (sin(alpha))^2. do I just put this into the formula

 

[lavinia]

It works out to be (sin(alpha))^2. do I just put this into the formula

I don't think that equation is right.

You need to figure out what x,y, and z are as functions of alpha.