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

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Discussion Overview

The discussion centers on finding the length of a circle at latitude 90 degrees on a unit sphere, specifically using the formula related to Riemannian manifolds. Participants explore the mathematical formulation and parametrization of the sphere to derive the length.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Some participants inquire about the application of the integral formula for calculating the length of the circle, suggesting the need to parametrize the sphere.
  • There is a proposal to use geometric reasoning to determine the radius of the circle instead of relying solely on the integral.
  • One participant expresses uncertainty about how to implement the formula and questions whether the final answer is pi, indicating a potential understanding of the problem.
  • Another participant suggests writing down the equation for the circle in trigonometric coordinates as a step towards solving the problem.
  • There is a challenge to the correctness of a proposed equation, with a suggestion to clarify the functions of x, y, and z in terms of alpha.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the correct approach or final answer, with multiple competing views and uncertainties expressed throughout the discussion.

Contextual Notes

Participants express uncertainty regarding the implementation of the formula and the correctness of certain equations, indicating potential limitations in their understanding of the parametrization and the integral involved.

whattttt
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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
 
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whattttt said:
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.
 
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
 
Last edited:
whattttt said:
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?
 
It works out to be (sin(alpha))^2. do I just put this into the formula
 
whattttt said:
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.
 

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