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    Let S be the unit sphere in Euclidean 3-space, and let d be the usual spherical distance function. Define a circle C with centre P and radius \rho by C = \{ Q \in S : d(P, Q) = \rho \}. How do I show that the circumference of C is 2\pi \sin \rho, and that its area is 2\pi (1 - \cos \rho)?

    I could cheat and say it's obviously true by drawing some diagrams. But I'd like to do this properly - and the first problem I'm coming up to is showing that this spherical circle is indeed a circle in Euclidean space. Even if I could show that, how do I know that it has the same circumference in the spherical metric as in the Euclidean metric? For the second part - how should I define spherical area?
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    Define them in terms of integrals?
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    Which integrals? At the moment I don't have the Riemannian metric yet, so I don't think I'm meant to use that. The definition of length I have involves taking a dissection of the curve, and summing the distances between points on the dissected curve, and taking the supremum of those estimates over all dissections.
 
 
 
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Updated: January 29, 2010

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