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What is the higher dimensional analogue for the arc of a circle?

I'd like to work with the set of all points lying within a certain distance of a given point on an n-sphere, and I'd like to describe these sets by the (generalised) solid angles which they subtend.

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    @J.M. Make that an answer, would you please?2012-08-01

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By request:

It seems you want either the spherical cap or the spherical wedge, or hyperspherical versions thereof.

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    Thanks, just what I was looking for. Thanks also to anon.2012-08-01