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first time posting in the math section, so please excuse my disregarding forum policies that I'm not aware of. I tried to find a solution to the problem by applying my own skills, looking up calculus books, different youtube lectures etc. Found some things for the three-dimensional special case, but that wasn't particularly helpful since I don't know how to generalize the solutions to an m-dimensional setting. Anyway, here's the problem:

Suppose we are given a cone and a sphere in m-dimensional Euclidean space. The sphere is the unit-sphere centered at the origin, as is the tip of the cone. The variables are the opening angle of the cone as well as its orientation (given as a point on its central axis). The intersection of those to primitives (assuming the cone's height is sufficiently large) is a cone with an outward curved basis. I need to know the area of the curved basis part.

Is there any obvious approach that I am missing?

Your help is appreciated.

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Why should the orientation matter? the sphere is m-symmetrical.. –  nbubis Jul 31 '12 at 14:45
    
You're right. I actually wanted to write that, but forgot :) The intersection is the same, so you way translate/rotate as you like. –  Michael Nett Jul 31 '12 at 14:49
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1 Answer

up vote 1 down vote accepted

Sounds like what you're looking for is the area of a hyperspherical cap. The cone itself is only relevant for calculating the height of the cap. See the paper "Concise formulas for the area and Volume of a Hyperspherical Cap" by Li here: http://docsdrive.com/pdfs/ansinet/ajms/2011/66-70.pdf Wikipedia also has a good page on hyperspheres. HTH, Mircea

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Hi, I managed to derive a formulation on my own after some time. Turns out there is no explicit formulation. However, the paper is a good resource, thanks for that. –  Michael Nett Sep 24 '12 at 15:27
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