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.

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

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

  • $\begingroup$ 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. $\endgroup$ – Michael Nett Sep 24 '12 at 15:27
  • $\begingroup$ "Explicit" is in the eye of the beholder. Formula (1) in the paper cited is the closed-form solution in terms of the incomplete Beta function. Numerous expressions for the incomplete Beta are known in terms of Gamma func., Hypergeometric func., continuous fractions, etc., and numerical calculators are widely available online. See for example the Wolfram website. $\endgroup$ – Mircea Apr 23 '17 at 18:21

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