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Solid Angles

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Computing the Continuous Discretely

Part of the book series: Undergraduate Texts in Mathematics ((UTM))

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Abstract

A natural generalization of a 2-dimensional angle to higher dimensions is called a solid angle. Given a pointed cone \(\mathcal{K}\subset \mathbb{R}^{d}\), the solid angle at its apex is the proportion of space that the cone \(\mathcal{K}\) occupies. In slightly different words, if we pick a point \(\mathbf{x} \in \mathbb{R}^{d}\) “at random,” then the probability that \(\mathbf{x} \in \mathcal{K}\) is precisely the solid angle at the apex of \(\mathcal{K}\). Yet another view of solid angles is that they are in fact volumes of spherical polytopes: the region of intersection of a cone with a sphere. There is a theory, which we will develop in this chapter and which goes back to I.G. Macdonald, that parallels the Ehrhart theory of Chapters 3 and 4, with some genuinely new ideas.

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Beck, M., Robins, S. (2015). Solid Angles. In: Computing the Continuous Discretely. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2969-6_13

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