The natural generalization of a two-dimensional angle to higher dimensions is called a *solid angle*. Given a pointed cone к ⊂ ℝ^{d}, the solid angle at its apex is the proportion of space that the cone к occupies. In slightly different words, if we pick a point х ∊ ℝ^{d} “at random,” then the probability that х ∊ к is precisely the solid angle at the apex of к. 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 here that parallels the Ehrhart theory of Chapters 3 and 4, but which has some genuinely new ideas.

## Keywords

Solid Angle Discrete Volume Ehrhart Polynomial Rational Convex Rational Polytopes
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© Springer Science+Business Media, LLC 2007