Geometry – Spatial Angles in Higher Dimensions

Question

Wikipedia gives an excellent treatise about solid angles in 1-2-3-Dimensions.
But how about n-D? I read once some notes from a seminar held
during WWII in Switzerland, and one result concerned spatial angles
in even dimensions (I have forgotten the reference), but I would like
to have a similar general definition, likely computationally nice.

So my question is: "What is a general expression for spatial angles in $\mathbb{R}^d$, given that origo-based vectors $x_1..x_{d+1}$ are known". It can be recursive of course.

And yes, I have an application on mind.

here is my reference:
http://en.wikipedia.org/wiki/Solid_angle

Thank you for any help 🙂

Answer 1

Let me cite a theorem by Ribando (Measuring Solid Angles Beyond Dimension Three, Discrete Comput Geom 36:479–487 (2006)), which is a rediscovery of a result of Aomoto (Analytic structure of Schlafli function, Nagoya Math. J. 68:1-16 (1977)) as described in the paper by Beck, Robins and Sam (Positivity Theorems for Solid-Angle Polynomials, Beitrage zur Algebra und Geometrie, Vol. 51, No. 2, 493-507 (2010)). I cite Ribando's result as its statement more to my taste:

Let $\Omega \subseteq \Bbb{R}^n$ be a solid-angle spanned by unit vectors $\lbrace v_1 , \dots , v_n \rbrace$, let $V$ be the matrix whose ith column is $v_i$ , and let $\alpha _{ij} = v_i \cdot v_j$ as above. Let $T_{\alpha}$ be the following infinite multivariable Taylor series: $$T_{\alpha} = \dfrac{det \ V}{(4 \pi )^{n/2}} \sum _{a \in \Bbb{N}^{{n \choose 2}}} \left[ \dfrac{(-2)^{\sum _{i < j} a_{ij}}}{ \Pi _{i<j} a_{ij}!} \Pi _{i} \Gamma \left( \dfrac{1 + \sum _{m \neq i} a_{im}}{2} \right) \right] \alpha^{a}$$ The series $T_{\alpha}$ agrees with the normalized measure of solid-angle $\Omega$ whenever $T_{\alpha}$ converges.