Solid angle via Ribando's multivariable hypergeometric series

Computes the normalized solid angle of a simplicial cone using the multivariable hypergeometric series of Ribando (2006), with the dimension- specific specialisations of Fitisone & Zhou (2023). Internally dispatches to a hand-rolled implementation for $n = 2$, a hand-rolled implementation for $n = 3$, and the general recursive enumerator hypergeometric_series_nd for $n \geq 4$.

Usage

hypergeometric_series(V, max_terms = 1000, tol = 1e-10, check_pd = TRUE)

Arguments

V: A square $n \times n$ numeric matrix whose columns are the cone generators. Need not be unit-normalized; columns are rescaled internally.
max_terms: Integer. Hard cap on the number of series terms accumulated before truncation. Default 1000.
tol: Numeric. Per-degree convergence tolerance. The series terminates when the contribution of an entire degree level falls below tol. Default 1e-10.
check_pd: Logical. When TRUE (default), the function verifies that the associated matrix $M_n(C)$ is positive definite before iterating, since this is the necessary and sufficient condition for absolute convergence of the series.

Value

A list with components

solid_angle: Normalized solid angle in $[0, 1]$.
n_terms: Total number of series terms accumulated.
converged: Logical, TRUE when the per-degree contribution fell below tol before max_terms was reached.
associated_matrix: The associated matrix $M_n(C)$.

Details

The Ribando-Aomoto series (equation 5 of Ribando 2006, equation 1 of Fitisone & Zhou 2023) is $$\Omega(C) = \frac{|\det V|}{(4\pi)^{n/2}} \sum_{a \in \mathbb{N}_0^N} \frac{(-2)^{|a|}}{a!}\, \prod_{i=1}^n \Gamma\!\left(\frac{1 + \deg_i(a)}{2}\right)\, \alpha^{a},$$ where $N = \binom{n}{2}$, $a = (a_{ij})_{i<j}$ is a multi-index over pairs of generator indices, $|a| = \sum_{i<j} a_{ij}$, $\deg_i(a) = \sum_{k \neq i} a_{ik}$, and $\alpha_{ij} = v_i \cdot v_j$. The series converges absolutely to $\Omega(C)$ if and only if the associated matrix $M_n(C) = (m_{ij})$ with $m_{ii} = 1$ and $m_{ij} = -|\alpha_{ij}|$ for $i \neq j$ is positive definite (Ribando 2006, Theorem 1.5).

The implementation enumerates terms by total degree $|a|$ so the truncation error after the cut-off can be bounded by the contribution of the next degree level. For $n = 2, 3$ the dispatcher delegates to specialised implementations that exploit the small number of pair indices.

References

Ribando, J. M. (2006). Measuring solid angles beyond dimension three. Discrete & Computational Geometry, 36(3), 479-487. doi:10.1007/s00454-006-1253-4

Aomoto, K. (1977). Analytic structure of Schlafli function. Nagoya Mathematical Journal, 68, 1-16. https://projecteuclid.org/euclid.nmj/1118786429

Fitisone, A., & Zhou, Y. (2023). Solid angle measure of polyhedral cones. arXiv:2304.11102 (math.CO). https://arxiv.org/abs/2304.11102

Examples

if (FALSE) { # \dontrun{
# 3D orthogonal cone: 1/8 of the sphere
V <- diag(3)
hypergeometric_series(V, max_terms = 100)$solid_angle    # 0.125

# 4D orthogonal cone: 1/16 of the sphere
V <- diag(4)
hypergeometric_series(V, max_terms = 200)$solid_angle    # 0.0625
} # }