Solid angle of a simplicial cone via signed decomposition

Computes the normalized solid angle of a simplicial cone by combining the Fitisone-Zhou line-based decomposition with the hypergeometric or tridiagonal series on each positive definite sub-cone, then aggregating the signed contributions. Handles the general case where the associated Gram matrix is not positive definite.

Usage

solid_angle_decomposition(V, max_terms = 500, tol = 1e-10)

Arguments

V

A square \(n \times n\) numeric matrix whose columns are linearly independent cone generators.

max_terms

Integer. Maximum number of series terms used on each positive definite sub-cone. Default 500.

tol

Numeric. Convergence tolerance for the series backends. Default 1e-10.

Value

A single numeric value in \([0, 1]\): the normalized solid angle of the input cone, equal to the sum of signed solid angles of the sub-cones returned by decompose_cone.

Details

The function applies the closed-form formula directly when \(n = 2\) or when \(n = 3\) with positive definite associated matrix. Otherwise it calls decompose_cone to obtain a signed family \(\{(C_i, s_i)\}\) and accumulates $$\Omega(C) = \sum_i s_i \, \Omega(C_i),$$ where each \(\Omega(C_i)\) is computed by the closed-form formula in two or three dimensions, by tridiagonal_series when the Gram matrix of \(C_i\) is tridiagonal, or by hypergeometric_series otherwise. Sub-cones whose rank falls below \(n\) contribute zero by Theorem 3.3 case (I).

Computational complexity is at most \(O((n-1)!)\) sub-cones by Corollary 3.4, but in practice the count is much smaller for well-conditioned cones. Deep recursion can be triggered by adversarial configurations and may exhaust the stack; the user-facing dispatcher compute_solid_angle guards against this by selecting the decomposition route only when the associated matrix is genuinely not positive definite.

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

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

See also

decompose_cone for the underlying signed decomposition; hypergeometric_series, tridiagonal_series for the per-sub-cone series backends; solid_angle_2d, solid_angle_3d for the closed-form formulas; compute_solid_angle for the dispatcher.

Examples

# 2D cone: closed-form short-circuit
V2 <- cbind(c(1, 0), c(0.5, sqrt(3) / 2))
solid_angle_decomposition(V2)
#> [1] 0.1666667

# 3D orthogonal cone: 1/8 of the sphere
V3 <- diag(3)
solid_angle_decomposition(V3)
#> [1] 0.125

if (FALSE) { # \dontrun{
# 3D cone requiring decomposition (non-PD associated matrix)
set.seed(123)
V3_npd <- normalize_vectors(matrix(rnorm(9), nrow = 3))
solid_angle_decomposition(V3_npd)
} # }