Signed decomposition of a simplicial cone into PD or low-dimensional pieces

Decomposes a simplicial cone into finitely many signed sub-cones using the line-based decomposition of Fitisone & Zhou (2023, Theorem 3.3, second decomposition). Each sub-cone is either lower-dimensional (and so contributes zero to the solid angle) or has a positive definite associated matrix (and so falls within the convergent regime of Ribando's hypergeometric series).

Usage

decompose_cone(V, method = "line")

Arguments

V

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

method

Character. Currently only "line" (the Theorem 3.3 line-based decomposition) is implemented.

Value

A list with components

cones

A list of \(n \times n\) numeric matrices, one per sub-cone in the decomposition.

signs

Integer vector of \(\pm 1\), the inclusion-exclusion sign of each sub-cone.

n_cones

Length of cones.

all_pd

Logical, TRUE when every full-rank sub-cone has a positive definite associated matrix.

Details

By Theorem 3.3 of Fitisone & Zhou (2023), any simplicial cone admits a decomposition into at most \((n-1)!\) sub-cones (Corollary 3.4), each falling into one of two terminal classes: (I) affine dimension strictly less than \(n\), contributing zero to the solid angle; or (II) positive definite associated matrix, so that Ribando's series (Theorem 1.5) converges and the sub-cone admits exact computation.

The reconstruction identity is $$\Omega_n(C) = \sum_{i} s_i \cdot \Omega_n(C_i),$$ where \(s_i \in \{-1, +1\}\) are the signs returned in signs and \(C_i\) are the cones returned in cones. The wrapper solid_angle_decomposition performs this aggregation.

Algorithmically, the decomposition is recursive. The current implementation chooses the last column \(w_n\) as the splitting line, computes the signs \(\delta_i = \mathrm{sign}(w_i \cdot w_n)\) for \(i < n\), and constructs the sub-cones using equations 14-17 of the source paper.

References

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

See also

solid_angle_decomposition for the aggregating wrapper that returns the scalar solid angle; compute_associated_matrix, is_positive_definite for the spectral tests used during decomposition; compute_solid_angle for the dispatcher.

Examples

if (FALSE) { # \dontrun{
# Decompose a 3D cone with non-positive-definite associated matrix
v1 <- c(1, 0, 0)
v2 <- c(-0.5, 0.8, 0.3)
v3 <- c(0.2, 0.3, 0.9)
V  <- normalize_vectors(cbind(v1, v2, v3))
decomp <- decompose_cone(V)
decomp$n_cones
decomp$all_pd
} # }