Construct a tridiagonal simplicial cone from consecutive angles

Constructs a set of \(n\) unit vectors in \(\mathbb{R}^n\) whose Gram matrix \(V^\top V\) is tridiagonal with the prescribed consecutive cosines on the off-diagonal. Supplies the canonical input to tridiagonal_series.

Usage

create_tridiagonal_cone(angles)

Arguments

angles

A numeric vector of length \(n - 1\) containing the angles, in radians, between consecutive generators. Each entry must lie strictly within \((0, \pi)\).

Value

An \(n \times n\) numeric matrix V whose columns are unit vectors satisfying \(v_i \cdot v_{i+1} = \cos(\text{angles}[i])\) and \(v_i \cdot v_j = 0\) for \(|i - j| > 1\).

Details

The construction builds the desired tridiagonal Gram matrix \(G_{ij} = \delta_{ij} + \cos(\text{angles}[i]) (\delta_{i, j-1} + \delta_{i, j+1})\) and recovers V as the Cholesky factor: \(V\) upper triangular with \(V^\top V = G\). If the constructed \(G\) is not positive definite (which happens when the consecutive angles are too large to admit a coherent set of generators), the function stops with an informative error.

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

tridiagonal_series for the natural consumer of the returned matrix; is_tridiagonal for the predicate it satisfies; compute_dot_product_matrix for inspecting the resulting Gram structure; compute_solid_angle for the dispatcher that auto-selects the tridiagonal backend.

Examples

if (FALSE) { # \dontrun{
angles <- rep(pi / 3, 3)             # consecutive 60-degree angles in R^4
V <- create_tridiagonal_cone(angles)
is_tridiagonal(t(V) %*% V)            # TRUE
tridiagonal_series(V)$solid_angle
} # }