Calculate solid angle of a 3D cone from extreme rays

Calculates the solid angle of a 3D cone deterministically using its extreme rays (V-representation). The method triangulates the spherical polygon formed by the rays on the unit sphere and sums the areas of the constituent spherical triangles using L'Huilier's theorem.

Usage

solid_angle_3d_from_rays(rays)

Arguments

rays

A numeric matrix with 3 rows, where each column represents a distinct extreme ray of the cone. At least 3 rays are required.

Value

A single numeric value representing the solid angle of the cone in steradians.

Details

The solid angle is a measure of the "visual size" of an object from a point. This function provides a precise value in steradians, avoiding the sampling error inherent in Monte Carlo methods. The total solid angle of a full sphere is 4*pi steradians.

Note

CRITICAL: This function assumes the input rays are ordered sequentially (e.g., counter-clockwise) around the cone's central axis. If the rays are unordered, the triangulation will be incorrect and the result will be invalid. Vertex enumeration algorithms (like in rcdd) do not guarantee such an ordering.

References

Malekmohammadi, N., & Mostafaee, A. (2017). Obtaining all extreme rays of a special cone using spanning trees in a complete digraph: application in DEA. Journal of the Operational Research Society, 69(3), 465-472. doi:10.1057/s41274-017-0265-9

Todhunter, I. (1886). Spherical Trigonometry, 5th edition. Macmillan and Co., London.

See also

generate_spanning_trees for the enumeration of extreme rays that this function consumes; lhuilier_angle for the per-triangle building block; angle_between for the arc-length helper; solid_angle_polyhedral for an alternative polyhedral computation; compute_solid_angle for the dispatcher.

Examples

if (FALSE) { # \dontrun{
# Octant in R^3: solid angle should be pi / 2 steradians
orthant <- matrix(c(1, 0, 0,
                    0, 1, 0,
                    0, 0, 1), nrow = 3)
solid_angle_3d_from_rays(orthant)        # ~ 1.5708

# Square-base wider cone with four ordered rays
rays <- cbind(c(1,  0.5,  0.5),
              c(1, -0.5,  0.5),
              c(1, -0.5, -0.5),
              c(1,  0.5, -0.5))
solid_angle_3d_from_rays(rays)
} # }