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Normalized solid angle of a polyhedral cone via fan triangulation

Source: R/geometric_methods.R
solid_angle_polyhedral.Rd

Computes the normalized solid angle of a polyhedral cone whose boundary is a spherical polygon with vertices given as rows of vertices. The polygon is fan-triangulated from the first vertex and each spherical triangle is summed via the Van Oosterom- Strackee formula.

Usage

solid_angle_polyhedral(vertices)

Arguments

vertices

Matrix where each row represents a unit vector defining vertices of the polyhedral cone. Must have dimensions n x 3 where n >= 3

Value

Normalized solid angle (0 to 1, where 1 = full sphere)

Details

For a spherical triangle defined by three unit vectors \(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\), the solid angle is computed using the Van Oosterom & Strackee (1983) formula:

$$\tan(\Omega/2) = \frac{|\mathbf{v}_1 \cdot (\mathbf{v}_2 \times \mathbf{v}_3)|}{1 + \mathbf{v}_1 \cdot \mathbf{v}_2 + \mathbf{v}_1 \cdot \mathbf{v}_3 + \mathbf{v}_2 \cdot \mathbf{v}_3}$$

For polygons with more than three vertices, the polygon is decomposed into triangular fans from the first vertex, and the solid angles are summed.

The input vertices are automatically normalized to unit length. The function handles both convex and non-convex polyhedral cones, though for non-convex cases the interpretation depends on the triangulation order.

References

Van Oosterom, A., & Strackee, J. (1983). The solid angle of a plane triangle. IEEE Transactions on Biomedical Engineering, 30(2), 125-126. doi:10.1109/TBME.1983.325207

See also

solid_angle_cone for a right circular cone; solid_angle_3d, spherical_triangle_area for the underlying triangle formula; solid_angle_3d_from_rays for an alternative L'Huilier-based aggregator; compute_solid_angle for the dispatcher.

Mazonka, O. (2012). Solid angle of conical surfaces, polyhedral cones, and intersecting spherical caps. arXiv:1205.1396v2. doi:10.48550/arXiv.1205.1396

Examples

# Equilateral triangle on unit sphere
vertices <- matrix(c(
  1, 0, 0,
  0, 1, 0,
  0, 0, 1
), nrow = 3, byrow = TRUE)
solid_angle_polyhedral(vertices)
#> [1] 0.125

# Square on unit sphere (cube face)
vertices <- matrix(c(
  1, 1, 1,
  -1, 1, 1,
  -1, -1, 1,
  1, -1, 1
), nrow = 4, byrow = TRUE)
vertices <- vertices / sqrt(rowSums(vertices^2))
solid_angle_polyhedral(vertices)
#> [1] 0.1666667

if (FALSE) { # \dontrun{
# Regular octahedron face (equilateral triangle)
theta <- acos(1/3)  # Angle for octahedron vertices
vertices <- matrix(c(
  cos(0), sin(0), 0,
  cos(2*pi/3), sin(2*pi/3), 0,
  cos(4*pi/3), sin(4*pi/3), 0
), nrow = 3, byrow = TRUE)
vertices[,3] <- sqrt(1 - rowSums(vertices[,1:2]^2))
omega <- solid_angle_polyhedral(vertices)
cat("Octahedron face solid angle:", omega, "\n")
cat("Should be 1/8 =", 1/8, "\n")
} # }