SolidAngleR: Computing normalized solid angles of polyhedral cones

The SolidAngleR package provides comprehensive methods for computing normalized solid angle measures of polyhedral cones in arbitrary dimensions. It implements state-of-the-art algorithms from computational geometry, offering multiple approaches tailored to different cone properties and dimensions.

Details

Overview

Solid angle measures quantify the "size" of a polyhedral cone as a fraction of the total space. This package provides closed formulas for 2D and 3D cones (Euler-Lagrange formula); hypergeometric series for positive definite cones (Ribando 2006); tridiagonal series for cones with special structure (Fitisone & Zhou 2023); decomposition methods for general cones (Fitisone & Zhou 2023); multivariate normal integration for ecological applications (Song et al. 2018); Monte Carlo estimation for complex geometries; geometric methods for circular cones and intersecting cones (Mazonka 2012); and spanning tree methods for extreme ray computation (Malekmohammadi & Mostafaee 2017).

Main functions

Unified interface

compute_solid_angle: Main function with automatic method selection
diagnose_cone: Diagnostic tool to suggest best method

Dimension-specific formulas

solid_angle_2d: 2D planar angle computation
solid_angle_3d: 3D Euler-Lagrange formula
spherical_triangle_area: Spherical triangle area

Series methods

hypergeometric_series: Ribando's hypergeometric series (Theorem 1.5)
tridiagonal_series: Optimized series for tridiagonal cones (Theorem 4.1)

Geometric methods

solid_angle_cone: Right circular cone
solid_angle_polyhedral: Van Oosterom & Strackee formula
solid_angle_intersecting_cones: Intersecting circular cones
plotIntersectingCones: Interactive 3D visualization

Extreme rays

generate_spanning_trees: Spanning tree generation
solid_angle_3d_from_rays: Solid angle from extreme rays

Utilities

compute_associated_matrix: Associated matrix M_n(C)
is_positive_definite: Positive definiteness check
create_tridiagonal_cone: Construct tridiagonal cone
angle_between: Angle between vectors
lhuilier_angle: L'Huilier's theorem for spherical triangles

Mathematical background

Normalized solid angle

The normalized solid angle measure $\omega$ quantifies the fraction of $n$-dimensional space occupied by a polyhedral cone. In 2D, $\omega = \theta/(2\pi)$ where $\theta$ is the planar angle; in 3D, $\omega = \Omega/(4\pi)$ where $\Omega$ is the solid angle in steradians; and in n-D, $\omega$ is the fraction of the unit sphere covered by the cone.

Key properties

The normalized solid angle $\omega$ has range $\omega \in [0, 1]$; for an orthant (axes-aligned cone), $\omega = 1/2^n$; for full space, $\omega = 1$; and for an empty cone, $\omega = 0$.

Associated matrix

For a simplicial cone $C$ generated by unit vectors $v_1, \ldots, v_n$, the associated matrix $M_n(C)$ is defined as:

$$M_n(C)_{ij} = \begin{cases} 1 & \text{if } i = j \\ -|v_i \cdot v_j| & \text{if } i \neq j \end{cases}$$

The hypergeometric series converges if and only if $M_n(C)$ is positive definite (Theorem 1.5, Ribando 2006).

Method selection guide

Cone property	Dimension	Method	Function
Any	2	Closed formula	`solid_angle_2d()`
Any	3	Euler-Lagrange	`solid_angle_3d()`
Positive definite	Any	Hypergeometric series	`method = "series"`
Tridiagonal V'V	Any	Tridiagonal series	`method = "tridiagonal"`
Any	Any	Decomposition	`method = "decomposition"`
Any	Any	Multivariate normal	`Omega()`

Use method = "auto" in compute_solid_angle for automatic selection, or diagnose_cone for detailed recommendations.

Typical workflows

Workflow 1: General cone computation


library(SolidAngleR)

# Define cone by generator matrix
V <- diag(4)  # 4D orthant

# Automatic method selection
omega <- compute_solid_angle(V)
print(omega)  # 0.0625 (1/16)

Workflow 2: Ecological feasibility domain


# Community interaction matrix
S <- 4
alpha <- matrix(runif(S * S, -1, 0), S, S)
diag(alpha) <- -1

# Feasibility domain size
result <- Omega(alpha)
print(result$Omega)  # Volume
print(result$Omega_scaled)  # Per-species measure

Workflow 3: Geometric shapes


# Circular cone
theta <- pi/3  # 60-degree apex
omega <- solid_angle_cone(theta)

# Intersecting cones
omega_int <- solid_angle_intersecting_cones(
  theta1 = pi/3,
  theta2 = pi/4,
  alpha = pi/6
)

# Visualize
fig <- plotIntersectingCones(
  theta1 = pi/3,
  theta2 = pi/4,
  phi = pi/3
)
fig

Workflow 4: Method comparison


V <- diag(3)

# Compare methods
omega_formula <- compute_solid_angle(V, method = "formula")
omega_series <- compute_solid_angle(V, method = "series")
omega_decomp <- compute_solid_angle(V, method = "decomposition")
omega_mvn <- Omega(V)$Omega

# All should give 0.125
c(omega_formula, omega_series, omega_decomp, omega_mvn)

Applications

Computational geometry

Applications include polyhedral cone volume computation; spherical geometry and geodesic calculations; and solid angle subtended by complex 3D objects.

Computer graphics

Applications include radiosity and lighting calculations; form factor computation; and visibility analysis.

Physics and engineering

Applications include radiation view factors; antenna beam solid angles; and crystallography and molecular geometry.

Note

This package is the original creation of the author in all conceptual, theoretical, and design aspects. Implementation was assisted by Anthropic's Claude Code v.2 (Opus 4.5) to streamline package development. All original ideas, creativity, and scientific contributions belong to the author, who maintains full responsibility for the package's correctness and reliability. While all code has been thoroughly reviewed and tested, users are encouraged to report any issues through the package's issue tracker.

References

Primary 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

Mazonka, O. (2012). Solid angle of conical surfaces, polyhedral cones, and intersecting spherical caps. arXiv:1205.1396v2 (math.MG). https://arxiv.org/abs/1205.1396

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

Song, C., Rohr, R. P., & Saavedra, S. (2018). A guideline to study the feasibility domain of multi-trophic and changing ecological communities. Journal of Theoretical Biology, 450, 30-36. doi:10.1016/j.jtbi.2018.04.030

Additional references

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

Arun, I., & Venkatapathi, M. (2025). An O(n) algorithm for generating uniform random vectors in n-dimensional cones. Sankhya A: The Indian Journal of Statistics, 87(2), 327-348. doi:10.1007/s13171-025-00387-9

Aomoto, K. (1977). Analytic structure of Schlafli function. Nagoya Mathematical Journal, 68, 1-16. https://projecteuclid.org/euclid.nmj/1118786429

Girard, A. (1626). Tables des sinus, tangentes, & secantes. Leiden.

Author

Pablo Almaraz pablo.almaraz@csic.es (ORCID)

Examples

# Quick start examples

# ========================================================================== #
# Example 1: 2D cone                                                      ####
# ========================================================================== #

v1 <- c(1, 0)
v2 <- c(1, 1) / sqrt(2)
V_2d <- cbind(v1, v2)
omega_2d <- compute_solid_angle(V_2d)
print(omega_2d)  # 0.125 (45° / 360°)
#> [1] 0.125

# ========================================================================== #
# Example 2: 3D orthogonal cone (octant)                                 ####
# ========================================================================== #

V_3d <- diag(3)
omega_3d <- compute_solid_angle(V_3d)
print(omega_3d)  # 0.125 (1/8 of space)
#> [1] 0.125

# ========================================================================== #
# Example 3: Diagnose cone properties                                    ####
# ========================================================================== #

V <- diag(4)
diag_report <- diagnose_cone(V)
print(diag_report)
#> Cone diagnostic report
#> ======================
#> 
#> Dimension:                4 
#> Linearly independent:     ✓ Yes 
#> Associated matrix PD:     ✓ Yes 
#> Tridiagonal structure:    ✓ Yes 
#> 
#> Eigenvalues:              1, 1, 1, 1 
#> Minimum eigenvalue:       1 
#> 
#> Suggested method:         tridiagonal series (most efficient) ✓ 

# ========================================================================== #
# Example 4: Circular cone                                               ####
# ========================================================================== #

theta <- pi/3  # 60-degree apex
omega_cone <- solid_angle_cone(theta)
print(omega_cone)
#> [1] 0.25

# ========================================================================== #
# Example 5: Ecological feasibility domain                               ####
# ========================================================================== #

if (FALSE) { # \dontrun{
# 4-species community
S <- 4
alpha <- matrix(runif(S * S, -1, 0), S, S)
diag(alpha) <- -1

result <- Omega(alpha)
cat("Feasibility volume:", result$Omega, "\n")
cat("Per-species measure:", result$Omega_scaled, "\n")
} # }