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Generic Monte Carlo estimator of a solid angle

Source: R/geometric_methods.R
solid_angle_monte_carlo.Rd

Estimates the unnormalized solid angle (in steradians) of an arbitrary region on the unit sphere by uniformly sampling points and counting those satisfying the user-supplied membership test. Returns the point estimate together with a binomial standard error.

Usage

solid_angle_monte_carlo(region_test, n_samples = 1e+05, seed = NULL)

Arguments

region_test

Function that takes a 3D point and returns TRUE if inside region

n_samples

Number of Monte Carlo samples (default: 100000)

seed

Random seed for reproducibility (optional)

Value

List containing:

estimate

Estimated solid angle in steradians

std_error

Standard error of the estimate

n_samples

Number of samples used

fraction_inside

Fraction of samples inside the region

Details

This function implements Monte Carlo integration on the unit sphere by:

  1. Generating n_samples points uniformly distributed on the unit sphere

  2. Testing each point against the region criterion

  3. Estimating solid angle as: \(\Omega = 4\pi \times (\text{fraction inside})\)

  4. Computing standard error using binomial statistics: \(\sigma = 4\pi\sqrt{p(1-p)/n}\) where p is the fraction inside

Uniform sampling on the sphere is achieved using the Marsaglia method: generate 3D Gaussian random vectors and normalize to unit length. This ensures equal probability density over the sphere surface.

The convergence rate is O(1/√n), which is typical for Monte Carlo methods. For 1% relative accuracy, approximately n = 10,000 samples are needed. For 0.1% accuracy, n = 1,000,000 samples are recommended.

This method is particularly useful when the region has complex geometry not amenable to analytical formulas; when verification of analytical results is needed; or when approximate solutions with known uncertainty bounds are acceptable.

References

Marsaglia, G. (1972). Choosing a point from the surface of a sphere. Annals of Mathematical Statistics, 43(2), 645-646. doi:10.1214/aoms/1177692644

Arvo, J. (2001). Stratified sampling of spherical triangles. Proceedings of ACM SIGGRAPH 2001, 437-438. doi:10.1145/383259.383300

Pharr, M., Jakob, W., & Humphreys, G. (2016). Physically based rendering: from theory to implementation, 3rd edition. Morgan Kaufmann. ISBN 978-0128006450.

See also

solid_angle_cone, solid_angle_cone_segment, solid_angle_intersecting_cones for analytic backends that avoid Monte Carlo error; generate_point_on_sphere, the single-sample analogue of the uniform-sphere sampling step; verify_cone_uniformity for goodness-of-fit testing on the samples; compute_solid_angle for the dispatcher.

Examples

# Estimate solid angle of a cone
cone_test <- function(point) {
  # Test if point is in cone with axis along z and apex angle pi/3
  z_component <- point[3]
  return(z_component >= cos(pi/3))
}
result <- solid_angle_monte_carlo(cone_test, n_samples = 10000)
cat("Estimated solid angle:", result$estimate, "±", result$std_error)
#> Estimated solid angle: 3.124 ± 0.05431203