Random planar angle by inverse-transform sampling

Draws a single random planar angle \(\theta \in [0, \theta_0]\) from the distribution required to produce uniform sampling on a spherical cap of half-angle \(\theta_0\) in dimension \(n\), using the inverse cumulative distribution map \(\theta = \Theta^{-1}(U)\) with \(U \sim \mathrm{Uniform}(0, \Omega_0)\).

Usage

generate_planar_angle_inverse(theta0, n)

Arguments

theta0

Numeric. Maximum planar angle (cone half-angle) in radians.

n

Integer. Dimension of the space.

Value

Numeric. Random angle \(\theta \in [0, \theta_0]\).

Details

The cumulative distribution function of \(\theta\) restricted to \([0, \theta_0]\) is \(F_\theta(\theta) = \Theta(\theta) / \Theta(\theta_0)\), where \(\Theta\) is implemented by theta_to_omega. Inverse-transform sampling generates one random angle per call by drawing \(U \sim \mathrm{Uniform}(0, \Omega_0)\) with \(\Omega_0 = \Theta(\theta_0)\) and returning \(\Theta^{-1}(U)\) through omega_to_theta.

This formulation requires both \(\Theta\) and \(\Theta^{-1}\), both of which can underflow for very large \(n\) and very narrow caps; in that regime generate_planar_angle_rejection is preferred.

References

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

See also

generate_planar_angle_rejection for the rejection-based alternative that is numerically stable in high dimensions; theta_to_omega, omega_to_theta for the underlying maps; generate_cone_sample for the downstream cone sampler that consumes this random angle.

Examples

# 30-degree cap in dimension 5: empirical density of planar angles
angles <- replicate(1000, generate_planar_angle_inverse(pi / 6, 5))
summary(angles)
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>  0.1131  0.3568  0.4281  0.4109  0.4801  0.5235