Random planar angle by one-dimensional rejection 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 one-dimensional rejection sampling with log-domain comparisons. Numerically stable in high dimensions where the inverse-transform variant underflows.

Usage

generate_planar_angle_rejection(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 probability density function of \(\theta\) is: $$f_\theta(\theta) = \frac{s_{n-1}}{s_n \Theta(\theta_0)} \sin^{n-2}(\theta)$$ This function uses one-dimensional rejection sampling with log transformations to avoid floating-point underflow. The acceptance condition is: $$\log(U) < (n-2)[\log(\sin\theta) - \log(\sin\min(\theta_0, \pi/2))]$$ where \(U \sim \text{Uniform}(0,1)\).

The average number of rejections is approximately linear in n, maintaining the O(n) complexity of the overall algorithm.

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_inverse for the inverse-transform alternative; rejection_cost for the expected number of rejections in the rejection-based cone sampler; generate_cone_sample for the downstream cone sampler.

Examples

# Narrow cap in dimension 100: rejection variant is preferred
angles <- replicate(1000, generate_planar_angle_rejection(pi / 12, 100))
summary(angles)
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>  0.2420  0.2580  0.2599  0.2591  0.2610  0.2618