Uniform random point on a hollow spherical cap

Returns a single point drawn uniformly from the hollow spherical cap of axis $\hat{\mu}$ and inner / outer planar angles $\theta_1 < \theta_2$ in dimension $n$.

Usage

generate_hollow_cone_sample(
  mu_hat,
  theta1,
  theta2,
  method = c("inverse", "rejection")
)

Arguments

mu_hat: Numeric vector. Central axis direction (will be normalized to unit vector).
theta1: Numeric. Inner planar angle in radians, $0 \le \theta_1 < \theta_2$.
theta2: Numeric. Outer planar angle in radians, $0 < \theta_2 \le \pi$.
method: Character. Method for generating planar angle: "inverse" for inverse transform sampling (default), or "rejection" for rejection sampling.

Value

Numeric vector. Unit vector uniformly distributed on the hollow cone surface.

Details

The hollow cone is defined as the set of all unit vectors $\hat{x}$ satisfying: $$\cos\theta_2 \le \hat{x} \cdot \hat{\mu} \le \cos\theta_1$$

The algorithm generates a random angle $\theta$ from: $$\theta = \Theta^{-1}(U(\Omega_2 - \Omega_1) + \Omega_1)$$ where $U \sim \text{Uniform}(0,1)$, $\Omega_1 = \Theta(\theta_1)$, and $\Omega_2 = \Theta(\theta_2)$.

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

Examples

mu_hat <- c(1, 0, 0)
samples <- replicate(1000, generate_hollow_cone_sample(mu_hat, pi / 6, pi / 3))
range(acos(samples[1, ]))   # angles fall in [pi/6, pi/3]
#> [1] 0.5261712 1.0466128