Planar angle to solid-angle fraction in n dimensions

Maps a planar (cross-sectional) angle \(\theta \in [0, \pi]\) to the fraction of the \((n-1)\)-sphere covered by a spherical cap of opening \(\theta\), expressed via the regularized incomplete beta function. Inverse of omega_to_theta.

Usage

theta_to_omega(theta, n)

Arguments

theta

Numeric. Planar angle in radians, \(0 \le \theta \le \pi\).

n

Integer. Dimension of the space.

Value

Numeric. Solid angle fraction \(\Omega \in [0, 1]\).

Details

The function \(\Theta: \mathbb{R} \to \mathbb{R}\) maps the planar cross-sectional angle \(\theta\) to the rotated solid angle fraction \(\Omega\) in n dimensions: $$\Theta(\theta) = \begin{cases} \frac{1}{2} I(\sin^2\theta; \frac{n-1}{2}, \frac{1}{2}) & \theta \in [0, \frac{\pi}{2}] \\ 1 - \frac{1}{2} I(\sin^2\theta; \frac{n-1}{2}, \frac{1}{2}) & \theta \in (\frac{\pi}{2}, \pi] \end{cases}$$ where \(I(x; \alpha, \beta)\) is the regularized incomplete beta function.

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

omega_to_theta for the inverse map; generate_planar_angle_inverse for inverse-transform sampling that uses both maps; generate_cone_sample for the downstream cone sampler.

Examples

theta_to_omega(pi / 4, 3)   # 45-degree cap in 3D
#> [1] 0.1464466
theta_to_omega(pi / 2, 3)   # hemisphere = 0.5
#> [1] 0.5