Maps a normalized solid-angle fraction \(\Omega \in [0, 1]\) back to
the corresponding planar opening angle \(\theta \in [0, \pi]\).
Inverse of theta_to_omega, evaluated through the
inverse regularized incomplete beta.
Details
The inverse function \(\Theta^{-1}\) is given by: $$\Theta^{-1}(\Omega) = \begin{cases} \arcsin\sqrt{I^{-1}(2\Omega; \frac{n-1}{2}, \frac{1}{2})} & \Omega \in [0, \frac{1}{2}] \\ \pi - \arcsin\sqrt{I^{-1}(2(1-\Omega); \frac{n-1}{2}, \frac{1}{2})} & \Omega \in (\frac{1}{2}, 1] \end{cases}$$ where \(I^{-1}(y; \alpha, \beta)\) is the inverse of 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
theta_to_omega for the forward map;
generate_planar_angle_inverse for inverse-transform
sampling that uses both maps.
Examples
omega_to_theta(0.5, 3) # hemisphere -> pi/2
#> [1] 1.570796
theta <- pi / 4
omega_to_theta(theta_to_omega(theta, 3), 3) # round-trip recovers theta
#> [1] 0.7853982