Normalized solid angle of the intersection of two circular cones

Computes the normalized solid angle of the spherical region in which two circular cones with common apex overlap. The user can pick the integral, analytic-special-case, or Monte Carlo backend, with an optional fallback when the chosen route fails.

Usage

solid_angle_intersecting_cones(
  theta1,
  theta2,
  alpha,
  method = c("integral", "analytic", "mc"),
  fallback = c("integral", "mc", "none"),
  mc_samples = 2e+05,
  mc_seed = NULL
)

Arguments

theta1: Apex angle of first cone in radians (0 < theta1 < pi)
theta2: Apex angle of second cone in radians (0 < theta2 < pi)
alpha: Angle between cone axes in radians (0 <= alpha <= pi)
method: Computation method: "integral" (default), "analytic", or "mc" for Monte Carlo.
fallback: Fallback method if analytic computation fails: "integral", "mc", or "none".
mc_samples: Number of Monte Carlo samples if method = "mc" or fallback = "mc" (default: 200000).
mc_seed: Optional random seed for Monte Carlo.

Value

Normalized solid angle of the intersection region (0 to 1, where 1 = full sphere)

Details

Compute solid angle of intersection of two cones

The intersection of two circular cones corresponds to the intersection of two spherical caps on the unit sphere. In a coordinate system where the first axis is the north pole and the second axis is separated by angle $\alpha$, a point on the sphere is $$x(\theta,\varphi) = (\sin\theta\cos\varphi,\; \sin\theta\sin\varphi,\; \cos\theta)$$ and belongs to the second cone if $$\sin\alpha \sin\theta \cos\varphi + \cos\alpha \cos\theta \ge \cos\theta_2.$$ For each $\theta \in [0,\theta_1]$, the admissible azimuth range is $$\varphi \in [-\arccos C(\theta),\, \arccos C(\theta)]$$ where $$C(\theta) = \frac{\cos\theta_2 - \cos\alpha \cos\theta}{\sin\alpha \sin\theta}.$$

The solid angle (steradians) of the intersection is then $$\Omega = \int_0^{\theta_1} L(\theta)\,\sin\theta\,d\theta$$ with $L(\theta)$ the admissible azimuth length. The result is normalized by $4\pi$.

All methods handle several special cases: when there is no overlap (if $\alpha \ge \theta_1 + \theta_2$), the function returns 0; when one cone contains the other (if $\alpha \le |\theta_1 - \theta_2|$), it returns the smaller cone; for co-directed axes ($\alpha \approx 0$), it returns the smaller cone; for opposite axes ($\alpha \approx \pi$), it uses a closed-form band area; when two hemispheres intersect, it returns $(\pi - \alpha)/(2\pi)$; and when one cone is a hemisphere, it reduces to solid_angle_cone_segment with $\gamma = \alpha$.

References

Mazonka, O. (2012). Solid angle of conical surfaces, polyhedral cones, and intersecting spherical caps. arXiv:1205.1396 (math.MG). Section 4 (Intersecting cones). doi:10.48550/arXiv.1205.1396

Examples

# Two 60-degree cones with 30-degree separation
solid_angle_intersecting_cones(pi/3, pi/3, pi/6)
#> [1] 0.1781148

# Two hemispheres at 90 degrees
solid_angle_intersecting_cones(pi/2, pi/2, pi/2)
#> [1] 0.25

# Two identical cones with varying separation
if (FALSE) { # \dontrun{
theta <- pi/4  # 45-degree cones
alpha_vals <- seq(0, pi, length.out = 50)
omega_vals <- sapply(alpha_vals, function(a) {
  solid_angle_intersecting_cones(theta, theta, a)
})

plot(alpha_vals * 180/pi, omega_vals,
     type = "l", lwd = 2,
     xlab = "Separation angle (degrees)",
     ylab = "Normalized intersection solid angle",
     main = "Intersection of two 45-degree cones")
abline(h = solid_angle_cone(theta), col = "red", lty = 2)
legend("topright", c("Intersection", "Single cone"),
       lty = c(1, 2), col = c("black", "red"))
} # }