Normalized solid angle of a circular cone cut by a plane

Computes the normalized solid angle of the segment obtained by cutting a right circular cone with apex half-angle \(\theta\) by a plane whose normal makes angle \(\gamma\) with the cone axis. Uses the Mazonka (2012, equations 35, 39, 41) closed form, with a numerical integration fallback when the analytic expression is invalid.

Usage

solid_angle_cone_segment(theta, gamma)

Arguments

theta

Apex angle of the cone in radians (0 < theta < pi)

gamma

Angle between the cutting plane normal and cone axis in radians (0 <= gamma <= pi)

Value

Normalized solid angle (0 to 1, where 1 = full sphere)

Details

A cone segment is the portion of a circular cone cut by a plane. The solid angle depends on the cone's apex angle \(\theta\) and the cutting plane's orientation \(\gamma\).

The calculation uses auxiliary angles (Equations 35, 39, 41 in Mazonka 2012):

$$\phi = \arccos\left(\frac{\cos\gamma}{\tan\theta}\right)$$ $$\beta = \arccos\left(\frac{\sin\gamma}{\sin\theta}\right)$$ $$\Omega = 2(\phi - \cos\theta \cdot \beta)$$ If the analytic expression yields a negative or invalid value due to numerical issues or boundary geometry, the function falls back to a robust numerical integration of the equivalent spherical-cap intersection.

Special cases are handled automatically: when \(\gamma = 0\), the function returns the full cone solid angle; when \(\gamma > \pi/2 + \theta\), the plane misses the cone entirely and the function returns 0; and when \(\phi\) or \(\beta\) are invalid, geometric constraints prevent intersection.

References

Mazonka, O. (2012). Solid angle of conical surfaces, polyhedral cones, and intersecting spherical caps. arXiv:1205.1396 (math.MG). Section 4.1 (Cone segment), equations 35, 39, 41. doi:10.48550/arXiv.1205.1396

See also

solid_angle_cone for the uncut cone; solid_angle_intersecting_cones for the related two-cone intersection (the special case \(\theta_2 = \pi/2\) reduces to this function with \(\gamma = \alpha\)); compute_solid_angle for the dispatcher.

Examples

# Full cone (no cut)
solid_angle_cone_segment(pi/3, 0)
#> [1] 0.25

# Cone segment with 45-degree cut
solid_angle_cone_segment(pi/3, pi/4)
#> [1] 0.1340916

if (FALSE) { # \dontrun{
# Visualize how cutting plane angle affects solid angle
theta <- pi/3  # 60-degree cone
gamma_vals <- seq(0, pi/2, length.out = 50)
omega_vals <- sapply(gamma_vals, function(g) solid_angle_cone_segment(theta, g))

plot(gamma_vals * 180/pi, omega_vals,
     type = "l", lwd = 2,
     xlab = "Cutting plane angle (degrees)",
     ylab = "Normalized solid angle",
     main = "Cone segment solid angle vs. cutting plane angle")
abline(h = solid_angle_cone(theta), col = "red", lty = 2)
legend("topright", c("Segment", "Full cone"), lty = c(1, 2), col = c("black", "red"))
} # }