Returns the normalized solid angle of a right circular cone with apex half-angle \(\theta\) via the closed-form Mazonka (2012, Equation 9) expression.
Details
A right circular cone with apex at the origin and apex half-angle \(\theta\) subtends a solid angle given by:
$$\Omega = 2\pi(1 - \cos\theta)$$
This implementation returns the normalized value \(\Omega/(4\pi)\), which represents the fraction of the full sphere (Equation 9 in Mazonka 2012).
For special cases: when \(\theta = \pi/2\) (hemisphere), the normalized solid angle equals 0.5; as \(\theta \to 0\) (narrow cone), it approaches 0; and as \(\theta \to \pi\) (full sphere), it approaches 1.
References
Mazonka, O. (2012). Solid angle of conical surfaces, polyhedral cones, and intersecting spherical caps. arXiv:1205.1396 (math.MG). Equation 9. doi:10.48550/arXiv.1205.1396
See also
solid_angle_cone_segment for a cone cut by a plane;
solid_angle_intersecting_cones for the intersection of
two cones; solid_angle_polyhedral for spherical polygon
regions; compute_solid_angle for the dispatcher.
Examples
# Solid angle of a 60-degree cone
solid_angle_cone(pi/3)
#> [1] 0.25
# Solid angle of a hemisphere (90-degree cone) - should be 0.5
solid_angle_cone(pi/2)
#> [1] 0.5
if (FALSE) { # \dontrun{
# Verify the formula for various cone angles
theta_values <- seq(0.1, pi - 0.1, length.out = 20)
omega_values <- sapply(theta_values, solid_angle_cone)
plot(theta_values, omega_values, type = "l",
xlab = "Apex angle (radians)", ylab = "Normalized solid angle",
main = "Solid angle vs. apex angle for circular cones")
abline(h = 0.5, v = pi/2, col = "red", lty = 2)
} # }