Normalized solid angle of a planar cone via inner product

Computes the normalized solid angle of a two-dimensional simplicial cone using the dot-product / acos variant of the planar angle formula. Provided as a complement to solid_angle_2d for numerical cross-checks.

Usage

solid_angle_2d_inner(v1, v2)

Arguments

v1

Numeric vector of length two; first cone generator. Need not be unit-normalized.

v2

Numeric vector of length two; second cone generator.

Value

A single numeric value in \([0, 0.5]\): the planar opening angle divided by \(2\pi\). Colinear generators return 0.

Details

For unit vectors \(v_1, v_2 \in \mathbb{R}^2\) the opening angle is $$\theta = \arccos(v_1 \cdot v_2),$$ giving \(\Omega = \theta / (2\pi)\). The dot product is clamped to \([-1, 1]\) before the acos call to prevent NaN from floating-point overshoot. The acos formulation loses precision for vectors very close to colinear or anti-colinear, so for production use solid_angle_2d (atan2-based) is preferred; this function is retained as a reference implementation and for verification tests.

References

Hanson, A. J. (2006). Visualizing Quaternions. Morgan Kaufmann. ISBN 978-0120884001. (Chapter on planar angle conventions.)

See also

solid_angle_2d for the numerically preferred atan2-based variant; solid_angle_3d for three dimensions; compute_solid_angle for the dispatcher.

Examples

# Right angle (quarter plane)
solid_angle_2d_inner(c(1, 0), c(0, 1))           # 0.25
#> [1] 0.25

# 45-degree opening: same numerical value as solid_angle_2d()
solid_angle_2d_inner(c(1, 0), c(1, 1) / sqrt(2)) # 0.125
#> [1] 0.125