Area of the spherical triangle on the unit sphere

Computes the unnormalized solid angle (in steradians) of the spherical triangle on $S^2$ whose vertices are three unit vectors. Equivalent to the spherical excess of the triangle.

Usage

spherical_triangle_area(v1, v2, v3)

Arguments

v1: Numeric vector of length three; first triangle vertex.
v2: Numeric vector of length three; second triangle vertex.
v3: Numeric vector of length three; third triangle vertex.

Value

A single non-negative numeric value: the area of the spherical triangle in steradians. The full sphere has total area $4\pi$, so the normalized solid angle is result / (4 * pi).

Details

The function evaluates the same Van Oosterom-Strackee formula used in solid_angle_3d but returns the unnormalized excess $$E = 2\, \mathrm{atan2}\!\left(|v_1 \cdot (v_2 \times v_3)|,\ 1 + v_1 \cdot v_2 + v_2 \cdot v_3 + v_3 \cdot v_1\right).$$ Mathematically, $E$ equals the spherical excess $(\alpha + \beta + \gamma) - \pi$ of the spherical triangle, where $\alpha, \beta, \gamma$ are the interior angles.

L'Huilier's theorem provides an equivalent formulation in terms of arc lengths; the function lhuilier_angle implements that alternative.

References

Van Oosterom, A., & Strackee, J. (1983). The solid angle of a plane triangle. IEEE Transactions on Biomedical Engineering, 30(2), 125-126. doi:10.1109/TBME.1983.325207

Todhunter, I. (1886). Spherical Trigonometry, 5th edition. Macmillan and Co., London. (Chapter on the spherical excess and L'Huilier's theorem.)

Examples

# Octant: spherical triangle area is pi/2 steradians
E <- spherical_triangle_area(c(1, 0, 0), c(0, 1, 0), c(0, 0, 1))
E                  # ~ 1.5708
#> [1] 1.570796
E / (4 * pi)       # 0.125 normalized
#> [1] 0.125