information:extraction, information:classification
Pyramid

The solid angle of a four-sided right rectangular pyramid with apex angles a and b (dihedral angles measured to the opposite side faces of the pyramid) is Ω = 4 arcsin (sin (a 2) sin (b 2)). {\displaystyle \Omega =4\arcsin \left(\sin \left({a \over 2}\right)\sin \left({b \over 2}\right)\right).}

If both the side lengths (α and β) of the base of the pyramid and the distance (d) from the center of the base rectangle to the apex of the pyramid (the center of the sphere) are known, then the above equation can be manipulated to give

Ω = 4 arctan α β 2 d 4 d 2 + α 2 + β 2. {\displaystyle \Omega =4\arctan {\frac {\alpha \beta }{2d{\sqrt {4d^{2}+\alpha ^{2}+\beta ^{2}}}}}.}

The solid angle of a right n -gonal pyramid, where the pyramid base is a regular n -sided polygon of circumradius r, with a pyramid height h is

Ω = 2 π − 2 n arctan (tan (π n) 1 + r 2 h 2). {\displaystyle \Omega =2\pi -2n\arctan \left({\frac {\tan \left({\pi \over n}\right)}{\sqrt {1+{r^{2} \over h^{2}}}}}\right).}

The solid angle of an arbitrary pyramid with an n -sided base defined by the sequence of unit vectors representing edges { s, s },... s can be efficiently computed by:

Ω = 2 π − arg ∏ j = 1 n ((s j − 1 s j) (s j s j + 1) − (s j − 1 s j + 1) + i). {\displaystyle \Omega =2\pi -\arg \prod _{j=1}^{n}\left(\left(s_{j-1}s_{j}\right)\left(s_{j}s_{j+1}\right)-\left(s_{j-1}s_{j+1}\right)+i\left\right).}

where parentheses (* *) is a scalar product and square brackets is a scalar triple product, and i is an imaginary unit. Indices are cycled: s = s and s = s. The complex products add the phase associated with each vertex angle of the polygon. However, a multiple of 2 π {\displaystyle 2\pi } is lost in the branch cut of arg {\displaystyle \arg } and must be kept track of separately. Also, the running product of complex phases must scaled occasionally to avoid underflow in the limit of nearly parallel segments.