In geometry, the polar sine generalizes the sine function of angle to the vertex angle of a polytope. It is denoted by psin.
Definition
n vectors in n-dimensional space
Let v1, ..., vn (n ≥ 1) be non-zero Euclidean vectors in n-dimensional space (Rn) that are directed from a vertex of a parallelotope, forming the edges of the parallelotope. The polar sine of the vertex angle is:
psin ( v 1 , … , v n ) = Ω Π , {\displaystyle \operatorname {psin} (\mathbf {v} _{1},\dots ,\mathbf {v} _{n})={\frac {\Omega }{\Pi }},}
where the numerator is the determinant
Ω = det [ v 1 v 2 ⋯ v n ] = | v 11 v 21 ⋯ v n 1 v 12 v 22 ⋯ v n 2 ⋮ ⋮ ⋱ ⋮ v 1 n v 2 n ⋯ v n n | , {\displaystyle {\begin{aligned}\Omega &=\det {\begin{bmatrix}\mathbf {v} _{1}&\mathbf {v} _{2}&\cdots &\mathbf {v} _{n}\end{bmatrix}}={\begin{vmatrix}v_{11}&v_{21}&\cdots &v_{n1}\\v_{12}&v_{22}&\cdots &v_{n2}\\\vdots &\vdots &\ddots &\vdots \\v_{1n}&v_{2n}&\cdots &v_{nn}\\\end{vmatrix}}\end{aligned}}\,,}
which equals the signed hypervolume of the parallelotope with vector edges
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