Concept
In geometry, the polar sine generalizes the sine function of angle to the vertex angle of a polytope. It is denoted by psin. 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: where the numerator is the determinant which equals the signed hypervolume of the parallelotope with vector edges and where the denominator is the n-fold product of the magnitudes of the vectors, which equals the hypervolume of the n-dimensional hyperrectangle with edges equal to the magnitudes of the vectors ||v1||, ||v2||, ... ||vn|| rather than the vectors themselves. Also see Ericksson. The parallelotope is like a "squashed hyperrectangle", so it has less hypervolume than the hyperrectangle, meaning (see image for the 3d case): as for the ordinary sine, with either bound being reached only in the case that all vectors are mutually orthogonal. In the case n = 2, the polar sine is the ordinary sine of the angle between the two vectors. A non-negative version of the polar sine that works in any m-dimensional space can be defined using the Gram determinant. The numerator is given as where the superscript T indicates matrix transposition. This can be nonzero only if m ≥ n. In the case m = n, this is equivalent to the absolute value of the definition given previously. In the degenerate case m < n, the determinant will be of a singular n × n matrix, giving Ω = 0, because it is not possible to have n linearly independent vectors in m-dimensional space. The polar sine changes sign whenever two vectors are interchanged, due to the antisymmetry of row-exchanging in the determinant; however, its absolute value will remain unchanged. The polar sine does not change if all of the vectors v1, ..., vn are scalar-multiplied by positive constants ci, due to factorization If an odd number of these constants are instead negative, then the sign of the polar sine will change; however, its absolute value will remain unchanged.