n vectors in n-dimensional space
Let v1, ..., vn, for n ≥ 2, 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
and in the denominator the n-fold product
of the magnitudes ||vi|| of the vectors equals the hypervolume of the n-dimensional hyperrectangle, with edges equal to the magnitudes of the vectors ||v1||, ||v2||, ... ||vn|| (not 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 only being reached in case all vectors are mutually orthogonal.
In case n = 2, the polar sine is the ordinary sine of the angle between the two vectors.
n vectors in m-dimensional space for m > n
A non-negative version of the polar sine exists for the case that the vectors lie in a space of higher dimension. In this case, the numerator in the definition is given as
where the superscript T indicates matrix transposition. In the case that m=n, the value of Ω for this non-negative definition of the polar sine is the absolute value of the Ω from the signed version of the polar sine given previously.
If the dimension of the space is more than n, then the polar sine is non-negative; otherwise it changes signs whenever two of the vectors vj and vk are interchanged - due to the antisymmetry of row-exchanging in the determinant:
- Invariance under scalar multiplication of vectors
The polar sine does not change if all of the vectors v1, ..., vn are 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.
- Trigonometric functions
- List of trigonometric identities
- Solid angle
- Law of sines
- Cross product and Seven-dimensional cross product
- Graded algebra
- Exterior derivative
- Differential geometry
- Volume integral
- Measure (mathematics)
- Product integral
- Gilad Lerman and Tyler Whitehouse On d-dimensional d-semimetrics and simplex-type inequalities for high-dimensional sine functions
- Eriksson, F. "The Law of Sines for Tetrahedra and n-Simplices." Geometriae Dedicata volume 7, pages 71–80, 1978.
- Leonhard Euler, "De mensura angulorum solidorum", in Leonhardi Euleri Opera Omnia, volume 26, pages 204–223.