Polar sine

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In geometry, the polar sine generalizes the sine function of angle to the vertex angle of a polytope. It is denoted by psin.

Definition[edit]

n vectors in n-dimensional space[edit]

The interpretations of 3D volumes for left: a parallelepiped (Ω in polar sine definition) and right: a cuboid (Π in definition). The interpretation is similar in higher dimensions.

Let v1, ..., vn (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

equal to the hypervolume of the parallelotope with vector edges[1]

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.[2]

The parallelotope is like a "squashed hyperrectangle", so it has less hypervolume than the hyperrectangle, meaning (see image for the 3d case):

and since this ratio can be negative, psin is always bounded between −1 and +1 by the inequalities:

as for the ordinary sine, with either bound only being reached in case all vectors are mutually orthogonal.

In the case n = 2, the polar sine is the ordinary sine of the angle between the two vectors.

In higher dimensions[edit]

A non-negative version of the polar sine which works in any m-dimensional space (mn) can be defined using the Gram determinant. The numerator is given as

where the superscript T indicates matrix transposition. In the case m = n this is equivalent to the absolute value of the definition given previously.

Properties[edit]

Interchange of vectors[edit]

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.

Invariance under scalar multiplication of vectors[edit]

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.

Vanishes with linear dependencies[edit]

If the vectors are not linearly independent, the polar sine will be zero. This will always be so in the degenerate case that the number of dimensions m is strictly less than the number of vectors n.

Relationship to pairwise cosines[edit]

The cosine of the angle between two non-zero vectors is given by

using the dot product. Comparison of this expression to the definition of the absolute value of the polar sine as given above gives:

In particular, for n = 2, this is equivalent to

which is the Pythagorean theorem.

History[edit]

Polar sines were investigated by Euler in the 18th century.[3]

See also[edit]

References[edit]

  1. ^ Lerman, Gilad; Whitehouse, J. Tyler (2009). "On d-dimensional d-semimetrics and simplex-type inequalities for high-dimensional sine functions". Journal of Approximation Theory. 156: 52–81. arXiv:0805.1430. doi:10.1016/j.jat.2008.03.005. S2CID 12794652.
  2. ^ Eriksson, F (1978). "The Law of Sines for Tetrahedra and n-Simplices". Geometriae Dedicata. 7: 71–80. doi:10.1007/bf00181352. S2CID 120391200.
  3. ^ Euler, Leonhard. "De mensura angulorum solidorum". Leonhardi Euleri Opera Omnia. 26: 204–223.

External links[edit]