Polar sine

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

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, ..., v_n (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:

$\operatorname {psin} (\mathbf {v} _{1},\dots ,\mathbf {v} _{n})={\frac {\Omega }{\Pi }},$

where the numerator is the determinant

${\begin{aligned}\Omega &=\det {\begin{bmatrix}\mathbf {v} _{1}&\mathbf {v} _{2}&\cdots &\mathbf {v} _{n}\end{bmatrix}}={\begin{vmatrix}v_{11}&v_{21}&\cdots &v_{n1}\\v_{12}&v_{22}&\cdots &v_{n2}\\\vdots &\vdots &\ddots &\vdots \\v_{1n}&v_{2n}&\cdots &v_{nn}\\\end{vmatrix}}\end{aligned}}$

equal to the hypervolume of the parallelotope with vector edges

${\begin{aligned}\mathbf {v} _{1}&=(v_{11},v_{12},\dots ,v_{1n})^{T}\\\mathbf {v} _{2}&=(v_{21},v_{22},\dots ,v_{2n})^{T}\\&\,\,\,\vdots \\\mathbf {v} _{n}&=(v_{n1},v_{n2},\dots ,v_{nn})^{T},\\\end{aligned}}$

and in the denominator the n-fold product

$\Pi =\prod _{i=1}^{n}\|\mathbf {v} _{i}\|$

of the magnitudes ||v_i|| of the vectors equals the hypervolume of the n-dimensional hyperrectangle, with edges equal to the magnitudes of the vectors ||v₁||, ||v₂||, ... ||v_n|| (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):

$\Omega \leq \Pi \implies {\frac {\Omega }{\Pi }}\leq 1$

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

$-1\leq \operatorname {psin} (\mathbf {v} _{1},\dots ,\mathbf {v} _{n})\leq 1,\,$

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

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

$\Omega ={\sqrt {\det \left({\begin{bmatrix}\mathbf {v} _{1}&\mathbf {v} _{2}&\cdots &\mathbf {v} _{n}\end{bmatrix}}^{T}{\begin{bmatrix}\mathbf {v} _{1}&\mathbf {v} _{2}&\cdots &\mathbf {v} _{n}\end{bmatrix}}\right)}}\,,$

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

Interchange of vectors

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.

${\begin{aligned}\Omega &=\det {\begin{bmatrix}\mathbf {v} _{1}&\mathbf {v} _{2}&\cdots &\mathbf {v} _{i}&\cdots &\mathbf {v} _{j}&\cdots &\mathbf {v} _{n}\end{bmatrix}}\\&=-\!\det {\begin{bmatrix}\mathbf {v} _{1}&\mathbf {v} _{2}&\cdots &\mathbf {v} _{j}&\cdots &\mathbf {v} _{i}&\cdots &\mathbf {v} _{n}\end{bmatrix}}\\&=-\Omega \end{aligned}}$

Invariance under scalar multiplication of vectors

The polar sine does not change if all of the vectors v₁, ..., v_n are scalar-multiplied by positive constants c_i, due to factorization

${\begin{aligned}\operatorname {psin} (c_{1}\mathbf {v} _{1},\dots ,c_{n}\mathbf {v} _{n})&={\frac {\det {\begin{bmatrix}c_{1}\mathbf {v} _{1}&c_{2}\mathbf {v} _{2}&\cdots &c_{n}\mathbf {v} _{n}\end{bmatrix}}}{\prod _{i=1}^{n}\|c_{i}\mathbf {v} _{i}\|}}\\[6pt]&={\frac {\prod _{i=1}^{n}c_{i}}{\prod _{i=1}^{n}|c_{i}|}}\cdot {\frac {\det {\begin{bmatrix}\mathbf {v} _{1}&\mathbf {v} _{2}&\cdots &\mathbf {v} _{n}\end{bmatrix}}}{\prod _{i=1}^{n}\|\mathbf {v} _{i}\|}}\\[6pt]&=\operatorname {psin} (\mathbf {v} _{1},\dots ,\mathbf {v} _{n}).\end{aligned}}$

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

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

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

$\cos(\mathbf {v} _{1},\mathbf {v} _{2})={\frac {\mathbf {v} _{1}\cdot \mathbf {v} _{2}}{\|\mathbf {v} _{1}\|\|\mathbf {v} _{2}\|}}\,$

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

$\left|\operatorname {psin} (\mathbf {v} _{1},\ldots ,\mathbf {v} _{n})\right|^{2}=\det \!\left[{\begin{matrix}1&\cos(\mathbf {v} _{1},\mathbf {v} _{2})&\cdots &\cos(\mathbf {v} _{1},\mathbf {v} _{n})\\\cos(\mathbf {v} _{2},\mathbf {v} _{1})&1&\cdots &\cos(\mathbf {v} _{2},\mathbf {v} _{n})\\\vdots &\vdots &\ddots &\vdots \\\cos(\mathbf {v} _{n},\mathbf {v} _{1})&\cos(\mathbf {v} _{n},\mathbf {v} _{2})&\cdots &1\\\end{matrix}}\right].$

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

$\sin ^{2}(\mathbf {v} _{1},\mathbf {v} _{2})=1-\cos ^{2}(\mathbf {v} _{1},\mathbf {v} _{2})\,,$

which is the Pythagorean theorem.

History

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

See also

• Trigonometric functions
• List of trigonometric identities
• Solid angle
• Simplex
• Law of sines
• Cross product and Seven-dimensional cross product
• Graded algebra
• Exterior derivative
• Differential geometry
• Volume integral
• Measure (mathematics)
• Product integral

External links

• Weisstein, Eric W. "Polar Sine". MathWorld.