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

Definition 1

n vectors in ndimensional space 1.1

n vectors in mdimensional space for m > n 1.2

Properties 2

History 3

See also 4

References 5

External links 6
Definition
n vectors in ndimensional 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 v_{1}, ..., v_{n}, for n ≥ 2, be nonzero Euclidean vectors in ndimensional space (ℝ^{n}) 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}(\bold{v}_1,\dots,\bold{v}_n) = \frac{\Omega}{\Pi},
where the numerator is the determinant

\begin{align} \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{align}
equal to the hypervolume of the parallelotope with vector edges^{[1]}

\begin{align} \mathbf{v}_1 & = ( v_{11}, v_{12}, \cdots v_{1n} )^T \\ \mathbf{v}_2 & = ( v_{21}, v_{22}, \cdots v_{2n} )^T \\ \vdots \\ \mathbf{v}_n & = ( v_{n1}, v_{n2}, \cdots v_{nn} )^T \\ \end{align}
and in the denominator the nfold product

\Pi = \prod_{i=1}^n \\bold{v}_i\
of the magnitudes v_{i} of the vectors equals the hypervolume of the ndimensional hyperrectangle, with edges equal to the magnitudes of the vectors v_{1}, v_{2}, ... v_{n} (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):

\Omega \leq \Pi \Rightarrow \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}(\bold{v}_1,\dots,\bold{v}_n) \leq 1,\,
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 mdimensional space for m > n
A nonnegative 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

\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 that m=n, the value of Ω for this nonnegative definition of the polar sine is the absolute value of the Ω from the signed version of the polar sine given previously.
Properties

Negation
If the dimension of the space is more than n, then the polar sine is nonnegative; otherwise it changes signs whenever two of the vectors v_{j} and v_{k} are interchanged  due to the antisymmetry of rowexchanging in the determinant:

\begin{align} \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{align}

Invariance under scalar multiplication of vectors
The polar sine does not change if all of the vectors v_{1}, ..., v_{n} are multiplied by positive constants c_{i}, due to factorization:

\begin{align} \operatorname{psin}(c_1 \bold{v}_1,\dots, c_n \bold{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 \bold{v}_i\} \\ & = \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 \\bold{v}_i\} \\ & = \operatorname{psin}(\bold{v}_1,\dots, \bold{v}_n) \\ \end{align}
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.
History
Polar sines were investigated by Euler in the 18th century.^{[3]}
See also
References

^ , volume 156, pages 52–81, 2009Journal of Approximation TheoryGilad Lerman and Tyler Whitehouse. "On ddimensional dsemimetrics and simplextype inequalities for highdimensional sine functions."

^ Eriksson, F. "The Law of Sines for Tetrahedra and nSimplices." Geometriae Dedicata, volume 7, pages 71–80, 1978.

^ Leonhard Euler, "De mensura angulorum solidorum", in Leonhardi Euleri Opera Omnia, volume 26, pages 204–223.
External links
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