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]

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, for n ≥ 2, be non-zero Euclidean vectors 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_1)^1 & (v_2)^1 & \cdots & (v_n)^1 \\
(v_1)^2 & (v_2)^2 & \cdots & (v_n)^2 \\
\vdots & \vdots & \ddots & \vdots \\
(v_1)^n & (v_2)^n & \cdots & (v_n)^n \\
\end{vmatrix}
\end{align}

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

 \begin{align}
\mathbf{v}_1 & = [ (v_1)^1, (v_1)^2, \cdots (v_1)^n ] \\
\mathbf{v}_2 & = [ (v_2)^1, (v_2)^2, \cdots (v_2)^n ] \\
\vdots \\
\mathbf{v}_n & = [ (v_n)^1, (v_n)^2, \cdots (v_n)^n ] \\
\end{align}

and in the denominator the n-fold product

 \Pi = \prod_{i=1}^n \|\bold{v}_i\|

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):

\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.

Properties[edit]

Negation

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:

 \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 absolute value of the polar sine does not change if all of the vectors v1, ..., vn are multiplied by positive constants ci, 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{\left(\prod_{i=1}^n c_i \right)}{\left(\prod_{i=1}^n c_i \right)}\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}

History[edit]

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

See also[edit]

References[edit]

  1. ^ Gilad Lerman and Tyler Whitehouse On d-dimensional d-semimetrics and simplex-type inequalities for high-dimensional sine functions
  2. ^ Eriksson, F. "The Law of Sines for Tetrahedra and n-Simplices." Geometriae Dedicata volume 7, pages 71–80, 1978.
  3. ^ Leonhard Euler, "De mensura angulorum solidorum", in Leonhardi Euleri Opera Omnia, volume 26, pages 204–223.

External links[edit]