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.

n vectors in n-dimensional space


Let v1, ..., vn (n ≥ 1) 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{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}\,,$$

which equals the signed hypervolume of the parallelotope with vector edges
 * $$ \begin{align}

\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{align}$$

and where the denominator is the n-fold product


 * $$\Pi = \prod_{i=1}^n \|\mathbf{v}_i\|$$

of the magnitudes of the vectors, which equals the hypervolume of the n-dimensional hyperrectangle with edges equal to the magnitudes of the vectors ||v1||, ||v2||, ... ||vn|| rather than 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):


 * $$|\Omega| \leq \Pi \implies \frac{|\Omega|}{\Pi} \leq 1 \implies -1 \leq \operatorname{psin}(\mathbf{v}_1,\dots,\mathbf{v}_n) \leq 1\,,$$

as for the ordinary sine, with either bound being reached only in the case that 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 that works in any $m$-dimensional space can be defined using the Gram determinant. It is a ratio where the denominator is as described above. The numerator is

\begin{bmatrix}\mathbf{v}_1 & \mathbf{v}_2 & \cdots & \mathbf{v}_n \end{bmatrix} \right)} \,, $$ where the superscript T indicates matrix transposition. This can be nonzero only if $m ≥ n$. In the case m = n, this is equivalent to the absolute value of the definition given previously. In the degenerate case $m < n$, the determinant will be of a singular $n × n$ matrix, giving $Ω = 0$ and $psin = 0$, because it is not possible to have $n$ linearly independent vectors in $m$-dimensional space when $m < n$.
 * \Omega| = \sqrt{\det \left(\begin{bmatrix}\mathbf{v}_1 & \mathbf{v}_2 & \cdots & \mathbf{v}_n \end{bmatrix}^T

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{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 v1, ..., vn are scalar-multiplied by positive constants ci, due to factorization


 * $$ \begin{align}

\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{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.

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.