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, for 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:

$\operatorname{psin}(\bold{v}_1,\dots,\bold{v}_n) = \frac{\Omega}{\Pi},$

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 n-fold product

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

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.^[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 m-dimensional space for m > n

A non-negative 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 non-negative 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 non-negative; otherwise it changes signs whenever two of the vectors v_j and v_k 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 polar sine does not change if all of the vectors v₁, ..., 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

• 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

References

1. Gilad Lerman and Tyler Whitehouse. "On d-dimensional d-semimetrics and simplex-type inequalities for high-dimensional sine functions." Journal of Approximation Theory, volume 156, pages 52–81, 2009
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

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