polar sine

File:3dvol.svg 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 ≥ 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:

:$\backslash operatorname\{psin\}(\backslash mathbf\{v\}\_1,\backslash dots,\backslash mathbf\{v\}\_n)\; =\; \backslash frac\{\backslash Omega\}\{\backslash Pi\},$

where the numerator is the determinant

:$\backslash begin\{align\}$

\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{{cite journal| doi=10.1016/j.jat.2008.03.005 | volume=156 | title= On d-dimensional d-semimetrics and simplex-type inequalities for high-dimensional sine functions | year=2009 | journal=Journal of Approximation Theory | pages=52–81 | last1 = Lerman | first1 = Gilad | last2 = Whitehouse | first2 = J. Tyler| arxiv=0805.1430 | s2cid=12794652 }}

:$\backslash begin\{align\}$

\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

:$\backslash Pi\; =\; \backslash prod\_\{i=1\}^n\; \backslash |\backslash mathbf\{v\}\_i\backslash |$

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||, ... ||v_n|| rather than the vectors themselves. Also see Ericksson.{{cite journal | last1 = Eriksson | first1 = F | year = 1978 | title = The Law of Sines for Tetrahedra and n-Simplices | journal = Geometriae Dedicata | volume = 7 | pages = 71–80 | doi=10.1007/bf00181352| s2cid = 120391200 }}

The parallelotope is like a "squashed hyperrectangle", so it has less hypervolume than the hyperrectangle, meaning (see image for the 3d case):

:$|\backslash Omega|\; \backslash leq\; \backslash Pi\; \backslash implies\; \backslash frac$

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.

A non-negative version of the polar sine that works in any {{math|m}}-dimensional space can be defined using the Gram determinant. It is a ratio where the denominator is as described above. The numerator is

:$$

|\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. This can be nonzero only if {{math|m ≥ n}}. In the case m = n, this is equivalent to the absolute value of the definition given previously. In the degenerate case {{math|m < n}}, the determinant will be of a singular {{math|n × n}} matrix, giving {{math|1=Ω = 0}} and {{math|1=psin = 0}}, because it is not possible to have {{mvar|n}} linearly independent vectors in {{mvar|m}}-dimensional space when {{math|m < n}}.

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.

:$\backslash begin\{align\}$

\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}

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

:$\backslash begin\{align\}$

\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.

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 {{math|m}} is strictly less than the number of vectors {{math|n}}.

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

:$\backslash cos(\backslash mathbf\{v\}\_1,\; \backslash mathbf\{v\}\_2)\; =\; \backslash frac\{\backslash mathbf\{v\}\_1\; \backslash cdot\; \backslash mathbf\{v\}\_2\}\{\backslash |\backslash mathbf\{v\}\_1\backslash |\; \backslash |\backslash mathbf\{v\}\_2\backslash |\}\backslash ,$

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

:$\backslash left|\backslash operatorname\{psin\}(\backslash mathbf\{v\}\_1,\; \backslash ldots,\; \backslash mathbf\{v\}\_n)\backslash right|^2$

= \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 {{math|1=n = 2}}, this is equivalent to

:$\backslash sin^2(\backslash mathbf\{v\}\_1,\; \backslash mathbf\{v\}\_2)\; =\; 1\; -\; \backslash cos^2(\backslash mathbf\{v\}\_1,\; \backslash mathbf\{v\}\_2)\backslash ,,$

which is the Pythagorean theorem.

Polar sines were investigated by Euler in the 18th century.{{cite journal | last1 = Euler | first1 = Leonhard | title = De mensura angulorum solidorum | journal = Leonhardi Euleri Opera Omnia | volume = 26 | pages = 204–223 }}

• 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

• {{MathWorld|PolarSine|Polar Sine}}

Category:Polytopes

Category:Trigonometry