Polar sine

In mathematics, the polar sine of a vertex angle of a polytope is defined as follows. Let "v"₁, ..., "v"_"n", "n" ≥ 2, be non-zero vectors from the vertex in the directions of the edges. The polar sine of the vertex angle is

: $operatorname\{psin\}(v\_1,dots,v\_2)\; =\; frac\{operatorname\{volume\}(v\_1,dots,v\_n)\}\{|v\_1|cdots|v\_n,$

the volume in the numerator being that of the parallelotope whose edges at the vertex in question are the given vectors "v"₁, ..., "v"_"n". [Eriksson, F. "The Law of Sines for Tetrahedra and "n"-Simplices." "Geometriae Dedicata" volume 7, pages 71–80, 1978.]

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 are interchanged.

The absolute value of the polar sine does not change if all of the vectors "v"₁, ..., "v"_"n" are multiplied by positive constants. In case "n" = 2, the polar sine is the ordinary sine of the angle between the two vectors.

As for the ordinary sine, the polar sign is bounded by the inequalities:$-1\; leq\; operatorname\{psin\}(v\_1,dots,v\_2)\; leq\; 1,,$with either bound only being reached in case all vectors are mutually orthogonal.

Polar sines were investigated by Euler in the 18th century. [Leonhard Euler, "De mensura angulorum solidorum", in "Leonhardi Euleri Opera Omnia", volume 26, pages 204–223.]

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