Polar sine

From Wikipedia, the free encyclopedia

In mathematics, the polar sine of a vertex angle of a polytope is defined as follows. Let v1, ..., vn, 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 v1, ..., vn.[1] The absolute value of the polar sine does not change if all of the vectors v1, ..., vn 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.[2]

[edit] References

  1. ^ Eriksson, F. "The Law of Sines for Tetrahedra and n-Simplices." Geometriae Dedicata volume 7, pages 71–80, 1978.
  2. ^ Leonhard Euler, "De mensura angulorum solidorum", in Leonhardi Euleri Opera Omnia, volume 26, pages 204–223.

[edit] External links