Polar sine
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
the volume in the numerator being that of the parallelotope whose edges at the vertex in question are the given vectors v1, ..., vn. Also see Ericksson.
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 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 sine is bounded by the inequalities
with either bound only being reached in case all vectors are mutually orthogonal.
Polar sines were investigated by Euler
in the 18th century.
Vertex angle
In geometry, a vertex angle is the angle associated with a vertex of a polygon.The vertex angle in a polygon is measured by the interior side of the vertex. For any simple n-gon, the sum of the interior angles is π radians or 180 degrees....
of a polytope
Polytope
In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...
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
the volume in the numerator being that of the parallelotope whose edges at the vertex in question are the given vectors v1, ..., vn. Also see Ericksson.
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 v1, ..., vn are multiplied by positive constants. In case n = 2, the polar sine is the ordinary sine
Sine
In mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....
of the angle between the two vectors.
As for the ordinary sine, the polar sine is bounded by the inequalities
with either bound only being reached in case all vectors are mutually orthogonal.
Polar sines were investigated by Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
in the 18th century.