Polar Sine

The polar sine is a function of a vertex angle of an -dimensional parallelotope or simplex. If the content of the parallelotope is and the lengths of the edges of the parallelotope that meet at vertex are , , ..., , then the value of the polar sine of that vertex is

Changing the length of an edge of the parallelotope by a factor changes the content by the same factor. Thus, a change in edge length does not affect the value of the right-hand side, and the polar sine function is dependent solely on the angles between the edges of the parallelotope, not their lengths. Also the polar sines of all of the vertex angles of the parallelotope are the same, since the right-hand side of the definition does not depend on the vertex chosen. If we extend the facets at a vertex, all of the vertex angles thus formed have the same polar sine, as they are simply translations of the vertex angles of the parallelotope.

If a sphere is centered at the vertex of an -dimensional angle, the rays of the angle intersect the sphere at points that are the vertices of an -dimensional spherical simplex that subtends the angle. We define the polar sine of that spherical simplex to be the polar sine of the angle it subtends. For an -dimensional spherical simplex , if the edges between vertices and have length , the value of its polar sine in space with Gaussian curvature is given by

Let an -dimensional spherical simplex lie on a sphere with center , and construct its polar simplex. Then the -dimensional angle at formed by the rays from through the vertices of the polar simplex has a sine value that is equal to the value of the polar sine of the original simplex.

The limit of the polar sine of an -dimensional spherical simplex as the curvature of the space approaches zero is , where is the content of the Euclidean simplex with the same edge lengths.