The polar simplex of an 
-dimensional
spherical simplex is the simplex having edges
that are supplements of the dihedral angles of
the original simplex. In elliptic n-space, the flat pole
of an 
-flat is a point located an arc length of 
radians distant from each point of the 
-flat. There are 
such poles, one for each of the 
facets of the original simplex. Passing an 
-flat through each subset of 
of these poles then divides the space into 
simplices.
There are twice as many simplexes in spherical n-space, with diametrically opposite simplexes being congruent, so the chosen simplex is the one located in the same hemisphere as the original simplex.
The polar simplex of a polar simplex is the original simplex. The principal circumcenter of a simplex is the incenter of its polar simplex, and the principal circumradius of a simplex is the complement of the inradius of its polar simplex. The altitudes of a simplex and its polar simplex lie on the lines connecting corresponding vertices.
