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Polar Simplex


In elliptic n-space, the flat pole of an (n-1)-flat is a point located an arc length of pi/2 radians distant from each point of the (n-1)-flat. For an n-dimensional spherical simplex, there are n+1 such poles, one for each of its n+1 facets. Passing an (n-1)-flat through each subset of n of these poles then divides the space into 2^n simplices. The polar simplex is the simplex having edges that are supplements of the dihedral angles of the original simplex.

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.


See also

Flat, Flat Pole, Polar Triangle

This entry contributed by Robert A. Russell

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Cite this as:

Russell, Robert A. "Polar Simplex." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/PolarSimplex.html

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