Simplex - The Standard Simplex

The Standard Simplex

The standard n-simplex (or unit n-simplex) is the subset of Rn+1 given by

The simplex Δn lies in the affine hyperplane obtained by removing the restriction t_i ≥ 0 in the above definition. The standard simplex is clearly regular.

The n+1 vertices of the standard n-simplex are the points e_i ∈ Rn+1, where

e₀ = (1, 0, 0, ..., 0),

e₁ = (0, 1, 0, ..., 0),

e_n = (0, 0, 0, ..., 1).

There is a canonical map from the standard n-simplex to an arbitrary n-simplex with vertices (v₀, …, v_n) given by

The coefficients t_i are called the barycentric coordinates of a point in the n-simplex. Such a general simplex is often called an affine n-simplex, to emphasize that the canonical map is an affine transformation. It is also sometimes called an oriented affine n-simplex to emphasize that the canonical map may be orientation preserving or reversing.

More generally, there is a canonical map from the standard -simplex (with n vertices) onto any polytope with n vertices, given by the same equation (modifying indexing):

These are known as generalized barycentric coordinates, and express every polytope as the image of a simplex: