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 ti ≥ 0 in the above definition. The standard simplex is clearly regular.
The n+1 vertices of the standard n-simplex are the points ei ∈ Rn+1, where
- e0 = (1, 0, 0, ..., 0),
- e1 = (0, 1, 0, ..., 0),
- en = (0, 0, 0, ..., 1).
There is a canonical map from the standard n-simplex to an arbitrary n-simplex with vertices (v0, …, vn) given by
The coefficients ti 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:
Read more about this topic: Simplex
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