Barycentric coordinate

Definition

That is to say E a vector Space Euclidean. Let us take in this space n+1 points:

$P\_0,\dots ,\; P\_n\; \backslash ,$

One also imposes that $dim\; (vect\; (P\_0,\dots ,\; P\_n))\; =\; n$ (i.e. these points all are not in same the Hyperplan).

Theorem

For all M of E , it is a (n+1) - uplet of Scalaire S $(x\_0,\dots ,\; x\_n)$ such as:

$\backslash \; sum\_\; \{i=0\}\; ^\; \{N\}\; \{x\_i\; \backslash \; vec\; \{MP\_i\}\}\; =\; \backslash \; vec\; \{0\}$

$\backslash \; sum\_\; \{i=0\}\; ^\; \{N\}\; \{x\_i\}\; =\; 1$

One calls this (n+1) - uplet the barycentric coordinated of M relative at the points $P\_0,\dots ,\; P\_n$.

Note: it is also possible to define the barycentric coordinates in the same way for a Espace refines.

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