= Barycentric coordinate system
{title2=$x=\sum_{i=0}^d\lambda_i v_i,\quad\sum_{i=0}^d\lambda_i=1$}
A <barycentric coordinate system> assigns to a point in the affine span of the vertices of a nondegenerate <simplex> its unique vertex weights $\lambda_0,\ldots,\lambda_d$, constrained to sum to one. Uniqueness follows because $v_1-v_0,\ldots,v_d-v_0$ form a <basis> of the translation space: expand $x-v_0$ in that basis and set $\lambda_0=1-\sum_{i=1}^d\lambda_i$. A point lies in the closed simplex exactly when every weight is nonnegative. An <affine function> satisfies $f(x)=\sum_i\lambda_i f(v_i)$, which makes these coordinates useful for interpolation and the <integral of an affine function over a simplex>.
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