Affine space/Barycentric coordinates/Definition

Let ${\displaystyle {}P_{i}}$, ${\displaystyle {}i\in I}$, denote an affine basis in an affine space ${\displaystyle {}E}$ over the ${\displaystyle {}K}$-vector space ${\displaystyle {}V}$. For a point ${\displaystyle {}P\in E}$, the uniquely determined numbers

${\displaystyle (a_{i},i\in I){\text{ with }}\sum _{i\in I}a_{i}=1}$

such that

${\displaystyle {}P=\sum _{i\in I}a_{i}P_{i}\,}$

holds, are called the barycentric coordinates of ${\displaystyle {}P}$.