Affine space/Point set/Barycentric combination/Definition

Barycentric combination

For a family $P_i$, $i\in I$, of points in an affine space $E$ and a tuple $a_i$, $i\in I$, in $K$ satisfying

$\sum _{i\in I}{a}_i=1$

(for $I$ infinite, only finitely many of the $a_i$ are allowed to be different from $0$), the sum $\sum _{i\in I}{a}_i{P}_i$ is called a barycentric combination of the $P_i$. The corresponding point in $E$ is given by

$\sum _{i\in I}{a}_i{P}_i=Q+\sum _{i\in I}{a}_i\overrightarrow{Q{P}_i},$

where $Q$ is an arbitrary point in $E$.