Affine space/Point set/Barycentric combination/Definition

For a family ${\displaystyle {}P_{i}}$, ${\displaystyle {}i\in I}$, of points in an affine space ${\displaystyle {}E}$ and a tuple ${\displaystyle {}a_{i}}$, ${\displaystyle {}i\in I}$, in ${\displaystyle {}K}$ satisfying

${\displaystyle {}\sum _{i\in I}a_{i}=1\,}$

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

${\displaystyle {}\sum _{i\in I}a_{i}P_{i}=Q+\sum _{i\in I}a_{i}{\overrightarrow {QP_{i}}}\,,}$

where ${\displaystyle {}Q}$ is an arbitrary point in ${\displaystyle {}E}$.