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

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

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