Affine combination

Definition

Let $K$ be a field and let $A$ be an affine space over a vector space $V$ over $K$. Let $p_1, \dots, p_k$ be points of $A$, and let $w_1, \dots, w_k$ be elements of $K$ with $w_1 + \dots + w_k = 1$. The affine combination of the points $p_i$ with the weights $w_i$ is the point $$\sum_{i=1}^{k} w_i p_i = o + \sum_{i=1}^{k} w_i (p_i - o),$$ where $o$ is a point of $A$. The result does not depend on the choice of $o$, because the weights add up to 1.