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.

Remarks

In a vector space with $o = 0$, the affine combination is the linear combination $w_1 p_1 + \dots + w_k p_k$. The midpoint of $p$ and $q$ is the affine combination with the weights $\frac{1}{2}$ and $\frac{1}{2}$. Affine maps keep affine combinations: $f\left(\sum_i w_i p_i\right) = \sum_i w_i f(p_i)$. An affine combination with all weights $w_i \geq 0$, over the real numbers, is a convex combination.

Explanations

L1 human

You cannot add two points of the plane in a sensible way without an origin. But a weighted average of points, with weights that sum to 1, gives the same point for each choice of origin. For example, $\frac{1}{3}A + \frac{1}{3}B + \frac{1}{3}C$ is the centroid of the triangle $ABC$.