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  "body": "\n## Definition\n\nLet $K$ be a field and let $A$ be an [[affine space|09XNAPTJQ7]] over a vector space $V$ over $K$.\nLet $p_1, \\dots, p_k$ be points of $A$, and let $w_1, \\dots, w_k$ be elements of $K$ with $w_1 + \\dots + w_k = 1$.\nThe affine combination of the points $p_i$ with the weights $w_i$ is the point\n$$\\sum_{i=1}^{k} w_i p_i = o + \\sum_{i=1}^{k} w_i (p_i - o),$$\nwhere $o$ is a point of $A$. The result does not depend on the choice of $o$, because the weights add up to 1.\n\n## Remarks\n\nIn a vector space with $o = 0$, the affine combination is the linear combination $w_1 p_1 + \\dots + w_k p_k$.\nThe [[midpoint|0S706JBZBG]] of $p$ and $q$ is the affine combination with the weights $\\frac{1}{2}$ and $\\frac{1}{2}$.\nAffine maps keep affine combinations: $f\\left(\\sum_i w_i p_i\\right) = \\sum_i w_i f(p_i)$.\nAn affine combination with all weights $w_i \\geq 0$, over the real numbers, is a convex combination.\n\n## Explanations\n\n### L1 human\n\nYou 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 $ABC$.\n",
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