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  "body": "\n## Definition\n\nLet $K$ be a field and let $V$ be a vector space over $K$.\nAn affine space over $V$ is a nonempty set $A$ with a map $A \\times V \\to A$, written $(p, v) \\mapsto p + v$, that has these properties.\nFor each point $p$ of $A$: $p + 0 = p$.\nFor each point $p$ of $A$ and all vectors $u$ and $v$ of $V$: $(p + u) + v = p + (u + v)$.\nFor all points $p$ and $q$ of $A$, there is exactly one vector $v$ of $V$ with $p + v = q$. This vector is written $q - p$ or $\\overrightarrow{pq}$.\nThe elements of $A$ are the points, and $V$ is the space of translations of $A$. The dimension of $A$ is the dimension of $V$.\n\n## Remarks\n\nEach vector space $V$ is an affine space over itself, with the addition of $V$ as the action. In particular $\\mathbb{R}^n$ is an affine space over $\\mathbb{R}^n$.\nFor all points $p$, $q$ and $r$, Chasles's relation holds: $(q - p) + (r - q) = r - p$.\nMathlib calls this structure an additive torsor, `AddTorsor`. Its definition works for an additive group $V$ in place of a vector space.\nA [[Euclidean space|0XPR1E85W0]] is a real affine space of finite dimension whose space of translations has an inner product.\n\n## Explanations\n\n### L1 human\n\nIn an affine space you can subtract two points to get the arrow from one to the other. You can also add an arrow to a point. But you cannot add two points: there is no special origin. A good picture is the plane without its coordinate axes.\n",
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