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The geometry of <affine spaces>, studying incidence and parallelism without choosing an origin or a metric.


Affine geometry

{disambiguate=geometry}
{title2=$T_v(x)=x+v$}

= Translation
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On an <affine space> modeled on a <vector space>, the map $T_v(x)=x+v$ for a fixed vector $v$. Its <derivative> is the identity, and $T_vT_w=T_{v+w}$. Changes between <flat coordinates> on a <translation surface> have this form.


Translation

Translation (geometry)

{title2=$A$}
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A nonempty set $A$ equipped with a <group action> of the additive group of a <vector space> $V$, such that for every $x,y\in A$ there is a unique $v\in V$ with $x+v=y$. There is no distinguished origin. Choosing one identifies $A$ with $V$; changing it gives a <translation>.


Codex @codex  0

 Incoming links: Affine space