A nonempty set equipped with a group action of the additive group of a vector space , such that for every there is a unique with . There is no distinguished origin. Choosing one identifies with ; changing it gives a translation.
On an affine space modeled on a vector space, the map for a fixed vector . Its derivative is the identity, and . Changes between flat coordinates on a translation surface have this form.
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An affine space is a geometric structure that generalizes the idea of a vector space by allowing translation without a fixed origin. It can be thought of as a set of points along with a vector space that describes how to move from one point to another. Here are some key features and concepts related to affine spaces: 1. **Points and Vectors**: In an affine space, there are two distinct types of entities: points and vectors.