The geometry of affine spaces, studying incidence and parallelism without choosing an origin or a metric.
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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Affine geometry is a branch of geometry that studies the properties of figures that remain invariant under affine transformations. These transformations include operations such as translation, scaling, rotation, and shearing, which can alter the size and orientation of shapes but do not change their basic structure or ratios of distances. Here are some key concepts in affine geometry: 1. **Affine Transformations**: An affine transformation is a function between affine spaces that preserves points, straight lines, and planes.