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For an introduction to the theory of schemes in algebraic geometry, see affine scheme, projective space, sheaf and scheme. en.wikipedia.org

Gauge symmetries of metric-affine gravitation theory are general covariant transformations. en.wikipedia.org

This also leads to difficulties since one can introduce somewhat pathological objects, e.g. an affine line with zero doubled. en.wikipedia.org

Note that on the "affine" plane, one might push off to a parallel line, so (thinking geometrically) the number of intersection points depends on the choice of push-off. en.wikipedia.org

These identifications are always given by affine transformations from one tangent plane to another. en.wikipedia.org

Once a set of basic primitives have been created, they can easily be combined with one another using operations such as sweep, shrink-wrap and affine transformation. en.wikipedia.org

In affine geometry, there is no metric structure but the parallel postulate does hold. en.wikipedia.org

Since sheaves have a local character and since any scheme is locally affine, the theorem admits a global statement of the same nature. en.wikipedia.org

There are many finite and infinite affine planes. en.wikipedia.org

In this viewpoint, an affine transformation geometry is a group of projective transformations that do not permute finite points with points at infinity. en.wikipedia.org

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