The polynomial space of the ipad

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In mathematics , an affine space is a geometric structure that generalizes the properties of Euclidean spaces that are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments . A Euclidean space is an affine space over the reals , equipped with a metric , the Euclidean distance . Therefore, in Euclidean geometry , an affine property is a property that may be proved in affine spaces.

In an affine space, there is no distinguished point that serves as an origin. Hence, no vector has a fixed origin and no vector can be uniquely associated to a point. In an affine space, there are instead displacement vectors , also called translation vectors or simply translations , between two points of the space. [1] Thus it makes sense to subtract two points of the space, giving a translation vector, but it does not make sense to add two points of the space. Likewise, it makes sense to add a displacement vector to a point of an affine space, resulting in a new point translated from the starting point by that vector.

The dimension of an affine space is defined as the dimension of the vector space of its translations. An affine space of dimension one is an affine line . An affine space of dimension 2 is an affine plane . An affine subspace of dimension n – 1 in an affine space or a vector space of dimension n is an affine hyperplane .

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