Affine space

In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these 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. Affine space is the setting for affine geometry. As in Euclidean space, the fundamental objects in an affine space are called points, which can be thought of as locations in the space without any size or shape: zero-dimensional. Through any pair of points an infinite straight line can be drawn, a one-dimensional set of points; through any three points that are not collinear, a two-dimensional plane can be drawn; and, in general, through k + 1 points in general position, a k-dimensional flat or affine subspace can be drawn. Affine space is characterized by a notion of pairs of parallel lines that lie within the same plane but never meet each other (non-parallel lines within the same plane intersect in a point). Given any line, a line parallel to it can be drawn through any point in the space, and the equivalence class of parallel lines are said to share a direction. Unlike for vectors in a vector space, in an affine space there is no distinguished point that serves as an origin. There is no predefined concept of adding or multiplying points together, or multiplying a point by a scalar number. However, for any affine space, an associated vector space can be constructed from the differences between start and end points, which are called free vectors, displacement vectors, translation vectors or simply translations. Likewise, it makes sense to add a displacement vector to a point of an affine space, resulting in a new point (of the same affine space)...