Affine

The adjective "affine" indicates everything that is related to the geometry of affine spaces. A coordinate system for the -dimensional affine space is determined by any basis of vectors, which are not necessarily orthonormal. Therefore, the resulting axes are not necessarily mutually perpendicular nor have the same unit measure. In this sense, affine is a generalization of Cartesian or Euclidean.

An example of an affine property is the average area of a random triangle chosen inside a given triangle (i.e., triangle triangle picking). Because this problem is affine, the ratio of the average area to the original triangle is a constant independent of the actual triangle chosen. Another example of an affine property is the areas (relative to the original triangle) of the regions created by connecting the side -multisectors of a triangle with lines drawn to the opposite vertices (i.e., Marion's theorem).

An example of a property that is not affine is the average length of a line connecting two points picked at random in the interior of a triangle (i.e., triangle line picking). For this problem, the average length depends on the shape of the original triangle, and is (apparently) not a simple function of the area or linear dimensions of original triangle.

An affine subspace of is a point , or a line, whose points are the solutions of a linear system

			(1)
			(2)

or a plane, formed by the solutions of a linear equation

(3)

These are not necessarily subspaces of the vector space , unless is the origin, or the equations are homogeneous, which means that the line and the plane pass through the origin. Hence, an affine subspace is obtained from a vector subspace by translation. In this sense, affine is a generalization of linear.

The distinction between affine and projective arises especially when comparing coordinates. For example, the triples and are the affine coordinates of two distinct points of the affine space , but are the homogeneous (or projective) coordinates of the same point of the projective plane , since homogeneous coordinates are determined up to proportionality.