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Affine


The adjective "affine" indicates everything that is related to the geometry of affine spaces. A coordinate system for the n-dimensional affine space R^n is determined by any basis of n 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 n-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 R^3 is a point P(x,y), or a line, whose points are the solutions of a linear system

a_1x+a_2y+a_3z=a_4
(1)
b_1x+b_2y+b_3z=b_4,
(2)

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

 ax+by+cz=d.
(3)

These are not necessarily subspaces of the vector space R^3, unless P 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 (1,-2,3) and (-2,4,-6) are the affine coordinates of two distinct points of the affine space R^3, but are the homogeneous (or projective) coordinates of the same point of the projective plane P^2, since homogeneous coordinates are determined up to proportionality.


See also

Affine Complex Plane, Affine Coordinates, Affine Equation, Affine Function, Affine Geometry, Affine Group, Affine Hull, Affine Plane, Affine Scheme, Affine Space, Affine Transformation, Affine Variety, Affinely Extended Real Numbers

This entry contributed by Margherita Barile

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Cite this as:

Barile, Margherita. "Affine." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Affine.html

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