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In geometry, an affine plane is a system of points and lines that satisfy the following axioms: [1]. Any two distinct points lie on a unique line. Given any line and any point not on that line there is a unique line which contains the point and does not meet the given line.
Intuitively, this means that an affine plane is a vector space of dimension two in which one has "forgotten" where the origin is. The second way occurs in incidence geometry, where an affine plane is defined as an abstract system of points and lines satisfying a system of axioms.
In traditional geometry, affine geometry is considered to be a study between Euclidean geometry and projective geometry. On the one hand, affine geometry is Euclidean geometry with congruence left out; on the other hand, affine geometry may be obtained from projective geometry by the designation of a particular line or plane to represent the ...
The affine plane of order three is a (9 4, 12 3) configuration. When embedded in some ambient space it is called the Hesse configuration. It is not realizable in the Euclidean plane but is realizable in the complex projective plane as the nine inflection points of an elliptic curve with the 12 lines incident with triples of these.
The last axiom ensures that the geometry is not trivial (either empty or too simple to be of interest, such as a single line with an arbitrary number of points on it), while the first two specify the nature of the geometry. The simplest affine plane contains only four points; it is called the affine plane of order 2. (The order of an affine ...
Origins from Alice's and Bob's perspectives. Vector computation from Alice's perspective is in red, whereas that from Bob's is in blue. The following characterization may be easier to understand than the usual formal definition: an affine space is what is left of a vector space after one has forgotten which point is the origin (or, in the words of the French mathematician Marcel Berger, "An ...
In an affine or Euclidean space of higher dimension, the points at infinity are the points which are added to the space to get the projective completion. [citation needed] The set of the points at infinity is called, depending on the dimension of the space, the line at infinity, the plane at infinity or the hyperplane at infinity, in all cases a projective space of one less dimension.
It is an affine space, which includes in particular the concept of parallel lines. ... In Euclidean geometry, a plane is a flat two-dimensional surface that extends ...