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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.
Typical examples of affine planes are Euclidean planes, which are affine planes over the reals equipped with a metric, the Euclidean distance.In other words, an affine plane over the reals is a Euclidean plane in which one has "forgotten" the metric (that is, one does not talk of lengths nor of angle measures).
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.
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 ...
The simplest affine plane contains only four points; it is called the affine plane of order 2. (The order of an affine plane is the number of points on any line, see below.) Since no three are collinear, any pair of points determines a unique line, and so this plane contains six lines.
Projective geometry is less restrictive than either Euclidean geometry or affine geometry. It is an intrinsically non-metrical geometry, meaning that facts are independent of any metric structure. Under the projective transformations, the incidence structure and the relation of projective harmonic conjugates are preserved.
Let X be an affine space over a field k, and V be its associated vector space. An affine transformation is a bijection f from X onto itself that is an affine map; this means that a linear map g from V to V is well defined by the equation () = (); here, as usual, the subtraction of two points denotes the free vector from the second point to the first one, and "well-defined" means that ...