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A similar construction, starting from the projective plane of order 3, produces the affine plane of order 3 sometimes called the Hesse configuration. An affine plane of order n exists if and only if a projective plane of order n exists (however, the definition of order in these two cases is not the same). Thus, there is no affine plane of order ...
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).
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 points at infinity. [19] In affine geometry, there is no metric structure but the parallel postulate does hold.
A finite projective plane will produce a finite affine plane when one of its lines and the points on it are removed. The order of a finite affine plane is the number of points on any of its lines (this will be the same number as the order of the projective plane from which it comes). The affine planes which arise from the projective planes PG(2 ...
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 ...
Projective geometry can be modeled by the affine plane (or affine space) plus a line (hyperplane) "at infinity" and then treating that line (or hyperplane) as "ordinary". [5] An algebraic model for doing projective geometry in the style of analytic geometry is given by homogeneous coordinates.
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 ...
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.