Search results
Results from the WOW.Com Content Network
An incidence structure more general than a finite affine plane is a k-net of order n. This consists of n 2 points and nk lines such that: Parallelism (as defined in affine planes) is an equivalence relation on the set of lines.
A finite plane of order n is one such that each line has n points (for an affine plane), or such that each line has n + 1 points (for a projective plane). One major open question in finite geometry is: Is the order of a finite plane always a prime power? This is conjectured to be true.
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, q) are denoted by AG(2, q). There is a projective plane of order N if and only if there is an affine plane of order N.
Every affine plane can be uniquely extended to a projective plane. The order of a finite affine plane is k, the number of points on a line. An affine plane of order n is an ((n 2) n + 1, (n 2 + n) n) configuration.
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).
A finite affine plane of order q, with the lines as blocks, is an S(2, q, q 2). An affine plane of order q can be obtained from a projective plane of the same order by removing one block and all of the points in that block from the projective plane. Choosing different blocks to remove in this way can lead to non-isomorphic affine planes.
A plane is said to have the "minor affine Desargues property" when two triangles in parallel perspective, having two parallel sides, must also have the third sides parallel. If this property holds in the affine plane defined by a ternary ring, then there is an equivalence relation between "vectors" defined by pairs of points from the plane. [14]
The affine span of X is the set of all (finite) affine combinations of points of X, and its direction is the linear span of the x − y for x and y in X. If one chooses a particular point x 0, the direction of the affine span of X is also the linear span of the x – x 0 for x in X.