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Given a point and a line, there is a unique line which contains the point and is parallel to the line. Parallelism is an equivalence relation on the lines of an affine plane. Since no concepts other than those involving the relationship between points and lines are involved in the axioms, an affine plane is an object of study belonging to ...
It has 15 points, 35 lines, and 15 planes and is the smallest three-dimensional projective space. [16] It also has the following properties: [17] Each point is contained in 7 lines and 7 planes. Each line is contained in 3 planes and contains 3 points. Each plane contains 7 points and 7 lines. Each plane is isomorphic to the Fano plane.
Adding four new points, each being added to all the lines of a single parallel class (so all of these lines now intersect), and one new line containing just these four new points produces the projective plane of order three, a (13 4) configuration. Conversely, starting with the projective plane of order three (it is unique) and removing any ...
Let L = (P, G, I) be an incidence structure, for which the elements of P are called points and the elements of G are called lines. L is a linear space if the following three axioms hold: (L1) two distinct points are incident with exactly one line. (L2) every line is incident to at least two distinct points. (L3) L contains at least two distinct ...
Lines that meet at the same point are said to be concurrent. The set of all lines in a plane incident with the same point is called a pencil of lines centered at that point. The computation of the intersection of two lines shows that the entire pencil of lines centered at a point is determined by any two of the lines that intersect at that point.
An incidence structure = (,,) consists of a set of points, a set of lines, and an incidence relation, or set of flags, ; a point is said to be incident with a line if (,) . It is a ( finite ) partial geometry if there are integers s , t , α ≥ 1 {\displaystyle s,t,\alpha \geq 1} such that: