Проективтік геометриядағы құрылымдар

Construction in projective geometry
In mathematics, an algebraic structure consisting of a non empty set and a ternary mapping may be called a ternary system. A planar ternary ring (PTR) or ternary field is special type of ternary system used by Marshall Hall to construct projective planes by means of coordinates. A planar ternary ring is not a ring in the traditional sense, but any field gives a planar ternary ring where the operation is defined by Thus, we can think of a planar ternary ring as a generalization of a field where the ternary operation takes the place of both addition and multiplication. There is wide variation in the terminology. Planar ternary rings or ternary fields as defined here have been called by other names in the literature, and the term "planar ternary ring" can mean a variant of the system defined here. The term "ternary ring" often means a planar ternary ring, but it can also simply mean a ternary system.

Define The structure is a loop with identity element 0.

Define The set is closed under this multiplication. The structure is also a loop, with identity element 1.

A planar ternary ring is said to be linear if For example, the planar ternary ring associated to a quasifield is (by construction) linear.

The connection between planar ternary rings (PTRs) and two dimensional geometries, specifically projective and affine geometries, is best described by examining the affine case first. In affine geometry, points on a plane are described using Cartesian coordinates, a method that is applicable even in non Desarguesian geometries — there, coordinate components can always be shown to obey the structure of a PTR. By contrast, homogeneous coordinates, typically used in projective geometry, are unavailable in non Desarguesian contexts. Thus, the simplest analytic way to construct a projective plane is to start with an affine plane and extend it by adding a "line at infinity"; this bypasses homogeneous coordinates. In an affine plane, when the plane is Desarguesian, lines can be represented in slope intercept form This representation extends to non Desarguesian planes through the ternary operation of a PTR, allowing a line to be expressed as Lines parallel to the y axis are expressed by
We now show how to derive the analytic representation of a general projective plane given at the start of this section. To do so, we pass from the affine plane, represented as , to a representation of the projective plane , by adding a line at infinity. Formally, the projective plane is described as , where represents an affine plane in Cartesian coordinates and includes all finite points, while denotes the line at infinity. Similarly, is expressed as Here, is an affine line which we give its own Cartesian coordinate system, and consists of a single point not lying on that affine line, which we represent using the symbol .