Answer to Question #204983 in Geometry for Umair

Question

Subject: Affine and Euclidean Geometry

Question: Generalized geometrical incidence? Explanation in detail with prove.

Answer 1

An affinity plane is a point and route incidence system so that there is precisely one line across each for two different points.

Given any line and point , there is precisely one line in P which does not reach the level in the line in question .Four points exist, so that there are no hills.

The Euclidean plane is the best known model for such axioms .

An example that fulfills the axioms is a model for a collection of axioms. Our axioms do not specify points, lines and incidences; instead, each specific model interprets these concepts.) More broadly, in the case of F, one constructs a refined plane over F using commanded pairs (x,y) as points, and y = mx + b or x = a (for fixed a,m,b to F) as lines;

${(x,mx+b) : x ∈ F} for m,b ∈ F;$

${(a,y) : y ∈ F} for a ∈ F.$

The effect is natural: a point is on a certain line so that the linear equation necessary is fulfilled. This plane is referred to as $A^2 (F)$ and is commonly referred to as $AG^2(F)$ and is also known as a classical affinity geometry over field F. Note that $A^2 (F)$ is merely the Euclidean, but is focused on the incidence of information, regardless of the extra distance structure, angle, topology.

There is a matching classical affinity plane $A^2$ for each Finite $F_q$ field (where q is a major power) ($F_q$ )

The number of points on any particular line of the plane is the order of the affine plane. From the axioms it is evident that the order is two or more. Note that the conventional finite $A^2Fq$ affine aircraft has the same order as the finite $F_q$ field used to build the aircraft, namely q. Moreover, the Euclidean plane is infinite in order (the cardinality of real numbers, namely |R| = 2 sides of the continuum, is more accurately its order). $A^2Q$ also has infinite order in countless ways.