An incidence structure
is called an affine plane if the following three axioms hold.
Two lines a and b of an affine plane are defined to be parallel if
either a=b or
We will write
in this case. The relation of parallelism
defines a parallelism
In fact, this is the unique parallelism on
- Each two distinct points are on a unique line.
- Given any point p and any line l not through p, there is exactly
one line g through p disjoint from l.
- There are three distinct points not on a line.
All point rows and all parallel classes of an affine plane
have the same cardinality n, called the order of
Moreover, if p is a point of
Primary examples of affine planes are the
affine planes over a vector space.
More general affine planes are koordinatized over
Contributed by Hauke Klein
Version $Id: affine.html,v 1.1 2000/12/26 18:27:34 hauke Exp $