Affine Plane

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Affine Plane is an incidence structure denoted by \(\sigma=(\mathscr{P,L,I}) \) satisfying following axioms:

1. [A 1] If p and q are two points, there is exactly one line on both.
2. [A 2] If L is a line and p be a point not on L, there is exactly one line on p parallel to L.
3. [A 3] If L is a line, there are at least two points on L.
4. [A 4] If L is a line, there is at least one point not on L.
5. [A 5] There is at least one line.

Example 1

The incidence structure with \(\mathscr{P}=\{a_0,a_1,a_2,a_3\}\) and \(\mathscr{L}=\{L_1, L_2,L_3,L_1',L_2',L_3'\}\) and incidence relation as given alongside is an affine plane.
It is called “a complete four-point” and is the smallest affine plane that exist.

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Example 2

Another example of affine plane is “Young’s configuration”. This structure has a total of nine points each on four lines and twelve lines each on three points.