Affine Plane

Definition

An affine plane is an incidence structure \(\sigma = (\mathscr{P}, \mathscr{L}, \mathrm{I})\) satisfying:

• A1: Two points determine exactly one line.
• A2: For line \(L\) and \(p \notin L\), exactly one line through \(p\) parallel to \(L\).
• A3: Every line has at least two points.
• A4: For every line, some point lies off it.
• A5: There exists at least one line.

Examples

The complete four-point with 4 points and 6 lines is the smallest affine plane. It has three pairs of parallel lines.

Young's configuration: 9 points, 12 lines, 4 sets of 3 parallel lines.

Theorems

An affine plane contains three non-collinear points.

Let \(\alpha = (\mathscr{P},\mathscr{L},\mathrm{I})\).

• By A5, there is a line \(L\).
• By A3, there are points \(p, q \in L\).
• By A4, there is a point \(r \notin L\).

Then, \(p, q, r\) are three non-collinear points. \(\blacksquare\)

An affine plane contains a four-point.

Let \(\alpha = (\mathscr{P},\mathscr{L},\mathrm{I})\) be an affine plane.

• Let \(L\) be a line (A5).
• Let \(a, b \in L\) (A3).
• Let \(c \notin L\) (A4).
• Then \(\exists!\) line \(M \parallel L\) through \(c\) (A2).
• \(M\) has another point \(d\) (A3).

So affine plane contains a four-point such as \(a,b,c,d\) . \(\blacksquare\)

Let \( \alpha=(\mathscr{P,L,I}) \) be an incidence structure satisfying

1. Two points determine a line
2. If \(L\) is a line and \(p\) be a point not on \(L\), there is exactly one line on \(p\) and parallel to \(L\)
3. There is a set of three non-collinear points

Then prove that \(\alpha \) is an affine plane.

1. (1) ⇒ This implies A1.
2. (2) ⇒ This implies A2.
3. (3) ⇒ This implies A5 (by 3, any two of three points define a line, by 1).
4. (3) ⇒ This implies A4 (third point lies off any line through the other two points).
5. A3 (≥2 points on each line):
   By (3), assume that \(p,q,r\) are three non-collinear points.
   Now Suppose, \(L\) is arbitrary line, then by (A4), there is a point not on \(L\)
   say \(p \notin L\).
   Case 1
   If \(pq\) and \(pr\) both meet at \(L\), then \(L\) has two points.
   Case 2
   If one of the lines, say \(pq\) does not meet \(L\),then by (2)-\(pr\) must meet \(L\).
   Also, by (2), there exists a line \(q \in M || pr\).
   This line \(M\) must meet \(L\), otherwise it contradicts (2).
   Hence \(L\) has two points.
   This satisfies (case 1 and case 2) axiom A3 of affine plane ⇒ (A3)

Hence all axioms hold → it's an affine plane. \(\blacksquare\)

Note: The three conditions are equivalent to the five axioms. This simplifies verification.