Incidence Structure

Projective Geometry

Projective Geometry

Projective geometry is a mathematical theory without any notion of distance but with very specific topological properties. It is "highly non-Euclidean", but nevertheless, the origin of Projective geometry is found inside Euclidean geometry.

Important Note

Projective geometry is rooted to the first postulate of Euclidean geometry, which states that we can draw a straight line from one point to another.

Projective geometry is a remarkable discovery with minimal invariants. It preserves straightness and continuity, but not distance, angle, intermediacy, or parallelism.

Incidence Geometry

Incidence geometry is an axiomatic system studying points, lines, and their incidences. The undefined terms are "point", "line", and "on". The axioms are:

• For each two distinct points, there is a unique line incident with both.
• Every line has at least two distinct points.
• There exist at least three distinct points.
• Not all points lie on the same line.

Common Notations in Incidence Geometry

• Points: lowercase letters like \( p, q, r, \dots \)
• Lines: uppercase letters like \( L, M, N, \dots \)
• Sets: script letters like \( \mathscr{P}, \mathscr{L}, \dots \)
• Planes: Greek letters like \( \sigma, \pi, \dots \)
• Notation \( L = pq \): line \( L \) passes through points \( p \) and \( q \).

Important Note

Incidence geometry is not metric-based — it does not preserve distance, angle, area, or volume.

Definition: Incidence Structure

An incidence structure is a triple \( \sigma = (\mathscr{P}, \mathscr{L}, \mathscr{I}) \) where:

• \( \mathscr{P} \): set of points
• \( \mathscr{L} \): set of lines
• \( \mathscr{I} \subseteq \mathscr{P} \times \mathscr{L} \): incidence relation
• \( \mathscr{P} \cap \mathscr{L} = \emptyset \)

\( \mathscr{I} = \{(p, L) \mid p \in \mathscr{P},\ L \in \mathscr{L}\} \)

If \( (p, L) \in \mathscr{I} \), we say "\( p \) is on \( L \)" or "\( L \) is on \( p \)".

A finite incidence structure with \( m \) points and \( n \) lines, where each point lies on \( r \) lines and each line contains \( s \) points, satisfies: \[ mr = ns \]

Example: Incidence Structure

Let \( \mathscr{P} = \{a, b, c\} \) and \( \mathscr{L} = \{A, B, C\} \), with incidence relation:

\( \mathscr{I} = \{(a,B), (a,C), (b,A), (b,C), (c,A), (c,B)\} \)

Then \( \sigma = (\mathscr{P}, \mathscr{L}, \mathscr{I}) \) is an incidence structure.

Geometric Representation

Matrix Representation

\[ \begin{array}{c|ccc} & A & B & C \\ \hline a & 0 & 1 & 1 \\ b & 1 & 0 & 1 \\ c & 1 & 1 & 0 \\ \end{array} \]

Important Notes

1. If \( p \) and \( q \) are two distinct points and there is a line incident with both, we say \( p \) and \( q \) are joined.
2. If there is exactly one line \( L \) on both \( p \) and \( q \), we say they determine a line, denoted \( L \).
3. If lines \( L \) and \( M \) share a common point, they intersect.
4. If there is exactly one point \( a \) on both \( L \) and \( M \), they determine a point: \( L \cap M = a \).
5. Points are collinear if all lie on the same line.
6. The set of all points on a line \( L \) is called the range of \( L \).
7. Lines are concurrent if all pass through a common point.
8. The set of all lines through a point \( p \) is the pencil of \( p \).
9. A plane is a flat, unbounded set of points and lines.
10. If two lines in a plane do not meet, they are parallel.
11. A four-point is a set of four points, no three collinear. E.g., \( a_0a_1a_2a_3 \).