Projective Plane - Degenerate Planes

Degenerate Planes

Degenerate planes do not fulfill the third condition in the definition of a projective plane. They are not structurally complex enough to be interesting in their own right, but from time to time they arise as special cases in general arguments. There are seven degenerate planes (Albert & Sandler 1968). They are:

1. the empty set;
2. a single point, no lines;
3. a single line, no points;
4. a single point, a collection of lines, the point is incident with all of the lines;
5. a single line, a collection of points, the points are all incident with the line;
6. a point P incident with a line m, an arbitrary (can be empty) collection of lines all incident with P and an arbitrary collection of points all incident with m;
7. a point P not incident with a line m, an arbitrary (perhaps empty) collection of lines all incident with P and all the points of intersection of these lines with m.

These seven cases are not independent, the fourth and fifth can be considered as special cases of the sixth, while the second and third are special cases of the fourth and fifth respectively. The seven cases can therefore be organized into two families of degenerate planes as follows (this representation is for finite degenerate planes, but may be extended to infinite ones in a natural way):

1) For any number of points P₁, ..., P_n, and lines L₁, ..., L_m,

L₁ = { P₁, P₂, ..., P_n}

L₂ = { P₁ }

L₃ = { P₁ }

...

L_m = { P₁ }

2) For any number of points P₁, ..., P_n, and lines L₁, ..., L_n, (same number of points as lines)

L₁ = { P₂, P₃, ..., P_n }

L₂ = { P₁, P₂ }

L₃ = { P₁, P₃ }

...

L_n = { P₁, P_n }