Projective plane

A projective plane is like a plane. The difference is that every line in a projective plane meet at some point. This is not true in a plane. There are some lines that do not meet there. These lines are called parallel lines. In a projective plane, two lines that look parallel meet at a really far away point. This point is called the point at infinity. This means that every line in a projective plane meet somewhere.

These lines look like they meet somewhere really far away. We say they meet at the vanishing point. The vanishing point is infinitely far away. In a projective plane, they actually meet.

The first people to think about projections are Renaissance artists. They want to draw perspective good, so they worked very hard. A common example of a projective plane is the real projective plane.

What is a projective plane?

A projective plane has lines and points. Points and lines can be incident. Incidence has these properties:

1. There is only one line incident with two different points.
2. There is only one point incident with two different lines.
3. There are four points on the plane, so that there cannot be a line incident with more than two of those points.

The second condition means that there are no parallel lines. The last condition means there are no degenerate projective planes (see below). We say "incident" because if a point is incident with a line, then that line is incident with the point.

Degenerate planes

File:Degenerate planes wec.svg

Degenerate projective planes

Degenerate planes are not projective planes. This is because they are very boring. Without the third condition, they would be projective planes. (Albert & Sandler 1968) says there are seven kinds of degenerate planes. They are:

1. The empty set (No lines, no points);
2. One point, no lines;
3. One line, no points;
4. One point, every line is incident with that point;
5. One line, every point is incident with that line;
6. One line incident with every point. One of the points has many lines incident with it. The other points are only incident to one line.;
7. A lot of points. There is one special point. Points that are not special are all incident with one special line. The special point is not incident with the special line. However, the point is incident with many other lines. All of the not special lines are incident with two points. One of those points is the special point. The other point is incident with the special line.

A degenerate plane can be multiple of these cases at once.