Projective Geometry - Projective Planes

When we apply projective geometry, we usually do it in two dimensions, reflecting the importance of projective planes. There are two ways to formulate a projective plane: with axioms or with linear algebra.

A projective plane can be defined as follows:

A projective plane consists of a set of line and a set of points under incidence which satisfy the following axioms:

1. There exists a unique line incident with two given points.
2. There exists a unique point incident with two given lines.
3. There are four points of which no three are collinear.

Another way to define a projective plane is to use linear algebra.

A projective plane over \(K\), where \(K\) is a skew field, consists of the vector spaces of \(K^3\).