# Projective Geometry - Projective Planes

###### This wiki is incomplete.

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.

## Axiomatic Formulation

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:

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

## Linear Algebraic Formulation

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 $K^3$.Let's think about what this means. Every vector space of dimension 1 corresponds to a point in the projective plane. Every vector space of dimension 2 corresponds to a line in the projective plane. Given two distinct spaces $V$ and $W$ of dimension 1, we have a unique space $V + W$ of dimension 2. Given two spaces $V$ and $W$ of dimesnion 2, we have a unique space $V\cap W$ of dimension 1.

**Cite as:**Projective Geometry - Projective Planes.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/projective-geometry-projective-planes/