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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:
- 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.
Another way to define a projective plane is to use linear algebra.
A projective plane over , where is a skew field, consists of the vector spaces of .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 and of dimension 1, we have a unique space of dimension 2. Given two spaces and of dimesnion 2, we have a unique space of dimension 1.