Biased graphs. VI. synthetic geometry

A biased graph is a graph with a class of selected circles (“cycles”, “circuits”), called balanced, such that no theta subgraph contains exactly two balanced circles. A biased graph $\Omega$ has two natural matroids, the frame matroid $G\left(\Omega \right)$ and the lift matroid $L\left(\Omega \right)$, and their extensions the full frame matroid $G^{?}\left(\Omega \right)$ and the extended (or complete) lift matroid $L_0\left(\Omega \right)$. In Part IV we used algebra to study the representations of these matroids by vectors over a skew field and the corresponding embeddings in Desarguesian projective spaces. Here we redevelop those representations, independently of Part IV and in greater generality, by using synthetic geometry.