Betweenness in Polar Coordinates

betweenness, geometry, lines, polar coordinates, trilinear coordinates We consider straight line drawings. From a graph theoretic point of view these are plane graphs with a set of preferred paths. This graph is expressed in geometric terms through the consideration of the betweenness of the points of intersection along each of the lines. If we fix a reference system for polar coordinates, we can identify a number of equalities and inequalities over the complete set of polar coordinates satisfied by this particular drawing. We show that every solution to these inequalities describes a drawing that is equivalent under graph theoretic considerations, in particular betweenness. We show that this equivalence is independent of the polar reference system.