Fig. 2

Deflection angles at junctions: the subfigures (\({{\bf a}}_{\star }\)) and (\({{\bf b}}_{\star }\)) show the deflection angles \(\delta _{*}^{}\) for the junctions \(j_{1}\) and \(j_{2}\), respectively, from the notional example in Fig. 1. A deflection angle of a street at a junction is basically the magnitude of the angular change experienced at the junction by the tangent of the street. In practice, the street can be arbitrarily oriented and the deflection angle becomes the magnitude of the angle between the incoming and outgoing tangents. The transposition to pairs of street-segments at junctions is obvious. Each subfigure corresponds to a possible incoming street-segment. The subfigures actually organize (index) themselves according to the cardinal direction (index) of their incoming street-segment. For each subfigure, the incoming tangent at the junction is in red and the outgoing tangents are in orange. Every double-arrow arc between the tangents of an incoming street-segments \({\bar{s}}_{i}\) and an outgoing one \({\bar{s}}_{o}\) indicates a deflection angle denoted by \(\delta _{i,o}\)—and has a radius linear with the supplementary angle \(\pi -\delta _{i,o}\). Realistic angular changes are assumed to be bounded above. We have set the deflection angle threshold to \(\pi /4\). The light-blue pie areas identify the forbidden deflection angles—and have the radius of any arc with the deflection angle threshold as deflection angle. An incoming street-segment might so continue its way only along any outgoing street-segment whose tangent or arc lies within the angular sector of the missing slice—and/or does not cross the pie area. Furthermore, realistic configurations of streets must obviously have no street overlap. No incoming street-segment can continue its way along an outgoing street-segment already passed through. The very basic idea behind the state-of-the-art for building configurations of streets is a loop: commit one choice of outgoing street-segment; move to its opposite junction; repeat. Figure 3 along with supplementary Animation A1 (Additional file 1) illustrate how streets can emerge from this approach. In contrast, our approach identifies at every junction all the combinations of incoming and outgoing street-segments and “flips” them. Figure 4 sketches why and how these combinations are actually maximum matchings, while Fig. 5 along with supplementary Animation A2 (Additional file 2) illustrate a short sequence of “flips”