Sites/generators in yellow. Line-edges in cyan, parabolic edges in red. Note that the parabolic edges are split at their apex, i.e. the closest point to the adjacent site.
Some problems with hooking up the half-edges of the two new faces that are associated with a new line-site remain. Hopefully soon to be fixed.
This code should in principle work for circular arc sites too. Adding arcs to the diagram results in elliptic and hyperbolic edges (in addition to line and parabola, which seem to work now).