Plane Multigraphs with One-Bend and Circular-Arc Edges of a Fixed Angle

For an angle $\alpha \in (0,\pi )$, we consider plane graphs and multigraphs in which the edges are either (i) one-bend polylines with an angle $\alpha$ between the two edge segments, or (ii) circular arcs of central angle $2(\pi -\alpha )$. We derive upper and lower bounds on the maximum density of such graphs in terms of $\alpha$. As an application, we improve upon bounds for the number of edges in $\alpha A{C}_1^{=}$ graphs (i.e., graphs that can be drawn in the plane with one-bend edges such that any two crossing edges meet at angle $\alpha$). This is the first improvement on the size of $\alpha A{C}_1^{=}$ graphs in over a decade.