In smooth orthogonal layouts of planar graphs, every edge is an alternating sequence of axis-aligned segments and circular arcs with common axis-aligned tangents. In this paper, we study the problem of finding smooth orthogonal layouts of low edge complexity, that is, with few segments per edge. We say that a graph has smooth complexity k—for short, an SCk-layout—if it admits a smooth orthogonal drawing of edge complexity at most k.
Our main result is that every 4-planar graph has an SC2-layout. While our drawings may have super-polynomial area, we show that for 3-planar graphs, cubic area suffices. We also show that any biconnected 4-outerplane graph has an SC1-layout. On the negative side, we demonstrate an infinite family of biconnected 4-planar graphs that require exponential area for an SC1-layout. Finally, we present an infinite family of biconnected 4-planar graphs that do not admit an SC1-layout.