Duncan, Christian A.; Eppstein, David; Goodrich, Michael T.; Kobourov, Stephen G.; Loeffler, Maarten; Noellenburg, Martin; Univ Arizona, Dept Comp Sci (CARLETON UNIV, DEPT MATHEMATICS & STATISTICS, 2018)
      In Lombardi drawings of graphs, edges are represented as circular arcs and the edges incident on vertices have perfect angular resolution. It is known that not every planar graph has a planar Lombardi drawing. We give an example of a planar 3-tree that has no planar Lombardi drawing and we show that all outerpaths do have a planar Lombardi drawing. Further, we show that there are graphs that do not even have any Lombardi drawing at all. With this in mind, we generalize the notion of Lombardi drawings to that of (smooth) k-Lombardi drawings, in which each edge may be drawn as a (differentiable) sequence of k circular arcs; we show that every graph has a smooth 2-Lombardi drawing and every planar graph has a smooth planar 3-Lombardi drawing. We further investigate related topics connecting planarity and Lombardi drawings.

      Arkin, Esther M.; Efrat, Alon; Knauer, Christian; Mitchell, Joseph S. B.; Polishchuk, Valentin; Rote, Guenter; Schlipf, Lena; Talvitie, Topi; Univ Arizona, Comp Sci (CARLETON UNIV, DEPT MATHEMATICS & STATISTICS, 2016)
      We show how to preprocess a polygonal domain with a fixed starting point s in order to answer efficiently the following queries: Given a point q, how should one move from s in order to see q as soon as possible? This query resembles the well-known shortest-path-to-a-point query, except that the latter asks for the fastest way to reach q, instead of seeing it. Our solution methods include a data structure for a different generalization of shortest-path-to-a-point queries, which may be of independent interest: to report efficiently a shortest path from s to a query segment in the domain.