Optimal Morphs of Planar Orthogonal Drawings

Abstract

We describe an algorithm that morphs between two planar orthogonal drawings $\Gamma_I$ and $\Gamma_O$ of a connected graph $G$, while preserving planarity and orthogonality. Necessarily $\Gamma_I$ and $\Gamma_O$ share the same combinatorial embedding. Our morph uses a linear number of linear morphs (linear interpolations between two drawings) and preserves linear complexity throughout the process, thereby answering an open question from Biedl et al. Our algorithm first unifies the two drawings to ensure an equal number of (virtual) bends on each edge. We then interpret bends as vertices which form obstacles for so-called wires: horizontal and vertical lines separating the vertices of $\Gamma_O$. These wires define homotopy classes with respect to the vertices of $G$ (for the combinatorial embedding of $G$ shared by $\Gamma_I$ and $\Gamma_O$). These homotopy classes can be represented by orthogonal polylines in $\Gamma_I$. We argue that the structural difference between the two drawings can be captured by the spirality of the wires in $\Gamma_I$, which guides our morph from $\Gamma_I$ to $\Gamma_O$.