Optimal Morphs of Planar Orthogonal Drawings

Abstract

We describe an algorithm that morphs between two planar orthogonal drawings $Γ_I$ and $Γ_O$ of a connected graph $G$, while preserving planarity and orthogonality. Necessarily $Γ_I$ and $Γ_O$ 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 $Γ_O$. We can find corresponding wires in $Γ_I$ that share topological properties with the wires in $Γ_O$. The structural difference between the two drawings can be captured by the spirality of the wires in $Γ_I$, which guides our morph from $Γ_I$ to $Γ_O$.