Morphing Rectangular Duals

TL;DR

This paper presents a method for morphing between rectangular duals of the same plane graph, allowing intermediate non-rectangular polygons, and provides an algorithm with polynomial complexity for the transformation.

Contribution

It introduces a polynomial-time algorithm for morphing between rectangular duals, accommodating intermediate non-rectangular polygons to ensure continuous contact representations.

Findings

Morphing is possible only if duals share the same REL.

Allows intermediate polygons of constant complexity.

Provides an $O(n^3)$ time algorithm with $O(n^2)$ steps.

Abstract

A rectangular dual of a plane graph $G$ is a contact representations of $G$ by interior-disjoint axis-aligned rectangles such that (i) no four rectangles share a point and (ii) the union of all rectangles is a rectangle. A rectangular dual gives rise to a regular edge labeling (REL), which captures the orientations of the rectangle contacts. We study the problem of morphing between two rectangular duals of the same plane graph. If we require that, at any time throughout the morph, there is a rectangular dual, then a morph exists only if the two rectangular duals realize the same REL. Therefore, we allow intermediate contact representations of non-rectangular polygons of constant complexity. Given an $n$-vertex plane graph, we show how to compute in $O(n^3)$ time a piecewise linear morph that consists of $O(n^2)$ linear morphing steps.