Morphing Schnyder drawings of planar triangulations

TL;DR

This paper presents an improved method for morphing between planar triangulation drawings using Schnyder embeddings, achieving bounded grid size and visually attractive morphs in quadratic steps.

Contribution

It introduces a new morphing technique based on Schnyder embeddings that reduces grid size and enhances visualization quality for triangulated graph drawings.

Findings

Achieves $O(n^2)$ morphing steps with bounded $O(n) imes O(n)$ grid size.

Uses linear morphs of Schnyder woods' flip operations.

Produces visually attractive morphs for weighted Schnyder drawings.

Abstract

We consider the problem of morphing between two planar drawings of the same triangulated graph, maintaining straight-line planarity. A paper in SODA 2013 gave a morph that consists of $O(n^2)$ steps where each step is a linear morph that moves each of the $n$ vertices in a straight line at uniform speed. However, their method imitates edge contractions so the grid size of the intermediate drawings is not bounded and the morphs are not good for visualization purposes. Using Schnyder embeddings, we are able to morph in $O(n^2)$ linear morphing steps and improve the grid size to $O(n)\times O(n)$ for a significant class of drawings of triangulations, namely the class of weighted Schnyder drawings. The morphs are visually attractive. Our method involves implementing the basic "flip" operations of Schnyder woods as linear morphs.