Planar and Toroidal Morphs Made Easier

TL;DR

This paper introduces simplified algorithms for planar and toroidal graph morphing problems based on barycentric interpolation, improving efficiency and naturalness of the morphs while avoiding classical methods.

Contribution

The paper presents new, simpler algorithms for planar straight-line graph morphing and extends barycentric interpolation to geodesic graphs on the torus, with improved computational efficiency.

Findings

Constructed $O(n)$ step morphs in $O(n^{1+rac{C6}{2}})$ time for planar graphs.

Extended barycentric interpolation to geodesic graphs on the flat torus with a simple scaling strategy.

Provided a simpler proof of a conjecture for geodesic torus triangulations.

Abstract

We present simpler algorithms for two closely related morphing problems, both based on the barycentric interpolation paradigm introduced by Floater and Gotsman, which is in turn based on Floater's asymmetric extension of Tutte's classical spring-embedding theorem. First, we give a much simpler algorithm to construct piecewise-linear morphs between planar straight-line graphs. Specifically, given isomorphic straight-line drawings $\Gamma_0$ and $\Gamma_1$ of the same 3-connected planar graph $G$, with the same convex outer face, we construct a morph from $\Gamma_0$ to $\Gamma_1$ that consists of $O(n)$ unidirectional morphing steps, in $O(n^{1+\omega/2})$ time. Our algorithm entirely avoids the classical edge-collapsing strategy dating back to Cairns; instead, in each morphing step, we interpolate the pair of weights associated with a single edge. Second, we describe a natural extension of barycentric interpolation to geodesic graphs on the flat torus. Barycentric interpolation cannot be applied directly in this setting, because the linear systems defining intermediate vertex positions are not necessarily solvable. We describe a simple scaling strategy that circumvents this issue. Computing the appropriate scaling requires $O(n^{\omega/2})$ time, after which we can can compute the drawing at any point in the morph in $O(n^{\omega/2})$ time. Our algorithm is considerably simpler than the recent algorithm of Chambers et al. (arXiv:2007.07927) and produces more natural morphs. Our techniques also yield a simple proof of a conjecture of Connelly et al. for geodesic torus triangulations.