## 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 very simple algorithm to construct piecewise-linear morphs between planar straight-line graphs. Specifically, given isomorphic straight-line drawings Γ_{0} and Γ_{1} of the same 3-connected planar graph G, with the same convex outer face, we construct a morph from Γ_{0} to Γ_{1} that consists of O(n) unidirectional morphing steps, in O(n^{1} ^{+} ^{ω} ^{/} ^{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^{ω} ^{/} ^{2}) time, after which we can compute the drawing at any point in the morph in O(n^{ω} ^{/} ^{2}) time. Our algorithm is considerably simpler than the recent algorithm of Chambers et al. and produces more natural morphs. Our techniques also yield a simple proof of a conjecture of Connelly et al. for geodesic torus triangulations.

Original language | English (US) |
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Title of host publication | Graph Drawing and Network Visualization - 29th International Symposium, GD 2021, Revised Selected Papers |

Editors | Helen C. Purchase, Ignaz Rutter |

Publisher | Springer |

Pages | 123-137 |

Number of pages | 15 |

ISBN (Print) | 9783030929305 |

DOIs | |

State | Published - 2021 |

Event | 29th International Symposium on Graph Drawing and Network Visualization, GD 2021 - Tübingen, Germany Duration: Sep 14 2021 → Sep 17 2021 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 12868 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 29th International Symposium on Graph Drawing and Network Visualization, GD 2021 |
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Country/Territory | Germany |

City | Tübingen |

Period | 9/14/21 → 9/17/21 |

## ASJC Scopus subject areas

- Theoretical Computer Science
- General Computer Science