Abstract morphing using the Hausdorff distance and Voronoi diagrams

TL;DR

This paper presents two novel abstract morphs for 2D shapes that utilize Hausdorff distance and Voronoi diagrams, ensuring properties like continuity and containment, with experimental analysis comparing their effectiveness and visual appeal.

Contribution

Introduction of two new shape morphs based on Hausdorff distance and Voronoi diagrams, with theoretical properties and experimental comparison to existing methods.

Findings

One morph is visually most attractive.

The morphs maintain continuity and containment.

Area and perimeter evolve smoothly during morphs.

Abstract

This paper introduces two new abstract morphs for two $2$-dimensional shapes. The intermediate shapes gradually reduce the Hausdorff distance to the goal shape and increase the Hausdorff distance to the initial shape. The morphs are conceptually simple and apply to shapes with multiple components and/or holes. We prove some basic properties relating to continuity, containment, and area. Then we give an experimental analysis that includes the two new morphs and a recently introduced abstract morph that is also based on the Hausdorff distance (Van Kreveld et al. Between shapes, using the Hausdorff distance. Computational Geometry 100:101817, 2022). We show results on the area and perimeter development throughout the morph, and also the number of components and holes. A visual comparison shows that one of the new morphs appears most attractive.