Elsevier

Computational Geometry

Volume 100, January 2022, 101817
Computational Geometry

Between shapes, using the Hausdorff distance

https://doi.org/10.1016/j.comgeo.2021.101817Get rights and content
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Abstract

Given two shapes A and B in the plane with Hausdorff distance 1, is there a shape S with Hausdorff distance 1/2 to and from A and B? The answer is always yes, and depending on convexity of A and/or B, S may be convex, connected, or disconnected. We show that our result can be generalized to give an interpolated shape between A and B for any interpolation variable α between 0 and 1, and prove that the resulting morph has a bounded rate of change with respect to α. Finally, we explore a generalization of the concept of a Hausdorff middle to more than two input sets. We show how to approximate or compute this middle shape, and that the properties relating to the connectedness of the Hausdorff middle extend from the case with two input sets. We also give bounds on the Hausdorff distance between the middle set and the input.

Keywords

Computational geometry
Hausdorff distance
Shape interpolation
Morphing

Cited by (0)

Research on the topic of this paper was initiated at the 4th Workshop on Applied Geometric Algorithms (AGA 2018) in Langbroek, The Netherlands, supported by the Netherlands Organisation for Scientific Research (NWO) under project no. 639.023.208. The second author is supported by the NWO under project no. 016.Veni.192.250. The first and fifth authors are supported by the NWO TOP grant no. 612.001.651.