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Between Shapes, Using the Hausdorff Distance

Authors Marc van Kreveld, Tillmann Miltzow, Tim Ophelders, Willem Sonke, Jordi L. Vermeulen

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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • computational geometry
  • Hausdorff distance
  • shape interpolation

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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 a generalization of this result on Hausdorff distances and middle shapes, and show some related properties. We also show that a generalization of such middle shapes implies a morph with a bounded rate of change. Finally, we explore a generalization of the concept of a Hausdorff middle to more than two sets and show how to approximate or compute it.

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Marc van Kreveld, Tillmann Miltzow, Tim Ophelders, Willem Sonke, and Jordi L. Vermeulen. Between Shapes, Using the Hausdorff Distance. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 13:1-13:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ISAAC.2020.13

Author Details

Marc van Kreveld
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands
Tillmann Miltzow
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands
Tim Ophelders
  • Department of Computational Mathematics, Science and Engineering, Michigan State University, East Lansing, MI, USA
Willem Sonke
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
Jordi L. Vermeulen
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands

Funding

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.
  • van Kreveld, Marc: supported by the NWO TOP grant no. 612.001.651.
  • Miltzow, Tillmann: supported by the NWO Veni grant EAGER.
  • Vermeulen, Jordi L.: supported by the NWO TOP grant no. 612.001.651.

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