Between Shapes, Using the Hausdorff Distance

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

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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

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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)


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.

Subject Classification

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


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