# Between Shapes, Using the Hausdorff Distance

arXiv: https://arxiv.org/abs/2009.14719 · HTML: https://mlsanity.com/paper/2009.14719/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Sep 2020.

## Built on by (most cited first, 3 of 3)

- [E(n) Equivariant Topological Neural Networks](https://mlsanity.com/paper/2405.15429/) (2024)
- [Quality assessment of 3D human animation: Subjective and objective evaluation](https://mlsanity.com/paper/2505.23301/) (2025)
- [Detection of Under-represented Samples Using Dynamic Batch Training for Brain Tumor Segmentation from MR Images](https://mlsanity.com/paper/2408.12013/) (2024)

## Abstract (arXiv)

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 generalised 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.