article published yes Herbert Edelsbrunner author 3FB178DA-F248-11E8-B48F-1D18A9856A870000-0002-9823-6833 Anton Nikitenko author 3E4FF1BA-F248-11E8-B48F-1D18A9856A870000-0002-0659-3201 HeEd department Alpha Shape Theory Extended project Mathematics, Computer Science project Persistence and stability of geometric complexes project The approximation of a circle with the edges of a fine square grid distorts the perimeter by a factor about 4/Pi. We prove that this factor is the same on average (in the ergodic sense) for approximations of any rectifiable curve by the edges of any non-exotic Delaunay mosaic (known as Voronoi path), and extend the results to all dimensions, generalizing Voronoi paths to Voronoi scapes. https://research-explorer.ista.ac.at/download/17149/19610/2025_DiscreteComputGeom_EdelsbrunnerHe.pdf application/pdfno https://creativecommons.org/licenses/by/4.0/ Springer Nature2025 eng 0179-5376 1432-0444 2012.03350 39974750 00123856620000410.1007/s00454-024-00660-y 73490-499 Edelsbrunner, H., &#38; Nikitenko, A. (2025). Average and expected distortion of Voronoi paths and scapes. <i>Discrete &#38; Computational Geometry</i>. Springer Nature. <a href="https://doi.org/10.1007/s00454-024-00660-y">https://doi.org/10.1007/s00454-024-00660-y</a> Edelsbrunner, Herbert, and Anton Nikitenko. “Average and Expected Distortion of Voronoi Paths and Scapes.” <i>Discrete &#38; Computational Geometry</i>, vol. 73, Springer Nature, 2025, pp. 490–99, doi:<a href="https://doi.org/10.1007/s00454-024-00660-y">10.1007/s00454-024-00660-y</a>. Edelsbrunner H, Nikitenko A. 2025. Average and expected distortion of Voronoi paths and scapes. Discrete &#38; Computational Geometry. 73, 490–499. H. Edelsbrunner, A. Nikitenko, Discrete &#38; Computational Geometry 73 (2025) 490–499. Edelsbrunner, Herbert, and Anton Nikitenko. “Average and Expected Distortion of Voronoi Paths and Scapes.” <i>Discrete &#38; Computational Geometry</i>. Springer Nature, 2025. <a href="https://doi.org/10.1007/s00454-024-00660-y">https://doi.org/10.1007/s00454-024-00660-y</a>. H. Edelsbrunner and A. Nikitenko, “Average and expected distortion of Voronoi paths and scapes,” <i>Discrete &#38; Computational Geometry</i>, vol. 73. Springer Nature, pp. 490–499, 2025. Edelsbrunner H, Nikitenko A. Average and expected distortion of Voronoi paths and scapes. <i>Discrete &#38; Computational Geometry</i>. 2025;73:490-499. doi:<a href="https://doi.org/10.1007/s00454-024-00660-y">10.1007/s00454-024-00660-y</a> 171492024-06-16T22:01:07Z2026-02-16T12:18:50Z