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Given a sufficiently dense sample of points on a surface (technically, an \epsilon-sample), reconstruct a surface homeomorphic to the original.
Amenta and Bern [?]
This has recently been accomplished for smooth surfaces [ACDL00], but remains open for surfaces with sharp edges and corners.
reconstruction; point sets
J. O’Rourke, 2 Aug. 2001.
N. Amenta, S. Choi, T. K. Dey, and N. Leekha. A simple algorithm for homeomorphic surface reconstruction. In Proc. 16th Annu. ACM Sympos. Comput. Geom., pages 213–222, 2000.
J. S. B. Mitchell and Joseph O’Rourke. Computational geometry column 42. Internat. J. Comput. Geom. Appl., 11(5):573–582, 2001. Also in SIGACT News 32(3):63-72 (2001), Issue 120.