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- Statement
How efficiently can one compute a polyhedral surface that is an \epsilon-approximation of a given triangulated surface in \R^3?

- Origin
Mitchell [?]

- Status/Conjectures
Open.

- Partial and Related Results
It is NP-hard to obtain the minimum-facet surface separating two nested convex polytopes [DG97], if they are allowed to have distinct structures; the complexity is open when one polytope is an offset of the other (e.g., if they arise as offsets of an underlying convex polytope that we seek to approximate). Polynomial-time approximation algorithms are known ([BG95], [MS95], [AS98]) for the case of nested convex polyhedra, and for separating two terrain surfaces, achieving factors within O(1) or O(\log n) of optimal. However, no polynomial-time approximation results are known for general surfaces.

- Appearances
- Categories
simplification

- Entry Revision History
J. O’Rourke, 2 Aug. 2001; J. Mitchell, 14 Oct 2020.

- [AS98]
P. K. Agarwal and S. Suri. Surface approximation and geometric partitions.

*SIAM J. Comput.*, 27:1016–1035, 1998.- [BG95]
H. Brönnimann and M. T. Goodrich. Almost optimal set covers in finite VC-dimension.

*Discrete Comput. Geom.*, 14:263–279, 1995.- [DG97]
G. Das and M. T. Goodrich. On the complexity of optimization problems for 3-dimensional convex polyhedra and decision trees.

*Comput. Geom. Theory Appl.*, 8:123–137, 1997.- [MO01]
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.- [MS95]
Joseph S. B. Mitchell and Subhash Suri. Separation and approximation of polyhedral objects.

*Comput. Geom. Theory Appl.*, 5:95–114, 1995.