Acute triangulations

Main Theorem (Burago-Zalgaller)
Every polygon (holes allowed) can be triangulated (subdivided without vertices on edges) into acute triangles. Moreover, such triangulation exist for every 2-dim polyhedral surface (with or without the boundary).

The original paper (long forgotten) is here:
Ju.D. Burago, V.A. Zalgaller, Polyhedral embedding of a net (in Russian), Vestnik Leningrad. Univ. 15 (1960), 66-80.

A recent relatively simple solution:
H. Maehara, Acute triangulations of polygons, European J. Combin. 23 (2002), 45-55.

I have my own solution, which is simpler than both of these but slightly more technical. Interestingly enough, if one allows right triangles there exit a plentiful literature:

B.S. Baker, E. Grosse, and C.S. Rafferty, Nonobtuse triangulation of polygons, Discrete Comput. Geom. 3 (1988), 147-168.

M. Bern, D. Eppstein, Polynomial-size nonobtuse triangulation of polygons, Internat. J. Comput. Geom. Appl. 2 (1992), 241-255; Errata 449-450.

M. Bern, S. Mitchell, and J. Ruppert, Linear-size nonobtuse triangulation of polygons, Discrete Comput. Geom. 14 (1995), 411-428.

Note: The last four papers are available on the web. Try MathSciNet and Google Scholar.

P.S. There are some attempts to generalize this results to higher (well, 3...) dimension. In this case one would like to make triangulations into tetrahedra with acute dihedral angles. Unfortunately, the results are not strong enough to list them here.

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Last updated 9/24/2005