Nonobtuse mesh

In computer graphics, a nonobtuse triangle mesh is a polygon mesh composed of a set of triangles in which no angle is obtuse, i.e. greater than 90°. If each (triangle) face angle is strictly less than 90°, then the triangle mesh is said to be acute. Every polygon with $n$ sides has a nonobtuse triangulation with $O(n)$ triangles (expressed in big O notation), allowing some triangle vertices to be added to the sides and interior of the polygon.^[1] These nonobtuse triangulations can be further refined to produce acute triangulations with $O(n)$ triangles.^[2]^[3]

Nonobtuse meshes avoid certain problems of nonconvergence or of convergence to the wrong numerical solution as demonstrated by the Schwarz lantern.^[1] The immediate benefits of a nonobtuse or acute mesh include more efficient and more accurate geodesic computation using fast marching, and guaranteed validity for planar mesh embeddings via discrete harmonic maps.

References

^ ^a ^b Bern, M.; Mitchell, S.; Ruppert, J. (1995), "Linear-size nonobtuse triangulation of polygons", Discrete & Computational Geometry, 14 (4): 411–428, doi:10.1007/BF02570715, MR 1360945
^ Maehara, H. (2002), "Acute triangulations of polygons", European Journal of Combinatorics, 23 (1): 45–55, doi:10.1006/eujc.2001.0531, MR 1878775
^ Yuan, Liping (2005), "Acute triangulations of polygons", Discrete & Computational Geometry, 34 (4): 697–706, doi:10.1007/s00454-005-1188-9, MR 2173934, S2CID 26601451

References

See also