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 [math]\displaystyle{ n }[/math] sides has a nonobtuse triangulation with [math]\displaystyle{ O(n) }[/math] 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 [math]\displaystyle{ O(n) }[/math] 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
- ↑ 1.0 1.1 Bern, M.; Mitchell, S.; Ruppert, J. (1995), "Linear-size nonobtuse triangulation of polygons", Discrete & Computational Geometry 14 (4): 411–428, doi:10.1007/BF02570715
- ↑ Maehara, H. (2002), "Acute triangulations of polygons", European Journal of Combinatorics 23 (1): 45–55, doi:10.1006/eujc.2001.0531
- ↑ Yuan, Liping (2005), "Acute triangulations of polygons", Discrete & Computational Geometry 34 (4): 697–706, doi:10.1007/s00454-005-1188-9
See also
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