Nonobtuse mesh: Difference between revisions

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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

@@ Line 1: / Line 1: @@
+{{Short description|Polygon mesh composed of triangles with all angles ≤ 90°}}
-{{Inline citations|date=April 2021}}
-A '''nonobtuse triangle mesh''' is composed of a set of triangles in which no angle is obtuse, ''i.e.'' greater than 90°.<ref>{{citation
+In [[computer graphics]], a '''nonobtuse triangle mesh''' is a [[polygon mesh]] composed of a set of [[triangle]]s 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>n</math> sides has a nonobtuse triangulation with <math>O(n)</math> triangles (expressed in [[big O notation]]), allowing some triangle vertices to be added to the sides and interior of the polygon.<ref name=bmr>{{citation
  | last1 = Bern | first1 = M.
  | last2 = Mitchell | first2 = S.
@@ Line 12: / Line 12: @@
  | title = Linear-size nonobtuse triangulation of polygons
  | volume = 14
+ | year = 1995| doi-access = free
- | year = 1995}}</ref> If each (triangle) face angle is strictly less than 90°, then the [[triangle mesh]] is said to be acute. The immediate benefits of a nonobtuse or acute mesh include more efficient and more accurate [[geodesic]] computation using [[fast marching method|fast marching]], and guaranteed validity for planar mesh embeddings via discrete harmonic maps.
+ }}</ref> These nonobtuse triangulations can be further refined to produce acute triangulations with <math>O(n)</math> triangles.<ref>{{citation
+ | last = Maehara | first = H.
+ | doi = 10.1006/eujc.2001.0531
+ | issue = 1
+ | journal = European Journal of Combinatorics
+ | mr = 1878775
+ | pages = 45–55
+ | title = Acute triangulations of polygons
+ | volume = 23
+ | year = 2002| doi-access = free
+ }}</ref><ref>{{citation
+ | last = Yuan | first = Liping
+ | doi = 10.1007/s00454-005-1188-9
+ | issue = 4
+ | journal = Discrete & Computational Geometry
+ | mr = 2173934
+ | pages = 697–706
+ | title = Acute triangulations of polygons
+ | volume = 34
+ | year = 2005| s2cid = 26601451
+ | doi-access = free
+ }}</ref>
+Nonobtuse meshes avoid certain problems of nonconvergence or of convergence to the wrong numerical solution as demonstrated by the [[Schwarz lantern]].<ref name=bmr/> The immediate benefits of a nonobtuse or acute mesh include more efficient and more accurate [[geodesic]] computation using [[fast marching method|fast marching]], and guaranteed validity for planar mesh embeddings via discrete harmonic maps.
 ==References==
+{{reflist}}
-*[http://www.cs.sfu.ca/~ysl/personal/publication/sgp06_electronic.pdf Nonobtuse Remeshing and Mesh Decimation]
-*[http://www.cs.sfu.ca/~ysl/personal/publication/TR-CMPT2006-13.pdf Guaranteed Nonobtuse Meshes via Constrainted Optimizations]
 ==See also==