Nonobtuse mesh

In <a href="/facts/Computer_graphics/lapBE8wv">computer graphics</a>, a nonobtuse triangle mesh is a <a href="/facts/Polygon_mesh/6WLD80ER">polygon mesh</a> composed of a set of <a href="/facts/Triangle/cRXUwsMn">triangles</a> in which no <a href="/facts/Angle/gFUTEQYq">angle</a> is obtuse, i.e. greater than 90°. If each (triangle) face angle is strictly less than 90°, then the <a href="/facts/Triangle_mesh/EphfTInp">triangle mesh</a> is said to be acute. Every <a href="/facts/Polygon/yAq7RjPz">polygon</a> with 
 
 
 
 n
 
 
 {\displaystyle n}
 
 sides has a nonobtuse triangulation with 
 
 
 
 O
 (
 n
 )
 
 
 {\displaystyle O(n)}
 
 triangles (expressed in <a href="/facts/Big_O_notation/weFFjSWg">big O notation</a>), allowing some triangle vertices to be added to the sides and interior of the polygon. These nonobtuse triangulations can be further refined to produce acute triangulations with 
 
 
 
 O
 (
 n
 )
 
 
 {\displaystyle O(n)}
 
 triangles.
Nonobtuse meshes avoid certain problems of nonconvergence or of convergence to the wrong numerical solution as demonstrated by the <a href="/facts/Schwarz_lantern/feQr4B7y">Schwarz lantern</a>. The immediate benefits of a nonobtuse or acute mesh include more efficient and more accurate <a href="/facts/Geodesic/fbWR6l80">geodesic</a> computation using <a href="/facts/Fast_marching_method/CHr830cQ">fast marching</a>, and guaranteed validity for planar mesh embeddings via discrete harmonic maps.

Nonobtuse mesh open-in-new

Nonobtuse mesh