Nonobtuse mesh

{{Short description|Polygon mesh composed of triangles with all angles ≤ 90°}}

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.{{citation

| last1 = Bern | first1 = M.

| last2 = Mitchell | first2 = S.

| last3 = Ruppert | first3 = J.

| doi = 10.1007/BF02570715

| issue = 4

| journal = Discrete & Computational Geometry

| mr = 1360945

| pages = 411–428

| title = Linear-size nonobtuse triangulation of polygons

| volume = 14

| year = 1995| doi-access = free

}} These nonobtuse triangulations can be further refined to produce acute triangulations with $O(n)$ triangles.{{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

}}{{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

}}

Nonobtuse meshes avoid certain problems of nonconvergence or of convergence to the wrong numerical solution as demonstrated by the Schwarz lantern. 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.