Survey of two-dimensional acute triangulations.

*(English)*Zbl 1261.52012A triangulation of a two-dimensional space is non-obtuse (acute), if all angles within its triangles are at most (strictly less than) \(\pi/2\). A balanced triangulation is an acute triangulation with all angles measuring more than \(\pi/6\).

This article gives an overview of recent results about acute and non-obtuse triangulations. The first part covers existence, asymptotic upper bounds, mesh generation algorithms, concrete upper bounds, and planar graphs. The second part discusses results on triangulating surfaces, platonic surfaces, double planar convex bodies, and Riemannian manifolds, in particular also the sphere and the Klein bottle. The article closes with a list of open problems regarding acute triangulations.

The results in the article are mostly given without proof. However, it contains an overview of the methods used by [S. Saraf, Eur. J. Comb. 30, No. 4, 833–840 (2009; Zbl 1171.52008)] to prove the Burago-Salgaller Theorem. In particular, a theorem in [L. Yuan, Discrete Comput. Geom. 34, No. 4, 697–706 (2005; Zbl 1112.52002)] providing a method to transform a non-obtuse triangulation of a polygon into an acute one is discussed in detail. Furthermore, the article reproduces a combinatorial proof from [T. Hangan, J. I. Itoh and T. Zamfirescu, Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 43(91), No. 3–4, 279–285 (2000; Zbl 1048.51501)] of the fact that the smallest acute triangulation of the boundary of the cube consists of 24 triangles. Some geometric aspects of these proofs are also exhibited by showing that every flat Möbius band admits an acute triangulation of size at most 9.

This article gives an overview of recent results about acute and non-obtuse triangulations. The first part covers existence, asymptotic upper bounds, mesh generation algorithms, concrete upper bounds, and planar graphs. The second part discusses results on triangulating surfaces, platonic surfaces, double planar convex bodies, and Riemannian manifolds, in particular also the sphere and the Klein bottle. The article closes with a list of open problems regarding acute triangulations.

The results in the article are mostly given without proof. However, it contains an overview of the methods used by [S. Saraf, Eur. J. Comb. 30, No. 4, 833–840 (2009; Zbl 1171.52008)] to prove the Burago-Salgaller Theorem. In particular, a theorem in [L. Yuan, Discrete Comput. Geom. 34, No. 4, 697–706 (2005; Zbl 1112.52002)] providing a method to transform a non-obtuse triangulation of a polygon into an acute one is discussed in detail. Furthermore, the article reproduces a combinatorial proof from [T. Hangan, J. I. Itoh and T. Zamfirescu, Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 43(91), No. 3–4, 279–285 (2000; Zbl 1048.51501)] of the fact that the smallest acute triangulation of the boundary of the cube consists of 24 triangles. Some geometric aspects of these proofs are also exhibited by showing that every flat Möbius band admits an acute triangulation of size at most 9.

Reviewer: Dimitris P. Vartziotis (Ioannina)

##### MSC:

52C20 | Tilings in \(2\) dimensions (aspects of discrete geometry) |