Survey of two-dimensional acute triangulations

Abstract We give a brief introduction to the topic of two-dimensional acute triangulations, mention results on related areas, survey existing achievements–with emphasis on recent activity–and list related open problems, both concrete and conceptual.

[1]  Verner E. Hoggatt,et al.  Acute Isosceles Dissection of an Obtuse Triangle , 1961 .

[2]  Damrong Guoy,et al.  Well-Centered Triangulation , 2008, SIAM J. Sci. Comput..

[3]  Victor Pambuccian,et al.  Acute Triangulation of a Triangle in a General Setting , 2010, Canadian Mathematical Bulletin.

[4]  David Eppstein,et al.  MESH GENERATION AND OPTIMAL TRIANGULATION , 1992 .

[5]  Elefterios A. Melissaratos,et al.  Coping with inconsistencies: a new approach to produce quality triangulations of polygonal domains with holes , 1992, SCG '92.

[6]  Damrong Guoy,et al.  A dihedral acute triangulation of the cube , 2009, Comput. Geom..

[7]  R. Gorenflo,et al.  Multi-index Mittag-Leffler Functions , 2014 .

[8]  R. H. MacNeal,et al.  An asymmetrical finite difference network , 1953 .

[9]  Tudor Zamfirescu,et al.  Acute triangulations of double planar convex bodies , 2012 .

[10]  Mamoru Watanabe,et al.  On the Number of Acute Triangles in a Straight-Line Embedding of a Maximal Planar Graph , 1999, J. Comb. Theory, Ser. B.

[11]  Marshall W. Bern,et al.  Linear-size nonobtuse triangulation of polygons , 1995, Discret. Comput. Geom..

[12]  J. O'Rourke,et al.  Geometric Folding Algorithms: Linkages , 2007 .

[13]  Carol T. Zamfirescu Acute triangulations of the double triangle , 2004 .

[14]  Carol T. Zamfirescu,et al.  Improved bounds for acute triangulations of convex polygons , 2010 .

[15]  Hiroshi Maehara,et al.  Acute Triangulations of Polygons , 2002, Eur. J. Comb..

[16]  Kenji Koyama,et al.  Acute triangles in triangulations on the plane with minimum degree at least 4 , 2005, Discret. Math..

[17]  Erik D. Demaine,et al.  Geometric folding algorithms - linkages, origami, polyhedra , 2007 .

[18]  Hiroshi Maehara,et al.  On a proper acute triangulation of a polyhedral surface , 2011, Discret. Math..

[19]  K. Wagner Bemerkungen zum Vierfarbenproblem. , 1936 .

[20]  Shubhangi Saraf,et al.  Acute and nonobtuse triangulations of polyhedral surfaces , 2009, Eur. J. Comb..

[21]  D. Chand,et al.  On Convex Polyhedra , 1970 .

[22]  Tudor Zamfirescu,et al.  Acute Triangulations of Flat Mobius Strips , 2007, Discret. Comput. Geom..

[23]  Joseph L. Gerver,et al.  The dissection of a polygon into nearly equilateral triangles , 1984 .

[24]  Jinichi Itoh Acute triangulations of sphere and icosahedron , 2001 .

[25]  Xiao Feng,et al.  Acute Triangulations of the Cuboctahedral Surface , 2010, CGGA.

[26]  Alper Üngör,et al.  Building spacetime meshes over arbitrary spatial domains , 2002, Engineering with Computers.

[27]  Charles Cassidy,et al.  A square acutely triangulated , 1980 .

[28]  Jon Louis Bentley,et al.  Quad trees a data structure for retrieval on composite keys , 1974, Acta Informatica.

[29]  Hiroshi Maehara,et al.  Plane graphs with straight edges whose bounded faces are acute triangles , 2003, J. Comb. Theory, Ser. B.

[30]  Erik D. Demaine,et al.  Generalized D-Forms Have No Spurious Creases , 2007, Discret. Comput. Geom..

[31]  Carol T. Zamfirescu,et al.  Acute Triangulations of Doubly Covered Convex Quadrilaterals , 2007 .

[32]  Maddalena Cavicchioli,et al.  Acute triangulations of convex quadrilaterals , 2012, Discret. Appl. Math..

[33]  Victor Pambuccian,et al.  Acute triangulation of a triangle in a general setting revisited , 2011 .

[34]  Martin Gardner,et al.  New Mathematical Diversions from "Scientific American" , 1971, The Mathematical Gazette.

[35]  S. Vavasis Stable finite elements for problems with wild coefficients , 1996 .

[36]  Jin-ichi Itoh,et al.  Acute Triangulations of the Regular Icosahedral Surface , 2004, Discret. Comput. Geom..

[37]  I. Fried Condition of finite element matrices generated from nonuniform meshes. , 1972 .

[38]  Igor Pak,et al.  Acute triangulations of polyhedra and ℝN , 2009, Comb..

[39]  A. Su,et al.  The National Council of Teachers of Mathematics , 1932, The Mathematical Gazette.

[40]  P. G. Ciarlet,et al.  Maximum principle and uniform convergence for the finite element method , 1973 .

[41]  Jorge Nuno Silva,et al.  Mathematical Games , 1959, Nature.

[42]  Liping Yuan,et al.  Acute triangulations of trapezoids , 2010, Discret. Appl. Math..

[43]  Alla Sheffer,et al.  Pitching Tents in Space-Time: Mesh Generation for Discontinuous Galerkin Method , 2002, Int. J. Found. Comput. Sci..

[44]  Brenda S. Baker,et al.  Nonobtuse triangulation of polygons , 1988, Discret. Comput. Geom..

[45]  Jin-ichi Itoh,et al.  Acute triangulations of flat tori , 2009, Eur. J. Comb..

[46]  Bath Ba THE GLOBAL DYNAMICS OF DISCRETE SEMILINEAR PARABOLIC EQUATIONS , 1993 .

[47]  David Eppstein,et al.  Polynomial-size nonobtuse triangulation of polygons , 1991, SCG '91.

[48]  Sergey Korotov,et al.  On Nonobtuse Simplicial Partitions , 2009, SIAM Rev..

[49]  Y. Colin de Verdière,et al.  TRIANGULATIONS PRESQUE EQUILATERALES DES SURFACES , 1990 .

[50]  J. A. Sethian,et al.  Fast Marching Methods , 1999, SIAM Rev..

[51]  Hiroshi Maehara On Acute Triangulations of Quadrilaterals , 2000, JCDCG.

[52]  Tudor Zamfirescu,et al.  Acute triangulations: a short survey , 2005 .

[53]  Jonathan Richard Shewchuk,et al.  What is a Good Linear Element? Interpolation, Conditioning, and Quality Measures , 2002, IMR.

[54]  Jin-ichi Itoh,et al.  Acute triangulations of the regular dodecahedral surface , 2007, Eur. J. Comb..

[55]  I. Babuska,et al.  ON THE ANGLE CONDITION IN THE FINITE ELEMENT METHOD , 1976 .

[56]  Ken-ichi Kawarabayashi,et al.  Acute triangles in 4-connected maximal plane graphs , 2005, Discret. Math..

[57]  Liping Yuan,et al.  Acute Triangulations of Pentagons , 2010 .

[58]  W. T. Tutte How to Draw a Graph , 1963 .

[59]  Tiow Seng Tan,et al.  An O(n2 log n) Time Algorithm for the Minmax Angle Triangulation , 1992, SIAM J. Sci. Comput..

[60]  Alper Üngör,et al.  Computing Acute and Non-obtuse Triangulations , 2007, CCCG.

[61]  David Eppstein,et al.  Provably good mesh generation , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[62]  David Eppstein,et al.  Tiling space and slabs with acute tetrahedra , 2003, Comput. Geom..

[63]  In-kyeong Choi On straight line representations of random planar graphs , 1992 .

[64]  S Kutateladze Existence Of A Convex Polyhedron With A Given Metric , 2005 .

[65]  Joseph O’Rourke How to Fold It: Folding Polygons to Convex Polyhedra , 2011 .

[66]  David Eppstein,et al.  Edge insertion for optimal triangulations , 1993, Discret. Comput. Geom..

[67]  B. McKay,et al.  Fast generation of planar graphs , 2007 .