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A dihedral acute triangulation of the cube
 Computational Geometry: Theory and Applications, (Accepted, and available online
, 2009
"... It is shown that there exists a dihedral acute triangulation of the cube. The method of constructing the acute triangulation is described, and symmetries of the triangulation are discussed. 1 ..."
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It is shown that there exists a dihedral acute triangulation of the cube. The method of constructing the acute triangulation is described, and symmetries of the triangulation are discussed. 1
NONOBTUSE TRIANGULATIONS OF PSLGS
, 2010
"... We show that any planar PSLG with n vertices has a conforming triangulation by O(n2.5) nonobtuse triangles; they may be chosen to be all acute or all right. This result also improves a previous O(n3) bound of Eldesbrunner and Tan for conforming Delaunay triangulations. In the special case that the ..."
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We show that any planar PSLG with n vertices has a conforming triangulation by O(n2.5) nonobtuse triangles; they may be chosen to be all acute or all right. This result also improves a previous O(n3) bound of Eldesbrunner and Tan for conforming Delaunay triangulations. In the special case that the PSLG is the triangulation of a simple polygon, we will show that only O(n2) elements are needed, improving an O(n4) bound of Bern and Eppstein. We also show that for any ǫ> 0, every PSLG has a conforming triangulation with O(n2 /ǫ2) elements and with all angles bounded above by 90 ◦ +ǫ. This improves a result of S. Mitchell when ǫ = 3 8π = 67.5 ◦ and Tan when ǫ = 7 30π = 42 ◦. Finally, we prove that any PSLG has a conforming quadrilateral mesh with O(n2) elements and all new angles between 60 ◦ and 120 ◦ (the complexity and angle bounds are both sharp). Moreover, all but O(n) of the angles may be taken in a smaller interval, say [89◦,91 ◦].
Acute triangulations of polyhedra and Rn
 COMBINATORICA 32 (1) (2012) 85–110
, 2012
"... We study the problem of acute triangulations of convex polyhedra and the space Rn. Here an acute triangulation is a triangulation into simplices whose dihedral angles are acute. We prove that acute triangulations of the ncube do not exist for n≥4. Further, we prove that acute triangulations of the ..."
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We study the problem of acute triangulations of convex polyhedra and the space Rn. Here an acute triangulation is a triangulation into simplices whose dihedral angles are acute. We prove that acute triangulations of the ncube do not exist for n≥4. Further, we prove that acute triangulations of the space Rn do not exist for n≥5. In the opposite direction, in R³, we present a construction of an acute triangulation of the cube, the regular octahedron and a nontrivial acute triangulation of the regular tetrahedron. We also prove nonexistence of an acute triangulation of R4 if all dihedral angles are bounded away from π/2.
A Dihedral Acute Triangulation of the Cube
"... It is shown that there exists a dihedral acute triangulation of the threedimensional cube. The method of constructing the acute triangulation is described, and symmetries of the triangulation are discussed. 1 ..."
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It is shown that there exists a dihedral acute triangulation of the threedimensional cube. The method of constructing the acute triangulation is described, and symmetries of the triangulation are discussed. 1
Efficient Construction and Simplification of Delaunay Meshes YongJin Liu∗
"... Figure 1: We present an efficient algorithm to convert an arbitrary manifold triangle meshM to a Delaunay mesh (DM), which has the same geometry of M. Our algorithm can also produce progressive Delaunay meshes, allowing a smooth choice of detail levels. Since DMs are represented using conventional m ..."
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Figure 1: We present an efficient algorithm to convert an arbitrary manifold triangle meshM to a Delaunay mesh (DM), which has the same geometry of M. Our algorithm can also produce progressive Delaunay meshes, allowing a smooth choice of detail levels. Since DMs are represented using conventional mesh data structures, the existing digital geometry processing algorithms can benefit the numerical stability of DM without changing any codes. For example, DMs significantly improve the accuracy of the heat method for computing geodesic distances. Delaunay meshes (DM) are a special type of triangle mesh where the local Delaunay condition holds everywhere. We present an efficient algorithm to convert an arbitrary manifold triangle mesh M into a Delaunay mesh. We show that the constructed DM has O(Kn) vertices, where n is the number of vertices in M and K is a modeldependent constant. We also develop a novel algorithm to simplify Delaunay meshes, allowing a smooth choice of detail levels. Our methods are conceptually simple, theoretically sound and easy to implement. The DM construction algorithm also scales well due to its O(nK logK) time complexity.