1,827 research outputs found
There are only two nonobtuse binary triangulations of the unit -cube
Triangulations of the cube into a minimal number of simplices without
additional vertices have been studied by several authors over the past decades.
For this so-called simplexity of the unit cube is now
known to be , respectively. In this paper, we study
triangulations of with simplices that only have nonobtuse dihedral
angles. A trivial example is the standard triangulation into simplices. In
this paper we show that, surprisingly, for each there is essentially
only one other nonobtuse triangulation of , and give its explicit
construction. The number of nonobtuse simplices in this triangulation is equal
to the smallest integer larger than .Comment: 17 pages, 7 figure
A tetrahedral space-filling curve for non-conforming adaptive meshes
We introduce a space-filling curve for triangular and tetrahedral
red-refinement that can be computed using bitwise interleaving operations
similar to the well-known Z-order or Morton curve for cubical meshes. To store
sufficient information for random access, we define a low-memory encoding using
10 bytes per triangle and 14 bytes per tetrahedron. We present algorithms that
compute the parent, children, and face-neighbors of a mesh element in constant
time, as well as the next and previous element in the space-filling curve and
whether a given element is on the boundary of the root simplex or not. Our
presentation concludes with a scalability demonstration that creates and adapts
selected meshes on a large distributed-memory system.Comment: 33 pages, 12 figures, 8 table
The Complexity of Finding Small Triangulations of Convex 3-Polytopes
The problem of finding a triangulation of a convex three-dimensional polytope
with few tetrahedra is proved to be NP-hard. We discuss other related
complexity results.Comment: 37 pages. An earlier version containing the sketch of the proof
appeared at the proceedings of SODA 200
Extremal properties for dissections of convex 3-polytopes
A dissection of a convex d-polytope is a partition of the polytope into
d-simplices whose vertices are among the vertices of the polytope.
Triangulations are dissections that have the additional property that the set
of all its simplices forms a simplicial complex. The size of a dissection is
the number of d-simplices it contains. This paper compares triangulations of
maximal size with dissections of maximal size. We also exhibit lower and upper
bounds for the size of dissections of a 3-polytope and analyze extremal size
triangulations for specific non-simplicial polytopes: prisms, antiprisms,
Archimedean solids, and combinatorial d-cubes.Comment: 19 page
Exact asymptotics of the uniform error of interpolation by multilinear splines
The question of adaptive mesh generation for approximation by splines has
been studied for a number of years by various authors. The results have
numerous applications in computational and discrete geometry, computer aided
geometric design, finite element methods for numerical solutions of partial
differential equations, image processing, and mesh generation for computer
graphics, among others. In this paper we will investigate the questions
regarding adaptive approximation of C2 functions with arbitrary but fixed
throughout the domain signature by multilinear splines. In particular, we will
study the asymptotic behavior of the optimal error of the weighted uniform
approximation by interpolating and quasi-interpolating multilinear splines
A Geometric Approach to Combinatorial Fixed-Point Theorems
We develop a geometric framework that unifies several different combinatorial
fixed-point theorems related to Tucker's lemma and Sperner's lemma, showing
them to be different geometric manifestations of the same topological
phenomena. In doing so, we obtain (1) new Tucker-like and Sperner-like
fixed-point theorems involving an exponential-sized label set; (2) a
generalization of Fan's parity proof of Tucker's Lemma to a much broader class
of label sets; and (3) direct proofs of several Sperner-like lemmas from
Tucker's lemma via explicit geometric embeddings, without the need for
topological fixed-point theorems. Our work naturally suggests several
interesting open questions for future research.Comment: 10 pages; an extended abstract appeared at Eurocomb 201
Tetrahedralization of a Hexahedral Mesh
Two important classes of three-dimensional elements in computational meshes
are hexahedra and tetrahedra. While several efficient methods exist that
convert a hexahedral element to a tetrahedral elements, the existing algorithm
for tetrahedralization of a hexahedral complex is the marching tetrahedron
algorithm which limits pre-selection of face divisions. We generalize a
procedure for tetrahedralizing triangular prisms to tetrahedralizing cubes, and
combine it with certain heuristics to design an algorithm that can triangulate
any hexahedra.Comment: The previous version had an error in the proof of Observation 2.1,
which has since been rectified in this version. Formatting and title change
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