On a theorem by Hsiang and Yu

In 1841, Delaunay classified surfaces of revolution with constant mean curvature in the Euclidean three space. As a byproduct of his result, one obtains: A surface of revolution has a periodic generating curve if and only if its mean curvature is non-zero. One hundred and forty years after Delaunay’s work, Hsiang and Yu extended this result to higher dimensions, by extending Delaunay’s idea of tracing the locus of a focus by rolling a given conic section along a line. In this paper, we give a new proof of their result using elementary ODE theory to obtain the periodicity of the solutions under consideration.      

一类三角域上的三次样条插值(英文)

In this paper, we consider spaces of cubic C^1-spline on a class of triangulations. By using the inductive algorithm, the posed Lagrange interpolation sets are constructed for cubic spline space. It is shown that the class of triangulations considered in this paper are nonsingular for S1/3 spaces. Moreover, the dimensions of those spaces exactly equal to L. L. Schuraaker＇s low bounds of the dimensions. At the end of this paper, we present an approach to construct triangulations from any scattered planar points, which ensures that the obtained triangulations for S1/3 space are nonsingular.     

Generating spherical Eulerian triangulations

We show that all spherical Eulerian triangulations can be inductively generated from the set of all even double wheels using just one of the two local transformations used in the algorithm by Brinkmann and McKay and originally proposed by Batagelj.     

5-Chromatic even triangulations on surfaces

A triangulation is said to be even if each vertex has even degree. It is known that every even triangulation on any orientable surface with sufficiently large representativity is 4-colorable [J. Hutchinson, B. Richter, P. Seymour, Colouring Eulerian triangulations, J. Combin. Theory, Ser. B 84 (2002) 225-239], but all graphs on any surface with large representativity are 5-colorable [C. Thomassen, Five-coloring maps on surfaces, J. Combin Theory Ser. B 59 (1993) 89-105]. In this paper, we shall characterize 5-chromatic even triangulations with large representativity, which appear only on nonorientable surfaces.     

On a property of minimal triangulations

A graph H has the property MT, if for all graphs G, G is H-free if and only if every minimal (chordal) triangulation of G is H-free. We show that a graph H satisfies property MT if and only if H is edgeless, H is connected and is an induced subgraph of P₅, or H has two connected components and is an induced subgraph of 2P₃.This completes the results of Parra and Scheffler, who have shown that MT holds for H=P_k, the path on k vertices, if and only if k?5 [A. Parra, P. Scheffler, Characterizations and algorithmic applications of chordal graph embeddings, Discrete Applied Mathematics 79 (1997) 171-188], and of Meister, who proved that MT holds for ?P₂, ? copies of a P₂, if and only if ??2 [D. Meister, A complete characterisation of minimal triangulations of 2K₂-free graphs, Discrete Mathematics 306 (2006) 3327-3333].     

Modeling wildfire propagation with Delaunay triangulation and shortest path algorithms

In this paper, a methodology for modeling surface wildfire propagation through a complex landscape is presented. The methodology utilizes a Delaunay triangulation to represent surface fire spread within the landscape. A procedure to construct the graph and estimate the rate of spread along the edges of a network is discussed. After the Delaunay data structure is constructed, a two pass shortest path algorithm is incorporated to estimate the minimum travel time paths and fire arrival times. Experimental results are also included.     

The Weil–Petersson Kähler Form and Affine Foliations on Surfaces

The space of broken hyperbolic structures generalizes the usual Teichmüller space of a punctured surface, and the space of projectivized broken measured foliations – or, equivalently, the space of projectivized affine foliations of a punctured surface – likewise generalizes the space of projectivized measured foliations. Just as projectivized measured foliations provide Thurston's boundary for Teichmüller space, so too do projectivized broken measured foliations provide a boundary for the space of broken hyperbolic structures. In this paper, we naturally extend the Weil–Petersson Kähler two-form to a corresponding two-form on the space of broken hyperbolic structures as well as Thurston's symplectic form to a corresponding two-form on the space of broken measured foliations, and we show that the former limits in an appropriate sense to the latter. The proof in sketch follows earlier work of the authors for measured foliations and depends upon techniques from decorated Teichmüller theory, which is also applied here to a further study of broken hyperbolic structures.     

Polynomial preserving recovery for meshes from Delaunay triangulation or with high aspect ratio

A newly developed polynomial preserving gradient recovery technique is further studied. The results are twofold. First, error bounds for the recovered gradient are established on the Delaunay type mesh when the major part of the triangulation is made of near parallelogram triangle pairs with ε‐perturbation. It is found that the recovered gradient improves the leading term of the error by a factor ε. Secondly, the analysis is performed for a highly anisotropic mesh where the aspect ratio of element sides is unbounded. When the mesh is adapted to the solution that has significant changes in one direction but very little, if any, in another direction, the recovered gradient can be superconvergent. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Improvements on Delaunay-based three-dimensional automatic mesh generator

This paper describes an automatic mesh generator providing tetrahedral meshes suitable in general for finite element simulations. The mesh generator is of the Delaunay type and the paper focuses on recent improvements relative to this a priori well-known method.     

Positively oriented ideal triangulations on hyperbolic three-manifolds

Let M³ be a non-compact hyperbolic 3-manifold that has a triangulation by positively oriented ideal tetrahedra. We show that the gluing variety defined by the gluing consistency equations is a smooth complex manifold with dimension equal to the number of boundary components of M³. Moreover, we show that the complex lengths of any collection of non-trivial boundary curves, one from each boundary component, give a local holomorphic parameterization of the gluing variety. As an application, some estimates for the size of hyperbolic Dehn surgery space of once-punctured torus bundles are given.