Construction Of Triangulations Research Articles

Diffusion maps for embedded manifolds with boundary with applications to PDEs

Given only a finite collection of points sampled from a Riemannian manifold embedded in a Euclidean space, in this paper we propose a new method to numerically solve elliptic and parabolic partial differential equations (PDEs) supplemented with boundary conditions. Since the construction of triangulations on unknown manifolds can be both difficult and expensive, both in terms of computational and data requirements, our goal is to solve these problems without a triangulation. Instead, we rely only on using the sample points to define quadrature formulas on the unknown manifold. Our main tool is the diffusion maps algorithm. We re-analyze this well-known method in a variational sense for manifolds with boundary. Our main result is that the variational diffusion maps graph Laplacian is a consistent estimator of the Dirichlet energy on the manifold. This improves upon previous results and provides a rigorous justification of the well-known relationship between diffusion maps and the Neumann eigenvalue problem. Moreover, using semigeodesic coordinates we derive the first uniform asymptotic expansion of the diffusion maps kernel integral operator for manifolds with boundary. This expansion relies on a novel lemma which relates the extrinsic Euclidean distance to the coordinate norm in a normal collar of the boundary. We then use a recently developed method of estimating the distance to boundary function (notice that the boundary location is assumed to be unknown) to construct a consistent estimator for boundary integrals. Finally, by combining these various estimators, we illustrate how to impose Dirichlet and Neumann conditions for some common PDEs based on the Laplacian. Several numerical examples illustrate our theoretical findings.

Braced Triangulations and Rigidity

We consider the problem of finding an inductive construction, based on vertex splitting, of triangulated spheres with a fixed number of additional edges (braces). We show that for any positive integer b there is such an inductive construction of triangulations with b braces, having finitely many base graphs. In particular we establish a bound for the maximum size of a base graph with b braces that is linear in b. In the case that b=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b=1$$\\end{document} or 2 we determine the list of base graphs explicitly. Using these results we show that doubly braced triangulations are (generically) minimally rigid in two distinct geometric contexts arising from a hypercylinder in R4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^4$$\\end{document} and a class of mixed norms on R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3$$\\end{document}.

Mating of Discrete Trees and Walks in the Quarter-Plane

We give a general construction of triangulations starting from a walk in the quarter plane with small steps, which is a discrete version of the mating of trees. We use a special instance of this construction to give a bijection between maps equipped with a rooted spanning tree and walks in the quarter plane. We also show how the construction allows to recover several known bijections between such objects in a uniform way.

FEATURES OF CONSTRUCTION OF DIGITAL RELIEF MODEL ACCORDING TO THE RESULTS OF GEODESIC SURVEYING LOCATIONS

Topographic maps in digital and electronic forms are created on the basis of available paper topographic maps or on the basis of primary materials of geodetic surveys. Geodetic surveys are performed both by ground methods, without the use of photogrammetric materials, and on the basis of materials obtained as a result of ground phototheodolite or aerial photography. The construction of a digital terrain model is a multi-stage process, which consists of a significant number of interconnected operations performed at the stage of in-house processing of the results of geodetic measurements carried out during engineering and geodetic surveys. The quality of the final result of modeling depends on the quality of each stage of construction of a digital terrain model, so it is extremely important to pay attention to all technological processes of model construction. The digital relief model is considered as an ordered set of triangular faces constructed by the Delaunay algorithm. The main condition of this type of triangulation is that in the middle of the circle described around any triangle can not be the vertex of another triangle. Construction of a digital terrain model based on the results of geodetic surveying of the area in the general case can be divided into several stages. At the beginning, an automated construction of triangulation is performed on the basis of the results of geodetic measurements, which carry information about three-dimensional coordinates of survey points. Allotments adjust the display of horizontals. Regardless of the selected surface display style, the surface model is a grid of triangles. At the next stage of construction of the digital model of a relief carry out visual control of the created model and if necessary carry out editing of elements of a surface and change of position of edges of triangulation for change of position of horizontals. The last stage of building a digital terrain model based on the results of geodetic surveying of the area is the design of modeling results, the application of individual styles of reflection for individual areas of the surface and the creation of mountain strokes and signatures of horizontals.

Triangulation de Delaunay et Triangulation des Tores

We present a construction of triangulations of the torus \({\mathbb{T}^n}\) with 2n+1−1 vertices, starting from the Delaunay triangulation of a generic lattices. The typical example is the Kuhnel–Lassmann triangulation, with help of the lattice \({A^{*}_{n}}\) . Concerning this triangulation, we also prove a result of unicity for some values of n.

The drawability problem for minimum weight triangulations

A graph is minimum weight drawable if it admits a straight-line drawing that is a minimum weight triangulation of the set of points representing the vertices of the graph. We study the problem of characterizing those graphs that are minimum weight drawable. Our contribution is twofold: We show that there exist infinitely many triangulations that are not minimum weight drawable. Furthermore, we present non-trivial classes of triangulations that are minimum weight drawable, along with corresponding linear time algorithms that take as input any graph from one of these classes and produce as output such a drawing. One consequence of our work is the construction of triangulations that are minimum weight drawable but not Delaunay drawable – that is, not drawable as a Delaunay triangulation.

Triangulations in CAGD

Triangulations, which play an important role in approximation theory, finite-element methods, numerical analysis, and computer-aided geometric design (CAGD), are defined. Triangulations in the plane, in space, and on the surface of a sphere are discussed. The storing and construction of triangulations are also discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>