Frequently, data in scientific computing is in its abstract form a finite point set in space, and it is sometimes useful or required to compute what one might call the "shape" of the set. For that purpose, this paper introduces the formal notion of the family of ff-shapes of a finite point set in R³. Each shape is a well-defined polytope, derived from the Delaunay triangulation of the point set, with a parameter ff 2 IR controlling the desired level of detail. An algorithm is presented that constructs the entire family of shapes for a given set of size n in time O(n²), worst case. A robust implementation of the algorithm is discussed and several applications in the area of scientific computing are mentioned.

Several algorithms for approximating terrains and other height fields using polygonal meshes are described, compared, and optimized. These algorithms take a height field as input, typically a rectangular grid of elevation data H(x; y), and approximate it with a mesh of triangles, also known as a triangulated irregular network, or TIN. The algorithms attempt to minimize both the error and the number of triangles in the approximation. Applications include fast rendering of terrain data for flight simulation and fitting of surfaces to range data in computer vision. The methods can also be used to simplify multi-channel height fields such as textured terrains or planar color images. The most successful method we examine is the greedy insertion algorithm. It begins with a simple triangulation of the domain and, on each pass, finds the input point with highest error in the current approximation and inserts it as a vertex in the triangulation. The mesh is updated either with Delaunay triangul...

This paper describes a fast algorithm for scattered data interpolation and approximation. Multilevel B-splines are introduced to compute a C²-continuous surface through a set of irregularly spaced points. The algorithm makes use of a coarse-tofine hierarchy of control lattices to generate a sequence of bicubic B-spline functions whose sum approaches the desired interpolation function. Large performance gains are realized by using B-spline refinement to reduce the sum of these functions into one equivalent B-spline function. Experimental results demonstrate that high-fidelity reconstruction is possible from a selected set of sparse and irregular samples.

Abstract not found

A new algorithm for the generation of anisotropic, unstructured triangular meshes in two dimensions is described. Inputs to the algorithm are the boundary geometry and a metric that specifies the desired element size and shape as a function of position. The algorithm is an example of what we call pliant mesh generation. It first constructs the constrained Delaunay triangulation of the domain, then iteratively smooths, refines, and retriangulates. On each iteration, a node is selected at random, it is repositioned according to attraction/repulsion with its neighbors, the neighborhood is retriangulated, and nodes are inserted or deleted as necessary. All operations are done relative to the metric tensor. This simple method generates high quality meshes whose elements conform well to the requested shape metric. The method appears particularly well suited to surface meshing and viscous flow simulations, where stretched triangles are desirable, and to time-dependent remeshing problems.

A plane geometric graph C in ! 2 conforms to another such graph G if each edge of G is the union of some edges of C. It is proved that for every G with n vertices and m edges, there is a completion of a Delaunay triangulation of O(m 2 n) points that conforms to G. The algorithm that constructs the points is also described. Keywords. Discrete and computational geometry, plane geometric graphs, Delaunay triangulations, point placement. Appear in: Discrete & Computational Geometry, 10 (2), 197--213 (1993) 1 Research of the first author is supported by the National Science Foundation under grant CCR-8921421 and under the Alan T. Waterman award, grant CCR-9118874. Any opinions, finding and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the view of the National Science Foundation. Work of the second author was conducted while he was on study leave at the University of Illinois. 2 Department of Computer Scienc...

In this paper we study the problem of flipping edges in triangulations of polygons and point sets. We prove that if a polygon Q n has k reflex vertices, then any triangulation of Q n can be transformed to another triangulation of Q n with at most O(n + k 2 ) flips. We produce examples of polygons with two triangulations T and T such that to transform T to T requires O(n 2 ) flips. These results are then extended to triangulations of point sets. We also show that any triangulation of an n point set always has n - 4 2 edges that can be flipped. 1. Introduction Let P n = {v 1 , ..., v n } be a collection of points on the plane. A triangulation of P n is a partitioning of the convex hull Conv(P n ) of P n into a set of triangles T = {t 1 , ..., t m } with disjoint interiors in such a way that the vertices of each triangle t of T are points of P n . The elements of P n will be called the vertices of T and the edges of the triangles t 1 , ..., t m of T will be called the edges...

Abstract. We show that a triangulation of a set of n points in the plane that minimizes the maximum edge length can be computed in time O(n 2). The algorithm is reasonably easy to implement and is based on the theorem that there is a triangulation with minmax edge length that contains the relative neighborhood graph of the points as a subgraph. With minor modi cations the algorithm works for arbitrary normed metrics. Key words. Computational geometry,point sets, triangulations, two dimensions, minmax edge length, normed metrics AMS(MOS) subject classi cations. 68U05, 68Q25, 65D05 Appear in: SIAM Journal on Computing, 22 (3), 527{551, (1993)

The application of the Natural Element Method (NEM) (Traversoni, 1994; Braun and Sambridge, 1995) to boundary value problems in two-dimensional small displacement elastostatics is presented. The discrete model of the domain \Omega consists of a set of distinct nodes N , and a polygonal description of the boundary @ In the Natural Element Method, the trial and test functions are constructed using natural neighbor interpolants. These interpolants are based on the Voronoi tessellation of the set of nodes N . The interpolants are smooth (C NEM is identical to linear finite elements. The NEM interpolant is strictly linear between adjacent nodes on the boundary of the convex hull, which facilitates imposition of essential boundary conditions. A methodology to model material discontinuities and non-convex bodies (cracks) using NEM is also described.

A strategy is presented to find a set of points which yields a Conforming Delaunay tetrahedralization of a three-dimensional Piecewise-Linear Complex (PLC). This algorithm is novel because it imposes no angle restrictions on the input PLC. In the process, an algorithm is described that computes a planar conforming Delaunay triangulation of a Planar StraightLine Graph (PSLG) such that each triangle has a bounded circumradius, which may be of independent interest. 1 Introduction In many two- and three-dimensional geometric modeling problems, notably the numerical approximation of the solution to a Partial Differential Equation with a Finite-Element type method [SF73], it is very desirable to obtain a triangulation (tetrahedralization) that respects the domain of interest. The task of forming such decompositions, along with ensuring that the elements of the decompositions satisfy application-specific quality requirements, is sometimes referred to as unstructured mesh generation. Se...