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...

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Petersen's theorem is a classic result in matching theory from 1891, stating that every 3-regular bridgeless graph has a perfect matching. Our work explores efficient algorithms for finding perfect matchings in such graphs. Previously, the only relevant matching algorithms were for general graphs, and the fastest algorithm ran in O(n^3/2) time for 3-regular graphs. We have developed an O(n log^4 n)-time algorithm for perfect matching in a 3-regular bridgeless graph. When the graph is also planar, we have as the main result of our paper an optimal O(n)-time algorithm. We present three applications of this result: terrain guarding, adaptive mesh refinement, and quadrangulation.

We study the approximation properties of the bivariate spline spaces S r 3r ( +) of smoothness r and degree 3r defined on triangulations + which are obtained from arbitrary nondegenerate convex quadrangulations by adding the diagonals of each quadrilateral.

Given a set S of n points in the plane, a quadrangulation of S is a planar subdivision whose vertices are the points of S, whose outer face is the convex hull of S, and every face of the subdivision (except possibly the outer face) is a quadrilateral. We show that S admits a quadrangulation if and only if S does not have an odd number of extreme points. If S admits a quadrangulation, we present an algorithm that computes a quadrangulation of S in O(n log n) time even in the presence of collinear points. If S does not admit a quadrangulation, then our algorithm can quadrangulate S with the addition of one extra point, which is optimal. We also provide an\Omega (n log n) time lower bound for the problem. Finally, our results imply that a k-angulation of a set of points can be achieved with the addition of at most k \Gamma 3 extra points within the same time bound.

Given a set S such as a polygon or a set of points, a quadrangulation of S is a partition of the interior of S, if S is a polygon, or the interior of the convex hull of S, if S is a set of points, into quadrangles (quadrilaterals) obtained by inserting edges between pairs of points (diagonals between vertices of the polygon) such that the edges intersect each other only at their end points. Not all polygons or sets of points admit quadrangulations, even when the quadrangles are not required to be convex (convex quadrangulations) . In this paper we briefly survey some recent results concerning the characterization of those planar sets that always admit quadrangulations (convex and non-convex) as well as some related computational problems. 1. Introduction In the field of computational geometry a triangulation of a finite planar set such as a set of points, line segments or polygon, is a well studied structure [O'R94], [PS85]. For one thing, a triangulation always exists and for anothe...

Multivariate spline functions are smooth piecewise polynomial functions over triangulations consisting of n-simplices in the Euclidean space R^n. We present a straightforward method for using these spline functions to numerically solve elliptic partial differential equations such as Poisson and biharmonic equations and fit given scattered data. This method does not require constructing macro-elements or locally supported basis functions nor computing the dimension of the finite element spaces or spline spaces. We have implemented the method in MATLAB using multivariate splines in R² and R³. Several numerical examples are presented to demonstrate the effectiveness and efficiency of the method.

Given a set S of n points in the plane, a quadrangulation of S is a planar subdivision whose vertices are the points of S, whose outer face is the convex hull of S, and every face of the subdivision (except possibly the outer face) is a quadrilateral. We show that S admits a quadrangulation if and only if S does not have an odd number of extreme points. If S admits a quadrangulation, we present an algorithm that computes a quadrangulation of S in O(n log n) time even in the presence of collinear points. If S does not admit a quadrangulation, then our algorithm can quadrangulate S with the addition of one extra point, which is optimal. We also provide an n log n) time lower bound for the problem. Our results imply that a k-angulation of a set of points can be achieved with the addition of at most k 3 extra points within the same time bound. Finally, we present an experimental comparison between three quadrangulation algorithms which shows that the Spiraling Rotating Calipers (SRC) algorithm (presented in Section 4) produces quadrangulations with the greatest number of convex quadrilaterals as well as those with the smallest difference between the average minimum and maximum angle over all quadrangles.

We study the multi-level method to precondition a linear system arising from discretizing an elliptic partial differential equation of order 2r by using Galerkin's method with spline spaces S 0 1 (4 n ) for r = 1 and S r\Gamma1 3r\Gamma1 (4 n ) or S r\Gamma1 3r\Gamma3 (3 + n ) for r 2, where S ae d denotes a spline space of smoothness ae and degree d, 4 n is the nth refinement of a given triangulation 4 0 by using standard uniform regular refinement or nonuniform refinement procedure, and 3 + n is the triangulation obtained from the nth refinement of a given quadrangulation by using uniform or nonuniform refinement procedure. We show we can alway construct a multi-level basis in spline space S 0 1 (4) or S r\Gamma1 3r\Gamma1 (4 n ) or S r\Gamma1 3r\Gamma3 (3 + n ) to precondition the linear system so that its condition number is O((n + 1) 2 ). A detail description of such a construction is given. AMS(MOS) Subject Classifications: 41A15, 41A63, 41A25, 65D10 Keywords and phrases: Bivariate Splines, B-net, Elliptic Equations, Finite Element Method, Full Approximation Order, Multi-level Basis, Preconditioning.

We consider the problem of obtaining "nice" quadrangulations of planar sets of points. For many applications "nice" means that the quadrilaterals obtained are convex if possible and as "fat" or as squarish as possible. For a given set of points a quadrangulation, if it exists, may not admit all its quadrilaterals to be convex. In such cases we desire that the quadrangulations have as many convex quadrangles as possible. Solving this problem optimally is not practical. Therefore we propose and experimentally investigate a heuristic approach to solve this problem by converting "nice" triangulations to the desired quadrangulations with the aid of maximum matchings computed on the dual graph of the triangulations. We report experiments on several versions of this approach and provide theoretical justification for the good results obtained with one of these methods. 1 Introduction Finite element mesh generation is a problem that has received considerable attention in recent years...