Quadrilateral Meshes for PSLGs

Discrete & Computational Geometry, 2010

We show that any simple planar n-gon can be meshed in linear time by O(n) quadrilaterals with all new angles bounded between 60 and 120 degrees.

Discrete & Computational Geometry, 2016

We show that any planar PSLG with n vertices has a conforming triangulation by O(n 2.5) nonobtuse triangles, answering the question of whether a polynomial bound exists. The triangles may be chosen to be all acute or all right. A nonobtuse triangulation is Delaunay, so this result improves a previous O(n 3) bound of Eldesbrunner and Tan for conforming Delaunay triangulations. In the special case that the PSLG is the triangulation of a simple polygon, we will show that only O(n 2) elements are needed, improving an O(n 4) bound of Bern and Eppstein. We also show that for any ǫ > 0, every PSLG has a conforming triangulation with O(n 2 /ǫ 2) elements and with all angles bounded above by 90 • + ǫ. This improves a result of S. Mitchell when ǫ = 3 8 π = 67.5 • and Tan when ǫ = 7 30 π = 42 • .

Proceedings of the twenty-fifth annual symposium on Computational geometry, 2009

In this paper, we present an algorithm that utilizes a quadtree data structure to construct a quadrilateral mesh for a simple polygonal region in which no newly created angle is smaller than 18.43 • (= arctan(1 3)) or greater than 171.86 • (= 135 • + 2 arctan(1 3)). This is the first known result, to the best of our knowledge, on a direct quadrilateral mesh generation algorithm with a provable guarantee on the angles.

2017

abstrakt. Polygonal meshes represent important geometric structures with a large number of applications. The study of polygonal meshes is motivated by many processing tasks in automotive/aerospace industry, egineering, architecture, engineering, construction industry, and industrial design. Much of literature on polygonal representations focuses on quadrilateral meshes which are composed of quadrilaterals as they possess several advantages compared to triangle meshes. In this short contribution we present a comparative study of known methods for constuctions of quadrangulations of various classes and for different purposes. We suggest a new method for computing all unique quadrilateral meshes of a certain class based on sequential construction.

International Journal for Numerical Methods in Engineering, 2012

A new indirect way of producing all-quad meshes is presented. The method takes advantage of a well known algorithm of the graph theory, namely the Blossom algorithm that computes the minimum cost perfect matching in a graph in polynomial time. The new Blossom-Quad algorithm is compared with standard indirect procedures. Meshes produced by the new approach are better both in terms of element shape and in terms of size field efficiency.

International Journal of …, 2005

a We should point out that the term constrained triangular mesh has been used in several other papers with a different meaning.

2006

In 1891, Peterson [Pet91] proved that every 3-regular bridgeless graph has a perfect matching. It is well-known that the dual of a triangular mesh on a compact manifolds is a 3-regular graph. M. Gopi and D. Eppstein [GE04] use Petersons theorem to solve the problem of constructing strips of triangles from triangular meshes on a compact manifold. P. Diaz-Gutierrez and M. Gopi [DG04] elaborate on the creation of strips of quadrilaterals when a perfect matching exists. In this paper, it is shown that the dual of a quadrilateral mesh on a 2-dimensional compact manifold with an even number of quadrilaterals (which is a 4-regular graph) also has a perfect matching. In general, however, not all 4-regular graphs have a perfect matching. Indeed, a counterexample is given that is planar.

International Journal of Computational Geometry & Applications, 2000

We use circle-packing methods to generate quadrilateral meshes for polygonal domains, with guaranteed bounds both on the quality and the number of elements. We show that these methods can generate meshes of several types: (1) the elements form the cells of a Voronoï diagram, (2) all elements have two opposite 90 • angles, (3) all elements are kites, or (4) all angles are at most 120 • . In each case the total number of elements is O(n), where n is the number of input vertices.

In 1891, Peterson [Pet91] proved that every 3-regular bridgeless graph has a perfect matching. It is well-known that the dual of a triangular mesh on a compact manifolds is a 3-regular graph. M. Gopi and D. Eppstein [GE04] use Petersons theorem to solve the problem of constructing strips of triangles from triangular meshes on a compact manifold. P. Diaz-Gutierrez and M. Gopi [DG04] elaborate on the creation of strips of quadrilaterals when a perfect matching exists. In this paper, it is shown that the dual of a quadrilateral mesh on a 2-dimensional compact manifold with an even number of quadrilaterals (which is a 4-regular graph) also has a perfect matching. In general, however, not all 4-regular graphs have a perfect matching. Indeed, a counterexample is given that is planar.