This paper presents an automatic method of creating surface models at several levels of detail from an original polygonal description of a given object. Representing models at various levels of detail is important for achieving high frame rates in interactive graphics applications and also for speeding-up the off-line rendering of complex scenes. Unfortunately, generating these levels of detail is a time-consuming task usually left to a human modeler. This paper shows how a new set of vertices can be distributed over the surface of a model and connected to one another to create a re-tiling of a surface that is faithful to both the geometry and the topology of the original surface. Themain contributions of this paper are: 1) a robust method of connecting together new vertices over a surface, 2) a way of using an estimate of surface curvature to distribute more new vertices at regions of higher curvature and 3) a method of smoothly interpolating between models that represent the same object at different levels of detail. The key notion in the re-tiling procedure is the creation of an intermediate model called the mutual tessellation of a surface that contains both the vertices from the original model and the new points that are to become vertices in the re-tiled surface. The new model is then created by removing each original vertex and locally re-triangulating the surface in a way that matches the local connectedness of the initial surface. This technique for surface retessellation has been successfully applied to iso-surface models derived from volume data, Connolly surface molecular models and a tessellation of a minimal surface of interest to mathematicians. CRCategoriesandSubjectDescriptors: I.3.3 [ComputerGraph- ics]: Picture/Image Generation -- Display algorithms

This paper describes a biologically motivated method of texture synthesis called reaction-diffusion and demonstrates how these textures can be generated in a manner that directly matches the geometry of a given surface. Reaction-diffusion is a process in which two or more chemicals diffuse at unequal rates over a surface and react with one another to form stable patterns such as spots and stripes. Biologists and mathematicians have explored the patterns made by several reaction-diffusion systems. We extend the range of textures that have previously been generated by using a cascade of multiple reaction-diffusion systems in which one system lays down an initial pattern and then one or more later systems refine the pattern. Examples of patterns generated by such a cascade process include the clusters of spots on leopards known as rosettes and the web-like patterns found on giraffes. In addition, this paper introduces a method by which reaction-diffusion textures are created to match the geometry of an arbitrary polyhedral surface. This is accomplished by creating a mesh over a given surface and then simulating the reactiondiffusion process directly on this mesh. This avoids the often difficult task of assigning texture coordinates to a complex surface. A mesh is generated by evenly distributing points over the model using relaxation and then determining which points are adjacent by constructing their Voronoi regions. Textures are rendered directly from the mesh by using a weighted sum of mesh values to compute surface color at a given position. Such textures can also be used as bump maps.

We survey the computational geometry relevant to finite element mesh generation. We especially focus on optimal triangulations of geometric domains in two- and three-dimensions. An optimal triangulation is a partition of the domain into triangles or tetrahedra, that is best according to some criterion that measures the size, shape, or number of triangles. We discuss algorithms both for the optimization of triangulations on a fixed set of vertices and for the placement of new vertices (Steiner points). We briefly survey the heuristic algorithms used in some practical mesh generators.

This paper presents a new computational method for fully automated triangular mesh generation, consistently applicable to wire-frame, surface, solid, and nonmanifold geometries. The method, called bubble meshing, is based on the observation that a pattern of tightly packed spheres mimics a Voronoi diagram, from which a set of well-shaped Delaunay triangles and tetrahedra can be created by connecting the centers of the spheres. Given a domain geometry and a node-spacing function, spheres are packed on geometric entities, namely, vertices, edges, faces, and volumes, in ascending order of dimension. Once the domain is filled with spheres, mesh nodes are placed at the centers of these spheres and are then connected by constrained Delaunay triangulation and tetrahedrization. To obtain a closely packed configuration of spheres, the authors devised a technique for physically based mesh relaxation with adaptive population control. The process of mesh relaxation significantly reduces the number of ill-shaped triangles and tetrahedra.

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...

In this paper we present an algorithm for detecting and repairing defects in the boundary of a polyhedron. These defects, usually caused by problems in CAD software, consist of small gaps bounded by edges that are incident to only one polyhedron face. The algorithm uses a partial curve matching technique for matching parts of the defects, and an optimal triangulation of 3-D polygons for resolving the unmatched parts. It is also shown that finding a consistent set of partial curve matches with maximum score, a subproblem which is related to our repairing process, is NP-Hard. Experimental results on several polyhedra are presented. Keywords: CAD, polyhedra, gap filling, curve matching, geometric hashing, triangulation. 1 Introduction The problem studied in this paper is the detection and repair of "gaps" in the boundary of a polyhedron. This problem usually appears in polyhedral approximations of CAD objects, whose boundaries are described using curved entities of higher leve...

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This paper proposes a computational method for fully automated quadrilateral meshing. Unlike previous methods, this new scheme can create a quadrilateral mesh whose directionality is precisely controlled. Given as input: (1) a 2D geometric domain, (2) a desired node spacing distribution as a scalar function defined over the domain, and (3) a desired mesh directionality as a vector field defined over the domain, the proposed method first packs square cells closely in the domain. The centers of the squares are then connected by Delaunay triangulation, yielding a triangular mesh topology. The triangular mesh is further converted into a quad-dominant mesh or an all-quad mesh that satisfies the given mesh directionality. Since the closely packed square cells mimic a pattern of Voroni polygons corresponding to a well-shaped graded quadrilateral mesh, the proposed method generates a high quality mesh whose element sizes and mesh directionality conform well to the given input. Keywords: quadri...

An algorithm for the generation of hexahedral element meshes is demonstrated. The algorithm starts by creating a structured grid in the interior of an object. Then the boundary region is meshed by means of an isomorphism technique that uses basically 2D-operations. Examples are given that demonstrate the quality of the algorithm. 1 Introduction In the last years considerable progress has been made in the numerical solution of partial differential equations that describe complex physical phenomena. The state of the art allows the use of the finite element method as a tool for solving many engineering problems. Numerous programs have been developed for that purpose. With many numerical problems being solved, more attention is paid in reducing the effort to perform such simulations. In this context mesh generation plays an important role, since usually models are constructed with CAD systems and stored in formats like IGES, STEP etc, whereas finite element simulations require that the ob...

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.