This paper proposes a new method for isotropic remeshing of triangulated surface meshes. Given a triangulated surface mesh to be resampled and a user-specified density function defined over it, we first distribute the desired number of samples by generalizing error diffusion, commonly used in image halftoning, to work directly on mesh triangles and feature edges. We then use the resulting sampling as an initial configuration for building a weighted centroidal Voronoi tessellation in a conformal parameter space, where the specified density function is used for weighting. We finally create the mesh by lifting the corresponding constrained Delaunay triangulation from parameter space. A precise control over the sampling is obtained through a flexible design of the density function, the latter being possibly low-pass filtered to obtain a smoother gradation. We demonstrate the versatility of our approach through various remeshing examples.

Figure 1: A view of a large urban model consisting of 26.8M triangles. In the left image, the parts visible from a region located at the junction of two streets (in green) are colored. In the right image, only the buildings with some visible parts are displayed. From-region visibility culling is considered harder than from-point visibility culling, since it is inherently four-dimensional. We present a conservative occlusion culling method based on factorizing the 4D visibility problem into horizontal and vertical components. The visibility of the two components is solved asymmetrically: the horizontal component is based on a parameterization of the ray space, and the visibility of the vertical component is solved by incrementally merging umbrae. The technique is designed so that the horizontal and vertical operations can be efficiently realized together by modern graphics hardware. Similar to image-based from-point methods, we use an occlusion map to encode visibility; however, the image-space occlusion map is in the ray space rather than in the primal space. Our results show that the culling time and the size of the computed potentially visible set depend on the size of the viewcell. For moderate viewcells, conservative occlusion culling of large urban scenes takes less than a second, and the size of the potentially visible set is only about two times larger than the size of the exact visible set.

A preliminary version of this paper has appeared as an extended abstract at CGI'98; see [26]. y

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In the manufacturing industry, finding an orientation for a mold that eliminates surface defects and insures a complete fill after the termination of the gravity casting process is an important and difficult problem which has not previously been investigated formally. We initiate the study of the gravity casting process from a geometric perspective and present an optimal `(n log n) time algorithm that solves this problem in two dimensions given an object of size n.

Given a simple polygon in the plane, a deflation is defined as the inverse of a flip in the Erdos-Nagy sense. In 1993 Bernd Wegner conjectured that every simple polygon admits only a finite number of deflations.

This thesis contains new results on the subject of polygonal structure reconfiguration. Specifically, the types of structures considered here are polygons, polygonal chains, triangulations, and polyhedral surfaces. A sequence of vertices (points), successively joined by straight edges, is a polygonal chain. If the sequence is cyclic, then the object is a polygon. A planar triangulation is a set of vertices with a maximal number of non-crossing straight edges joining them. A polyhedral surface is a three-dimensional structure consisting of flat polygonal faces that are joined by common edges. For each of these structures there exist several methods of reconfiguration. Any such method must provide a well-defined way of transforming one instance of a struc-ture to any other. Several types of reconfigurations are reviewed in the introduction, which is followed by new results. We begin with efficient algorithms for comparing monotone chains. Next, we prove that flat chains with unit-length edges and an-gles within a wide range always admit reconfigurations, under the dihedral model of motion. In this model, angles and edge lengths are preserved. For the universal

The Graham scan is a fundamental backtracking technique in computational geometry which was originally designed to compute the convex hull of a set of points in the plane and has since found application in several different contexts. In this note we show how to use the Graham scan to triangulate a simple polygon. The resulting algorithm triangulates an n vertex polygon P in O(kn) time where k-1 is the number of concave vertices in P. Although the worst case running time of the algorithm is O(n 2 ), it is easy to implement and is therefore of practical interest. 1. Introduction A polygon P is a closed path of straight line segments. A polygon is represented by a sequence of vertices P = (p 0 ,p 1 ,...,p n-1 ) where p i has real-valued x,y-coordinates. We assume that no three vertices of P are collinear. The line segments (p i ,p i+1 ), 0 i n-1, (subscript arithmetic taken modulo n) are the edges of P. A polygon is simple if no two nonconsecutive edges intersect. A simple polygon part...

Finite element models used in crash worthiness simulations now contain more than one million mostly quadrilateral elements. Interactive visualization of those models for pre- and postprocessing applications cannot be achieved by brute force rendering of all triangles on even advanced graphics hardware. Well-known mesh simplification algorithms are rarely applied, since the finite element structure needs to be retained, which is usually accomplished by drawing an additional wire frame around each single element. We propose a new visualization technique which combines the advantages of a specifically adapted mesh reduction algorithm with a texture based rendering of surface details. We start out with a two-stage segmentation based on continuity features of the FE model. The resulting patches form a parameterization domain where lost surface details and element boundaries can be stored in a normal map and in a densely encoded wire frame texture. The triangulated patches are rendered with a fragment shader performing perpixel lighting and high quality outline generation for each finite element in a single pass. We achieve a significant speedup in the visualization while retaining the visual details of the FE model and we expect our method to be applicable to all areas of Computer-Aided Engineering. 1

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