Modeling two-dimensional and three-dimensional objects is an important theme in computer graphics. Two main types of models are used in both cases: boundary representations, which represent the surface of an object explicitly but represent its interior only implicitly, and constructive solid geometry representations, which model a complex object, surface and interior together, as a boolean combination of simpler objects. Because neither representation is good for all applications, conversion between the two is often necessary. We consider the problem of converting boundary representations of polyhedral objects into constructive solid geometry (CSG) representations. The CSG representations for a polyhedron P are based on the half-spaces supporting the faces of P . For certain kinds of polyhedra this problem is equivalent to the corresponding problem for simple polygons in the plane. We give a new proof that the interior of each simple polygon can be represented by a monotone...

We investigate 3-d visibility problems in which the viewing position moves along a straight flightpath. Specifically we focus on two problems: determining the points along the flightpath at which the topology of the viewed scene changes, and answering ray-shooting queries for rays with origin on the flightpath. Three progressively more specialized problems are considered: general scenes, terrains, and terrains with vertical flightpaths. 1.

Partitioning trees, a multi-dimensional generalization of binary search trees, is alone among the principal methods for representing geometry in combining the representation of a set with the geometric search structure required for efficient spatial operations such as set operations and visibility. Since a partitioning tree may be interpreted as specifying a program for exploring the structure induced on a space (e.g. by objects), there are many trees which represent the same spatial structure but provide different searches of the space. The issue of generating a good program to determine spatial relations between sets is then transformed into the issue of constructing good partitioning trees. The metric we choose for characterizing goodness is the expected cost of various elementary operations calculated using simple probability models. However, choosing the optimal from at least n! different trees by enumeration is not viable. Consequently, we employ heuristics that make local decisions based on the expected cost models. In addition to this quantitative methodology, we develop a qualitative understanding of what constitutes a good representation. This leads us to the notion of a good tree as one that provides a sequence or set of approximations, each obtained by various prunings of a single tree.

In this paper we present the Brep-index, a multidimensional space partitioning data structure that provides quick spatial access to the vertices, edges and faces of a boundary representation (Brep), thus yielding a single unified representation for polyhedral solids. We give an algorithm for the construction of the Brep-index and prove its correctness. We show that its size is \Omega\Gamma v + e + f), where v, e, and f are the number of vertices, edges, and faces of the Brep. The lower bound can be achieved for some Breps by compressing the structure using simple rewrite rules. We then demonstrate robust point and line/Brep classification methods given an implementation that uses finite-precision arithmetic. Keywords: Classification, Brep, BSP Trees, Data Structures 1. Introduction Most data structures that exist for representing polyhedral solids can be categorized either as boundary-based or as volume-based. Each category has certain benefits not found in the other and therefore, ...

this paper, except Figures 4.8 and 4.9, are two dimensional, representing polyhedra as polygons. 1 2 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 4 * vp Figure 4.1: Visibility ordering of the cells of a mesh relative to viewpoint vp. can be computed and stored in a preprocessing step. The MPVO algorithm can be extended to order many nonconvex meshes; this is described in detail in Section 4.4.2. 4.3 Preliminary Definitions A convex polyhedron in E

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This paper presents an algorithm for shadow calculation in dynamic polyhedral scenes illuminated by point light sources. It is based on a modification of Shadow Volume Binary Space Partition trees, to allow these be constructed from the original scene polygons in arbitrary order and to support for fast reconstruction after a change in scene geometry. Timings using sample scenes are presented that indicate substantial savings both in terms of computation time and shadows produced. KEY WORDS: shadows, BSP Trees, SVBSP Trees, dynamic modification. 1

We present a system for incorporating multiple level of detail (LOD) models of 3D objects within a single hierarchical data structure. This system was designed for a scientific visualization application involving terrain and volume rendering. Our data structure is a modified Binary Space Partitioning (BSP) tree. We describe how our tree construction and traversal routines may be used with a variety of LOD methods. This is demonstrated with two different LOD methods: a new method specialized for terrain elevation height fields, and an existing method for general objects. Images, timings, and storage data for our implementation are provided. Keywords: BSP trees, Virtual reality, Real-time graphics, Multiple levels-of-detail. 1 Introduction This research was motivated by a scientific visualization project in which we were asked to produce an interactive display combining renderings of terrain and volume data. The volume data, derived from simulations of radio frequency (RF) propagation...

Abstract: We present a novel sorting algorithm, Vis-Sort, to sort 1D and 3D geometric elements. Given a set of acyclic and non-intersecting 3D geometric primitives, Vis-Sort computes the visibility ordering from a viewpoint. The running time of our algorithm is dependent upon the degree of sortedness in the 3D sequence and is bounded by O(�Y �n), where n is the number of primitives and �Y � is the Knuth’s measure of disorder. The Knuth’s measure of disorder computes the minimum number of elements that need to be removed from the sequence for the remaining sequence to be sorted [35]. Vis-Sort exploits the spatial and temporal coherence between successive instances in a dynamic environment and performs incremental computations. Our algorithm requires no preprocessing and is applicable to all kind of models, including polygon soups and deformable models. We have used our algorithm for order-independent transparency computations in high-depth complexity environments and performing N-body collision culling in dynamic environments. We have implemented our algorithm and tested the system on a PC with a 3.4 GHz Pentium IV CPU with an NVIDIA GeForce FX 6800 Ultra GPU and applied it to complex environments with tens or hundreds of thousands of polygons. Our algorithm can compute a visibility ordering among the objects and triangles at interactive frame rates.

Let S be a set of polygons in the plane. A translation ordering for S (in direction d) is an ordering of the polygons such that, if the polygons are moved one by one to infinity in direction d according to this ordering, no collisions occur between the polygons. We show that, after O(nlogn) preprocessing using O(n) space, it is possible to determine, for any given d, in O(log n) time whether such an ordering exists and, if so, to compute an ordering in O(n) time.