We consider the problem of transmitting huge triangle meshes in the context of a Web-like client-server architecture. Approximations of the original mesh are transmitted by applying selective refinement. A multiresolution geometric model is maintained by the server. A client may query the server for a mesh at an arbitrary, continuously variable, level of detail. The client makes repeated queries over time with different query parameters. The server answers to queries by traversing the multiresolution model and transmitting updates to the client, which uses them to progressively modify a current mesh. We study this problem in the context of a vertex-based multiresolution model, which is a special instance of the Multi-Triangulation [Pup96, DFPM97], based on vertex insertion and removal. We define a compact data structure for such a model which exploits the specific update rule. We propose a dynamic algorithm for selective refinement and we discuss in detail its implementation as a...

Given a terrain defined as a piecewise-linear function with n triangles, and m point sites on it, we would like to identify the location on the terrain that minimizes the maximum distance to the sites. The distance is measured as the length of the Euclidean shortest path along the terrain. To simplify the problem somewhat, we extend the terrain to (the surface of) a polyhedron. To compute the optimum placement, we compute the furthest-site Voronoi diagram of the sites on the polyhedron. The diagram has maximum combinatorial complexity Θ(mn²), and the algorithm runs in O(mn² log² mlogn) time.

For many geometric problems, there are ecient algorithms that surprisingly use very little extra space other than the given array holding the input. For many geometric query problems, there are ecient data structures that need no extra space at all other than an array holding a permutation of the input. In this paper, we obtain the rst such space-economical solutions for a number of fundamental problems, including three-dimensional convex hulls, two-dimensional Delaunay triangulations, xed-dimensional range queries, and xed-dimensional nearest neighbor queries.

Transmission of geographic data over the Internet, rendering at different resolutions /levels of detail, or processing at unnecessarily fine detail pose interesting challenges and opportunities. In this paper we explore the applicability to GIS of the notion of progressive meshes, introduced by Hoppe [13] to the field of computer graphics. In particular, we describe progressive TINs as an alternative to hierarchical TINs, design algorithms for solving GIS tasks such as selective refinement, point location, visibility or line of sight queries, isoline/contour line extraction and provide empirical results which show that our algorithms are of considerable practical relevance. Moreover, the selective refinement data structure and refinement algorithm solves a question posed by Hoppe.

We address the problem of designing compact data structures for encoding a Triangulated Irregular Network (TIN). In particular, we study the problem of compressing connectivity, i.e., the information describing the topological structure of the TIN, and we propose two new compression methods which have different purposes. The goal of the first method is to minimize the number of bits needed to encode connectivity information: it encodes each vertex once, and at most two bits of connectivity information for each edge of a TIN; algorithms for coding and decoding the corresponding bitstream are simple and efficient. A practical evaluation shows compression rates of about 4.2 bits per vertex, which are comparable with those achieved by more complex methods. The second method compresses a TIN at progressive levels of detail and it is based on a strategy which iteratively removes a vertex from a TIN according to an error-based criterion. Encoding and decoding algorithms are presented and c...

Given m points (sites) and n obstacles (barriers) in the plane, we address the problem of finding a straight-line minimum cost spanning tree on the sites, where the cost is proportional to the number of intersections (crossings) between tree edges and barriers. If the barriers are infinite lines then there is a spanning tree where every barrier is crossed by O ( √ m) tree edges (connectors), and this bound is asymptotically optimal (spanning tree with low stabbing number). Asano et al. showed that if the barriers are pairwise disjoint line segments, then there is a spanning tree such that every barrier crosses at most 4 tree edges and so the total cost is at most 4n. Constructionswith3crossings per barrier and 2n total cost provide a lower bound. We obtain tight bounds on the minimum cost spanning tree in the most exciting special case where the barriers are interior disjoint line segments that form a convex subdivision and there is a point in every cell. In particular, we show that there is a spanning tree such that every barrier is crossed by at most 2 tree edges, and there is a spanning tree of total cost 5n/3. Both bounds are tight.

In this paper we investigate decimation algorithms for simplifying triangulated terrain models in order to support progressive transmission of GIS terrain models over the web. We report on experiments with heuristics that select an independent set of vertices to be deleted, while trying to preserve terrain characteristics.

We address the problem of designing compact data structures for encoding a Triangulated Irregular Network (TIN) as a sequential bitstream. In particular, we study the problem of compressing connectivity, i.e., the information describing the topological structure of the TIN and we propose two new compression methods which have different purposes. The goal of the first method is to minimize the number of bits needed to encode connectivity information: it encodes each vertex once, and requires two bits of connectivity information for each edge of a TIN. We present efficient algorithms for coding and decoding the corresponding bitstream and show some practical evaluation of the method. The second method compresses a TIN at progressive levels of detail and is based on a strategy which iteratively removes a vertex from a TIN according to an error-based criterion. Encoding and decoding algorithms are presented and compared with other approaches to progressive compression. 2 Introduction Hug...

We consider the problem of finding low-cost spanning trees for sets of n points in the plane, where the cost of a spanning tree is defined as the total number of intersections of tree edges with a given set of m barriers. We obtain the following results: (i) if the barriers are possibly intersecting line segments, then there is always a spanning tree of cost O(min(m 2, mx/)); (ii) if the barriers are disjoint line segments, then there is always a spanning tree of cost O(m); Off) if the barriers are disjoint convex objects, then there is always a spanning tree of cost O(n + m).

Given a triangulation T of n points in the plane, we are interested in the minimal set of edges in T such that T can be reconstructed from this set (and the vertices of T ) using constrained Delaunaytriangulation.