An (n; s) Davenport-Schinzel sequence, for positive integers n and s, is a sequence composed of n distinct symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly non-contiguous) subsequence, any alternation a \Delta \Delta \Delta b \Delta \Delta \Delta a \Delta \Delta \Delta b \Delta \Delta \Delta of length s + 2 between two distinct symbols a and b. The close relationship between Davenport-Schinzel sequences and the combinatorial structure of lower envelopes of collections of functions make the sequences very attractive because a variety of geometric problems can be formulated in terms of lower envelopes. A near-linear bound on the maximum length of Davenport-Schinzel sequences enable us to derive sharp bounds on the combinatorial structure underlying various geometric problems, which in turn yields efficient algorithms for these problems.

Recent progress in acquiring shape from range data permits the acquisition of seamless million-polygon meshes from physical models. In this paper, we present an algorithm and system for converting dense irregular polygon meshes of arbitrary topology into tensor product B-spline surface patches with accompanying displacement maps. This choice of representation yields a coarse but efficient model suitable for animation and a fine but more expensive model suitable for rendering. The first step in our process consists of interactively painting patch boundaries over a rendering of the mesh. In many applications, interactive placement of patch boundaries is considered part of the creative process and is not amenable to automation. The next step is gridded resampling of eachboundedsection of the mesh. Our resampling algorithm lays a grid of springs acrossthe polygonmesh, then iterates between relaxing this grid and subdividing it. This grid provides a parameterization for the mesh section, w...

Introduction A natural and well-studied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of the edges that comprise it. Efficient algorithms are well known for this problem, as briefly summarized below. The shortest path problem takes on a new dimension when considered in a geometric domain. In contrast to graphs, where the encoding of edges is explicit, a geometric instance of a shortest path problem is usually specified by giving geometric objects that implicitly encode the graph and its edge weights. Our goal in devising efficient geometric algorithms is generally to avoid explicit construction of the entire underlying graph, since the full induced graph may be very large (even exponential in the input size, or infinite). Computing an optimal

Abstract We describe simple greedy algorithms to construct the shortest set of loops that generates either the fundamental group (with a given basepoint) or the first homology group (over any fixed coefficient field) of any oriented 2-manifold. In particular, we show that the shortest set of loops that generate the fundamental group of any oriented combinatorial 2-manifold, with any given basepoint, can be constructed in O(n log n) time using a straightforward application of Dijkstra's shortest path algorithm. This solves an open problem of Colin de Verdi`ere and Lazarus.

Three open problems on folding/unfolding are discussed: (1) Can every convex polyhedron be cut along edges and unfolded at to a single nonoverlapping piece? (2) Given gluing instructions for a polygon, construct the unique 3D convex polyhedron to which itfolds. (3) Can every planar polygonal chain be straightened?

Consider a simple polyhedron P, possibly non-convex, composed of n triangular regions (faces), each assigned a positive weight indicating the cost of travel in that region. We present and experimentally study several algorithms to compute an approximate weighted geodesic shortest path, ß 0 (s; t), between two points s and t on the surface of P. Our algorithms are simple, practical, less prone to numerical problems, adaptable to a wide spectrum of weight functions, and use only elementary data structures. An additional feature of our algorithms is that execution time and space utilization can be traded off for accuracy; likewise, a sequence of approximate shortest paths for a given pair of points can be computed with increasing accuracy (and execution time) if desired. Dynamic changes to the polyhedron (removal, insertions of vertices or faces) are easily handled. The key step in these algorithms is the construction of a graph by introducing Steiner points on the edges of the given p...

Given a convex polytope P with n faces in IR 3 , points s; t 2 @P , and a parameter 0 ! " 1, we present an algorithm that constructs a path on @P from s to t whose length is at most (1+ ")d P (s; t), where dP (s; t) is the length of the shortest path between s and t on @P . The algorithm runs in O(n log 1=" + 1=" 3 ) time, and is relatively simple to implement. The running time is O(n+1=" 3 ) if we only want the approximate shortest path distance and not the path itself. We also present an extension of the algorithm that computes approximate shortest path distances from a given source point on @P to all vertices of P . Work by the first and the fourth authors has been supported by National Science Foundation Grant CCR-93--01259, by an Army Research Office MURI grant DAAH04-96-1-0013, by a Sloan fellowship, by an NYI award, and by matching funds from Xerox Corporation. Work by the first three authors has been supported by a grant from the U.S.--Israeli Binational Science ...

A new algorithm is proposed for calculating the approximate shortest path on a polyhedral surface. The method mainly uses Dijkstra’s algorithm and is based on selective refinement of the discrete graph of a polyhedron. Although the algorithm is an approximation, it has the significant advantages of being fast, easy to implement, high approximation accuracy, and numerically robust. The approximation accuracy and computation time are compared between this approximation algorithm and the extended Chen & Han (ECH) algorithm that can calculate the exact shortest path for non-convex polyhedra. The approximation algorithm can calculate shortest paths within 0.4 % accuracy to roughly 100-1000 times faster than the ECH algorithm in our examples. Two applications are discussed of the approximation algorithm to geometric modeling.

Shortest path problems are among the... In this paper we propose several simple and practical algorithms (schemes) to compute an approximated weighted shortest path Π'(s, t) points s and t on the surface of a polyhedron P.

In this paper, we propose a complete framework for 3D geometry modeling and processing that uses only fast geodesic computations. The basic building block for these techniques is a novel greedy algorithm to perform a uniform or adaptive remeshing of a triangulated surface. Our other contributions include a parameterization scheme based on barycentric coordinates, an intrinsic algorithm for computing geodesic centroidal tessellations, and a fast and robust method to flatten a genus-0 surface patch. On large meshes (more than 500,000 vertices), our techniques speed up computation by over one order of magnitude in comparison to classical remeshing and parameterization methods. Our methods are easy to implement and do not need multilevel solvers to handle complex models that may contain poorly shaped triangles.