Showing papers presented at "Symposium on Computational Geometry in 1998"

Surface reconstruction by Voronoi filtering

Abstract: We give a simple combinatorial algorithm that computes a piecewise-linear approximation of a smooth surface from a finite set of sample points. The algorithm uses Voronoi vertices to remove triangles from the Delaunay triangulation. We prove the algorithm correct by showing that for densely sampled surfaces, where density depends on a local feature size function, the output is topologically valid and convergent (both pointwise and in surface normals) to the original surface. We briefly describe an implementation of the algorithm and show example outputs.

Tetrahedral mesh generation by Delaunay refinement

Abstract: Given a complex of vertices, constraining segments, and planar straight-line constraining facets in E3, with no input angle less than 90’. an algorithm presented herein can generate a conforming mesh of Delaunay tetrahedra whose circumradius-to-shortest edge ratios are no greater than two. The sizes of the tetrahedra can provably gr

Raising roofs, crashing cycles, and playing pool: applications of a data structure for finding pairwise interactions

Abstract: The straight skeleton of a polygon is a variant of the medial axis introduced by Aichholzer et al., defined by a shrinking process in which each edge of the polygon moves inward at a fixed rate. We construct the straight skeleton of an n -gon with r reflex vertices in time O(n 1+e + n 8/11+e r 9/11+e ) , for any fixed e >0 , improving the previous best upper bound of O(nr log n) . Our algorithm simulates the sequence of collisions between edges and vertices during the shrinking process, using a technique of Eppstein for maintaining extrema of binary functions to reduce the problem of finding successive interactions to two dynamic range query problems: (1) maintain a changing set of triangles in R3 and answer queries asking which triangle is first hit by a query ray, and (2) maintain a changing set of rays in R3 and answer queries asking for the lowest intersection of any ray with a query triangle. We also exploit a novel characterization of the straight skeleton as a lower envelope of triangles in R3 . The same time bounds apply to constructing non-self-intersecting offset curves with mitered or beveled corners, and similar methods extend to other problems of simulating collisions and other pairwise interactions among sets of moving objects.

Construction of contour trees in 3D in O(n log n) steps

Abstract: We outline an O(n log n) algorithm for computing the contour trees for simplicial meshes with n elements in 3D. As a byproduct we describe an O(nlog n) algorithm for “resolution” of singularities of piecewise-linear functions in 3D (i.e., transforming singularities into simple Morse-like ones by subdividing the mesh).

A condition guaranteeing the existence of higher-dimensional constrained Delaunay triangulations

Abstract: Let X be a complex of vertices and piecewise linear constraining facets embedded in Ed. Say that a simplex is strongly Delaunay if its vertices are in X and there exists a sphere that passes through its vertices but passes through and encloses no other vertex. Then X has a d-dimensional constrained Delaunay triangulation if each k-dimensional constraining facet in X with k d 2 is a union of strongly Delaunay k-simplices. This theorem is especially useful in E3 for forming tetrahedralizations that respect specified planar facets. If the bounding segments of these facets are subdivided so that the subsegments are strongly Delaunay, then a constrained tetrahedralization exists. Hence, fewer vertices are needed than in the most common practice in the literature, wherein additional vertices are inserted in the relative interiors of facets to form a conforming (but unconstrained) Delaunay tetrahedralization.

Resource-constrained geometric network optimization

Abstract: WC study a variety of geometric network optimization prob lcms on a set of points, in which we are given a resource bound, a, on the total length of the network, and our ob jcctivc is to maximize the number of points visited (or the total “value” of points visited), In particular, we resolve the well-publicized open problem on the approximabiity of the rooted “orienteering problem” for the case in which the sites are given as points in the plane and the network required is a cycle. We obtain a 2approximation for this problem, We also obtain approximation algorithms for variants of this problem in which the network required is a tree (S-approximation) or a path Q-approximation). No prior approximation bounds were known for any of these problems, We also obtain improved approximation algorithms for geometric instances of the unrooted orienteering problem, where we obtain a 2-approximation for both the cycle and tree versions of the problem on points in the plane, as well as a G-approximation for the tree version in edge-weighted graphs, E’urther, we study generalizations of the basic orienteering problem, to the case of multiple roots, sites that are polygonnl regions, etc., where we again give the first known approximation results. Our methods are based on some new tools which may be of interest in their own right: (1) some new results on m-

Interval arithmetic yields efficient dynamic filters for computational geometry

Abstract: We discuss interval techniques for speeding up the exact evaluation of geometric predicates and describe an efficient implementation of interval arithmetic that is strongly influenced by the rounding modes of the widely used IEEE 754 standard. Using this approach we engineer an efficient floating point filter for the computation of the sign of a determinant that works for arbitrary dimensions. Furthermore we show how to use our interval techniques for exact linear optimization problems of low dimension as they arise in geometric computing. We validate our approach experimentally, comparing it with other static, dynamic and semi-static filters.

Improved incremental randomized Delaunay triangulation

Abstract: We propose a new data structure to compute the Delaunay triangulation of a set of points in the plane. It combines good worst case complexity, fast behavior on real data, and small memory occupation. The location structure is organized into several levels. The lowest level just consists of the triangulation, then each level contains the triangulation of a small sample of the levels below. Point location is done by marching in a triangulation to determine the nearest neighbor of the query at that level, then the march restarts from that neighbor at the level below. Using a small sample (3 %) allows a small memory occupation; the march and the use of the nearest neighbor to change levels quickly locate the query.

Geometric applications of a randomized optimization technique

Abstract: We propose a simple, general, randomized technique to reduce certain geo- metric optimization problems to their corresponding decision problems. These reductions increase the expected time complexity by only a constant factor and eliminate extra log- arithmic factors in previous, often more complicated, deterministic approaches (such as parametric searching). Faster algorithms are thus obtained for a variety of problems in computational geometry: finding minimal k-point subsets, matching point sets under trans- lation, computing rectilinear p-centers and discrete 1-centers, and solving linear programs with k violations.

Asymmetric rendezvous on the plane

Abstract: Wo consider rendezvous problems in which two players move on Ihe plane and wish to cooperate in order to minimise their firat meeting time. We begin by considering the case when they know that they are a distance d apart, but they do not know the direction in which they should travel. We alao conoider a situation in which player 1 knows the initial position of player 2, while player 2 is only given information on the initial distance of player 1. Finally we give some reaulls for the case where one of the players is placed at an initial position chosen equiprobably from a finite set of pointn,

Designing a data structure for polyhedral surfaces

Abstract: Design solutions for a program library are presented for combinatorial data structures in computational geometry, such as planar maps and polyhedral surfaces. Design issues considered are genericity, flcsibility, time and space efficiency, and ease-of-use. We focus on topological aspects of polyhedral surfaces. Edge-based reprew%ations for polyhedrons are evaluated with respect to the design goals. A design for polyhedral surfaces in a halfedge data structure is developed following the generic programming paradigm known from the Standard Template Library STL for C++. Connections arc shown to planar maps and face-based structures managing holes in facets.

A unified approach to labeling graphical features

Abstract: The automatic placement of text or symbol labels corresponding to graphical objects is critical in several application areas such as Cartography, Geographical Information Systems, and Graph Drawing. In this paper we present a general framework for solving the problem of assigning text or symbol labels to a set of graphical features in two dimensional drawings or maps. Our approach does not favor the labeling of one type of graphical feature (such as a node, edge, or area) over another. Additionally, the labcls arc allowed to have arbitrary size and orientation. We have applied our framework to drawings of graphs. We have implemented our techniques and have performed extensive experimentation on hierarchical and orthogonal drawings of graphs. The resulting label assignments are very practical and indicate the effectiveness of our approach.

Constructing approximate shortest path maps in three dimensions

Abstract: We present a new technique for constructing a data structure that approximates shortest path maps inR d . By applying this technique, we get the following two results on approximate shortest path maps in R 3 . (i) Given a polyhedral surface or a convex polytopeP with n edges in R 3 , a source point s on P, and a real parameter 0 <" 1, we present an algorithm that computes a subdivision ofP of size O((n=") log(1=")) which can be used to answer eciently approximate shortest path queries. Namely, given any point t onP, one can compute, in O(log (n=")) time, a distanceP;s(t), such that dP;s(t) P;s(t) (1 + ")dP;s(t), where dP;s(t) is the length of a shortest path between s and t onP. The map can be computed in O(n2 logn +( n=") log (1=") log (n=")) time, for the case of a poly- hedral surface, and in O((n=" 3 ) log(1=" )+( n=" 1:5 ) log (1=") logn) time ifP is a convex polytope. (ii) Given a set of polyhedral obstacles O with a total of n edges in R 3 , a source point s in R 3 n int(O2OO, and a real parameter 0 <" 1, we present an algorithm that computes a subdivision of R 3 , which can be used to answer eciently approximate shortest path queries. That is, for any point t2 R 3 , one can compute, in O(log (n=")) time, a distanceO;s(t) that "-approximates the length of a shortest path from s to t that avoids the interiors of the obstacles. This subdivision can be computed in roughly O(n 4 =" 6 ) time.

Results on k-sets and j-facets via continuous motion

Abstract: Let P be a set ofn points inIRd in general position, i.e., no i + 1 points on a common(i 1)-flat, 1 i d. A k-set of P is a setS of k points inP that can be separated from P nS by a hyperplane. Aj-facet ofP is an oriented(d 1)simplex spanned by d points inP which has exactlyj points fromP on the positive side of its affine hull. If P is a planar point set and n is even, ahalving edge is an undirected edge between two points, such that the connecting line has the same number of points on either side. The number of(n=2)-sets is twice the number of halving edges. Inspired by Dey’s recent proof of a new bound on the number ofk-sets we show that C +X p2P (deg p+ 1)=2 2 = n=2 2 wheredeg p is the number of halving edges incident to point p andC is the number of crossing pairs of halving edges. The identity allows us, among other things, to determine the maximum number of halving edges in a set of 12 points. An analogous identity holds for j-facets. ForP in IR3 we show that forj n=4 2 the number of ( j)-facets (i.e.,i-facets with0 i j) is maximized for sets in convex position, where this number is known to be (j + 1)(j + 2)n 2(j + 1)(j + 2)(j + 3)=3 : For k n=4 1, k2n k(k 1)(2k + 5)=3 is the tight upper bound for the number of ( k)-sets (i.e.,i-sets with 1 i k). Part of this work was performed while R.S. and E.W. were visit ing he DIMACS center in November 1989, while R.S. visited FU Berlin in 1992, while E.W. visited Tel Aviv University in February 1994, whi le A.A. and E.W. were still at FU Berlin, and while B.A. was visiting ETH i n April 1997. B.A. has been partially supported by a Sloan Research F ellowship. E.W. has been partially supported by a Max-Planck Research P rize. Finally we discuss the relation between the vector of numbers ofk-sets,k = 1; : : : ; n 1 and the vector of numbers ofj-facets,j = 0; : : : ; n d for a given point set. In the plane the number of k-sets equals the number of (k 1)facets. InIR3 thek-set vector determines the j-facet vector (and vice versa) by a linear relation. There is no such relation in IRd for d exceeding 3. These results can be obtained by arguments via continuous motion of one point set to another while observing certain quantities related to k-sets andj-facets. For the relation betweenk-sets andj-facets inIR3, we give a more direct argument via so-called k-set polytopes.

A general framework for assembly planning: the motion space approach

Abstract: Assembly planning is the problem of finding a sequence of motions to assemble a product from its parts. We present a general framework for finding assembly motions based on the concept of motion space . Assembly motions are parameterized such that each point in motion space represents a mating motion that is independent of the moving part set. For each motion we derive blocking relations that explicitly state which parts collide with other parts; each subassembly (rigid subset of parts) that does not collide with the rest of the assembly can easily be derived from the blocking relations. Motion space is partitioned into an arrangement of cells such that the blocking relations are fixed within each cell. We apply the approach to assembly motions of several useful types, including one-step translations, multistep translations, and infinitesimal rigid motions. Several efficiency improvements are described, as well as methods to include additional assembly constraints into the framework. The resulting algorithms have been implemented and tested extensively on complex assemblies. We conclude by describing some remaining open problems.

Randomized external-memory algorithms for some geometric problems

Abstract: We show that the well-known random incremental constructlon of Clarkson and Shor [14] can be adapted via gradations to provide efficient external-memory algorithms for some geomctric problems. In particular, as the main result, we obtain an optimal randomized algorithm for the problem of computing the trapezoidal decomposition determined by a set of N line scgmcnts in the plane with K pairwise intersections, that requires G($$ logMjB Q + 5) expected disk accesses (I/OS), where M is the size of the available internal memory and B is the size of the block transfer. The approach is sufficiently general to obtain algorithms for the problems of 2-d and 3-d convex hulls, 2-d abstract Voronoi diagrams and batched point location in a planar subdivision, which require an optimal expected number of I/OS and are olmplcr than the ones previously known. The results extend to a external-memory model with multiple disks.

Exact geometric predicates using cascaded computation

Abstract: In this paper we talk about a new efficient numerical approach to deal with inaccuracy when implementing eometric algorithms. Using various floating-point filters together with arbitrary precision packages, we develop an easy-to-use xpression compiler called EXPCOMP. EXPCOMP supports all common operations +Ibines , ‘, /, ,/, Applying a new semi-static filter, EXPCOMP comthe speed of static filters with the power of dynamic filters. The filter stages deal with all kinds of floating-point exceptions, including underflow The resulting programs how a very good runtime behaviour.

Improved algorithms for robust point pattern matching and applications to image registration

Abstract: Given two images of roughly the same scene, image registration is the process of determining the transformation that most nearly maps one image to another. This problem is of particular interest in remote sensing applications, where it is known that two images correspond to roughly the same gecgraphic region, but the exact alignment between the images io not known. There are many approaches to image registration. We will consider an approach based on extracting a Ret of point features from each of the two images, and thus reducing the problem to a point pattern matching problem. Because of measurement errors and the presence of outlying data points in either of the images, it is important that the diotance measure between two point sets be robust to theeo cffecto. We will measure distances using the partial Hauodorff distance, An important element of image registration applications is that the search begins with a priori information on the bounds of transformation, and a good algorithm should be able to take advantage of this information. Point matching can be a computationally intensive task, and there have been a number of algorithms and approaches proposed for solving this problem, both from theoretical and applied standpoints. One common approach is based on a *Dopartmont of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, MaryInnd. Email: mountQce.umd.edu. The support of the Nationnl Science Foundation under grant CCR-9712379 is gratefully acknowledged. tCentcr for Automation Research, University of Maryland, Collego Park, and Center of Excellence in Space Data and Information Sciences (CESDIS), Code 930.5, Space Data and Computing Division, NASA Goddard Space Flight Center, Greenbolt, Mnr.vlnnd. EmaiL nathanocf ar .umd. edu. The SUP port of tha Applied Information Sciences Branch (AISB), Code 935, NASAIGSFC, under contract NAS 5555-37 is greatfully acknowledged. SUniversities Space Research Association/CESDIS, Code 030,5, Space Data and Computing Division, NASA/GSFC, Grconbolt, Maryland. Email: lemoigneQcesdis.gsfc.nasa.gov. The nupport of the AISB, Code 935, NASA/GSFC is greatfully acknowledged, jkr&sjon tomnko dj&l orhardcopies ofnjl orpartoftiworkfm pcmond or clormom we in grated without fee qrovided that copies rue not mndc or &ributcd for profit or comn~ercld advantage and that copjm beor Ihin notiw and the full citation on the f~page. Tf copy odwwi~rice, to rcpubliQ to post on servers or to redlstnbue to W require3 prior rpeoitio permission Mdlor a fffi. SCCJ 98 Minneapolis M~IUV.XO~~ USA Copyri&t /KM 1998 O-89791-9734/98/ 6...$5.00 geometric branch-and-bound search of transformation space and another is based on using point alignments to derive the matching transformation. The former has the advantage that it can provide guarantees on the accuracy of the final match, and that it naturally uses any a priori information to bound the search. The latter has the advantage of simplicity and speed. We introduce a new practical approximation algorithm, which we call bounded alignment, for robust point pattern matching. This algorithm is a novel combination of these two approaches. Our algorithm operates within the framework of the branch-and-bound, but employs point-to-point alignments to accelerate the search. We show that this combination retains many of the strengths of branch-and-bound search, but provides significantly faster search times by esplaiting alignments. We have implemented the algorithm and have demonstrated its performance on both synthetically generated data points and actual satellite images.

Drawing planar partitions I: LL-drawings and LH-drawings

Abstract: In this paper, we study how to draw a planar partition, i.e., a planar graph with a given partition of the vertices. The goal is to obtain a drawing without crossings such that the partition is clearly visible. Previously, only the special case of bipartite graphs has been studied. Par two models of displaying the partition, we show necessary and sufficient conditions for the existence of a drawing, and how to test them in linear time. We also present linear-time algorithms to create such drawings, if possibIe.

Implementations of the LMT heuristic for minimum weight triangulation

Abstract: No polynomial-time algorithm is kno-ivn to compute t,he minimum weight triangulation (MWT) of a finite planar point set. In this paper xve present efficient implementat,ions of the LMT-skeleton heurist’ic, xvhich identifies edges that must be, and cannot be, in an MWT. For uniformly distributed points, v:e can compute the esact MWT of tens of thousands of points in minutes. These results are obtained by improving the asymptot,ic time and memory usage of the LMTskeleton heurist.ic and of filters that identify initial candidate edges, and also by bucketing and further t,uning for evenly distributed points. Further details and an implementation as a macro for the IPE dran;ng prog7cam are available on the web: http://www.cs.ubc.ca/spider/snoeyink/mwt.

Motion planning for multiple robots

Abstract: We study the motion-planning problem for pairs and triples of robots operating in a shared workspace containing n obstacles. A standard way to solve such problems is to view the collection of robots as one composite robot, whose number of degrees of freedom is d, the sum of the numbers of degrees of freedom of the individual robots. We show that it is su cient to consider a constant number of robot systems whose number of degrees of freedom is at most d 1 for pairs of robots, and d 2 for triples. (The result for a pair assumes that the sum of the number of degrees of freedom of the robots constituting the pair reduces by at least one if the robots are required to stay in contact; for triples a similar assumption is made. Moreover, for triples we need to assume that a solution with positive clearance exists.) We use this to obtain an O(n) time algorithm to solve the motion-planning problem for a pair of robots; this is one order of magnitude faster than what the standard method would give. For a triple of robots the running time becomes O(n ), which is two orders of magnitude faster than the standard method. We also apply our method to the case of a collection of bounded-reach robots moving in a low-density environment. Here the running time of our algorithm becomes O(n log n) both for pairs and triples.

Efficiently approximating polygonal paths in three and higher dimensions

Abstract: We present efficient algorithms for solving polygonal-path approximation problems in three and higher dimensions. Given an n-vertex polygonal curve P inRd , d‚ 3, we approximate P by another poly- gonal curve P0 of mn vertices in Rd such that the vertex sequence of P0 is an ordered subsequence of the vertices of P. The goal is either to minimize the size m of P0 for a given error tolerance " (called the min-# problem), or to minimize the deviation error " between P and P 0 for a given size m of P 0 (called the min-" problem). Our techniques enable us to develop efficient near-quadratic-time algorithms in three dimensions and subcubic-time algorithms in four dimensions for solving the min-# and min-" problems. We discuss extensions of our solutions to d-dimensional space, where d > 4, and for the L1 and L1 metrics.

Point set labeling with sliding labels

Abstract: This paper discusses algorithms for labeling sets of points in the plane, where labels are not restricted to some finite number of positions. We show that continuously sliding labels allows more points to be labeled both in theory and in practice, WC define six different models of labeling, and analyze how much better-more points get a label-ne model can be than another. Maximizing the number of labeled points is N&hard, but we show that all models have a polynomiallimo approximation scheme, and all models have a simple and eflicient factor-4 approximation algorithm. Finally, we give experimental results based on the factor-i approximation algorithm to compare the models in practice.

Rotational polygon containment and minimum enclosure

Abstract: An algorithm and implementation is given for rotats’onal poly~gon wntainment: given polygons A, Pz, P3,. . . , A and a container polygon C, find rotation8 and translations for the k polygons that place them into the container without overlapping. A version of the algorithm and implementation also oolves rotational minimum enclosure: given a class C of container polygons, find a container C E C of minimum area for which containment has a solution. Minimum enclosure algorithm8 are given for the following classes: 1) rcctanglco of fixed width, 2) scaled copies of a fixed convex polygon, 3) arbitrary rectangles. Containment and minimum cnclosuro arc NP-hard (even in the purely translational ca~o). The minimum enclosure is approximate: it bounds the the minimum area between (1 e)A and A. Experiments arc done. to determine the largest practical value of 1; for both containment and minimum enclosure. Important applicntions for these algorithm to industrial problems are dincunsed. The paper also give8 practical algorithms and numoricnl techniques for robustly calculating polygon set intcracction, Minkowski sum, and range intersection: the intcrscction of a polygon with itself as it rotates through a range of angles.

Matching 2D patterns of protein spots

Abstract: A new algorithmic approach to comparing 2D patterns of protein spots obtained by the 2D gel electrophoresis technique is presented. Both the matching of a local pattern vs. a full 2D gel image and the global matching between full images are discussed. Preset slope and length tolerances of pattern edges serve as matching criteria. The local matching algorithm relies on a data structure derived from the incremental Delaunay triangulation of a point set and a 2–step hashing technique. The approach for the global matching uses local matching for landmark settings, which in most previous algorithmic solutions has been done interactively by the user.

Multi-criteria geometric optimization problems in layered manufacturing

Abstract: In Lnycred Manufacturing, the choice of the build direction for the model influences several design criteria, including the number of layers, the volume and contact-area of the support r&ructures, and the surface finish. These, in turn, impact the throughput and cost of the process. In this papcr, efficient geometric algorithms are given to reconcile two or more of these criteria simultaneously, under three formulations of multi-criteria optimization: Finding a build direction which (i) optimizes the criteria sequentially, (ii) optimizes their weighted sum, or (ii) allows the criteria to meet designer-prescribed thresholds. While the algorithms involving “support volume” or %ontact area” apply only to convex models, the solutions for “‘surface finish” and ‘%mmbcr of layers” are applicable to any polyhedral model. Some of the latter algorithms have also been implemented and tested on real-world models obtained from industry. The geometric techniques used include construction and searching of certain arrangements on the unit-sphere, 3dlmcnsional convex hulls, Voronoi diagrams, point location, and hierarchical representations. Additionally, solutions are also provided, for the first time, for the constrahted versions of two fundamental geometric problems, namely polyhedron width and large& empty diik on the unit-sphere.

Constructing cuttings in theory and practice

Abstract: We present a new randomized incremental algorithm for computing a cutting in an arrangement of lines in the plane. The nlgorithm produce cuttings whose expected size is 0 P”), and the expected running time of the algorithm is 0 I r-w). Both bounds are asymptotically optimal for nondegenerate arrangements. The algorithm is also simple to implement, and we present empirical results showing that the algorithm and some of its variants perform well in practice. We alao present another efficient algorithm (with slightly worse time bound) that generates small cuttings whose size is guaranteed to be close to the known upper bound of [9].

Geometric graphs with few disjoint edges

Abstract: A geometric graph is a graph drawn in the plane so that the vertices are represented by points in general position, the edges are represented by straight line segments connecting the corresponding points.

On the Union of k -Curved Objects.

Abstract: A (not necessarily convex) object C in the plane is κ -curved for some constant 0 , if it has constant description complexity, and for each point p on the boundary of C , one can place a disk B⫅C of radius κ· diam (C) whose boundary passes through p . We prove that the combinatorial complexity of the boundary of the union of a set C of n κ -curved objects (e.g., fat ellipses or rounded heart-shaped objects) is O (λ s (n) log n) , for some constant s .

Curvature-constrained shortest paths in a convex polygon (extended abstract)

Abstract: Let B be a point robot moving in the plane, whose path is constrained to have curvature at most 1, and let P be a convex polygon with n vertices. We study the collision-free, optimal path-planning problem for B moving between two configurations inside P (a configuration specifies both a location and a direction of travel). We present an O(n2 log n) time algorithm for determining whether a collision-free path exists for B between two given configurations. If such a path exists, the algorithm returns a shortest one. We provide a detailed classification of curvature-constrained shortest paths inside a convex polygon and prove several properties of them, which are interesting in their own right. Some of the properties are quite general and shed some light on curvature-constrained shortest paths amid obstacles.