Showing papers presented at "Symposium on Computational Geometry in 1999"

Small distortion and volume preserving embeddings for planar and Euclidean metrics

Abstract: A finite metric space, (S,d) , contains a finite set of points and a distance function on pairs of points. A contraction is an embedding, h, of a finite metric space (S, d) into Rd where for any u, v E S, the Euclidean (&) distance between h(u) and h(v) is no more than d(u, v). The distortion of the embedding is the maximum over pairs of the ratio of d(u, w) and the Euclidean distance between h(u) and h(v). Bourgain showed that any graphical metric could be embedded with distortion O(logn). Linial, London and Rabinovich and Aumman and Rabani used such embeddings to prove an O(log k) approximate max-flow min-cut theorem for k commodity flow problems. A generalization of embeddings that preserve distances between pairs-of points are embeddings that preserve volumes of larger sets. In particular, A (k, c)-volume respecting embedding of n-points in any metric space is a contraction where every subset of k points has within an ck-’ factor of its maximal possible k l-dimensional volume. Feige invented these embeddings in devising a polylogarithmic approximation algorithm for the bandwidth problem using these embeddings. Feige’s methods have subsequently been used by Vempala for approximating versions of the VLSI layout problem. Feise showed that a (k, O(10,g~‘~ n,/m)) volume r&ecting embedding‘ eksted.” Be -recently found improved (k, 0( mdk log k + log n)) volume respecting embeddings. For metrics arising from planar graphs (planar metrics), we give (k,O(m)) volume respecting contractions. As a corollary, we give embeddings for Permission to makkr digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. XC’99 Miami Beach Florida Copyright ACM 1999 I-581 13-068-6/99/06...$5.00 planar metrics with distortion O(e). This gives rise to an O(e)-approximate max-flow min-cut theorem for multicommodity flow problems in planar graphs. We also give an improved bound for volume respecting embeddings for Euclidean metrics. In particular, we give an (k,O(dog klog D)) volume respecting embedding where D is the ratio of the largest distance to the smallest distance in the metric. Our results give improvements for Feige’s and Vempala’s approximation algorithms for planar and Euclidean metrics. For volume respecting embeddings, our embeddings do not degrade very fast when preserving the volumes of large subsets. This may be useful in the future for approximation algorithms or if volume .respecting embeddings prove to be of independent interest.

Sliver exudation

Abstract: A sliver is a tetrahedron whose four vertices lie close to a plane and whose projection to that plane is a convex quadrilateral with no short edge. Slivers are notoriously common in 3-dimensional Delaunay triangulations even for well-spaced point sets. We show that if the Delaunay triangulation has the ratio property introduced in [15] then there is an assignment of weights so the weighted Delaunay triangulation contains no slivers. We also give an algorithm to compute such a weight assignment.

A core library for robust numeric and geometric computation

Abstract: Nonrobustness is a well-known problem in many areas of computational science. Until now, robustness techniques and the construction of robust algorithms have been the province of experts in this field of research. We describe a new C/C++ library (CORE) for robust numeric and geometric computation based on the principles of Exact Geometric Computation (EGC). Through our library, for the first time, any programmer can write robust and efficient algorithms. The Core Library is based on a novel numerical core that is powerful enough to support EGC for algebraic problems. This is coupled with a simple delivery mechanism which transparently extends conventional C/C++ programs into robust codes. We are currently addressing efficiency issues in our library: (a) at the compiler and language level, (b) at the level of incorporating EGC techniques, as well as the (c) the system integration of both (a) and (b). Pilot experimental results are described. The basic library is availableathttp://cs.nyu.edu/exact/core/andthe C++-to-C compiler is under development.

Temporally coherent conservative visibility

Abstract: Efficiently identifying polygons that are visible from a changing synthetic viewpoint is an important problem in computer graphics. Even with hardware support, simple algorithms like depth-buffering cannot achieve interactive frame rates when applied to geometric models with many polygons. However, a visibility algorithm that exploits the occlusion properties of the scene to identify a superset of visible polygons, without touching most invisible polygons, could achieve fast frame rates while viewing such models. In this paper, we present a new approach to the visibility problem. The novel aspects of our algorithm are that it is temporally coherent and conservative ; for all viewpoints the algorithm overestimates the set of visible polygons. As the synthetic viewpoint moves, the algorithm reuses visibility information computed for previous viewpoints. It does so by computing visual events at which visibility changes occur, and efficiently identifying and discarding these events as the viewpoint changes. In essence, the algorithm implicitly constructs and maintains a linearized portion of an aspect graph , a data structure for representing visual events. We demonstrate that the visibility algorithm significantly accelerates rendering of several test models.

Crust and anti-crust: a one-step boundary and skeleton extraction algorithm

Abstract: We wish to extract the topology from scanned maps. In previous work [15] this was done by extracting a skeleton from the Voronoi diagram, but this required vertex labelling and was only useable for polygon maps. We wished to take the crust algorithm of Amenta, Bern and Eppstein [3] and modify it to extract the skeleton from unlabelled vertices. We find that by reducing the algorithm to a local test on the simple Voronoi diagram of point sites we may extract both the crust and the skeleton simultaneously. We show that this produces the same results as the original algorithm, and illustrate its utility with various cartographic applications.

Folding flat silhouettes and wrapping polyhedral packages: new results in computational origami

Abstract: We show a remarkable fact about folding paper: From a single rectangular sheet of paper, one can fold it into a ∞at origami that takes the (scaled) shape of any connected polygonal region, even if it has holes. This resolves a long-standing open problem in origami design. Our proof is constructive, utilizing tools of computational geometry, resulting in e‐cient algorithms for achieving the target silhouette. We show further that if the paper has a difierent color on each side, we can form any connected polygonal pattern of two colors. Our results apply also to polyhedral surfaces, showing that any polyhedron can be \wrapped" by folding a strip of paper around it. We give three methods for solving these problems: the flrst uses a thin strip whose area is arbitrarily close to optimal; the second allows wider strips to be used; and the third varies the strip width to optimize the number or length of visible \seams" subject to some restrictions.

Curve reconstruction: connecting dots with good reason

Abstract: Curve reconstruction algorithms are supposed to reconstruct curves from point samples. Recent papers present algorithms that come with a guarantee: Given a sufficiently dense sample of a closed smooth curve, the algorithms construct the correct polygonal reconstruction. Nothing is claimed about the output of the algorithms, if the input is not a dense sample of a closed smooth curve, e.g., a sample of a curve with endpoints. We present an algorithm that comes with a guarantee for any setP of input points. The algorithm constructs a polygonal reconstruction G and a smooth curve that justifies G as the reconstruction fromP . © 2000 Elsevier Science B.V. All rights reserved.

On deletion in Delaunay triangulations

Abstract: This paper presents how the space of spheres and shelling may be used to delete a point from a d-dimensional triangu- lation efficiently. In dimension two, if k is the degree of the deleted vertex, the complexity is O( k log k), but we notice that this number only applies to low cost operations, while time consuming computations are only done a linear number of times. This algorithm may be viewed as a variation of Heller’s algorithm [He190, Mid93], which is popular in the geographic information system community. Unfortunately, Heller algorithm is false, as explained in this paper.

Motion planning for a rigid body using random networks on the medial axis of the free space

Abstract: Several motion planning methods using networks of randomly generated nodes in the free space have been shown to perform well in a number of cases, however their performance degrades when paths are required to pass through narrow passages in the free space. In previous work we proposed MAPRM, a method of sampling the configuration space in which randomly generated configurations, free or not, are retracted onto the medial axis of the free space without having to first compute the medial axis; this was shown to increase sampling in narrow passages. In this paper we give details of the MAPRM algorithm for the case of a rigid body moving in three dimensions, and show that the retraction may be carried out without explicitly computing the C-obstacles or the medial axis. We give theoretical and experimental results to show this improves performance on problems involving narrow corridors and compare the performance to uniform random sampling from the free space.

H-Walk: hierarchical distance computation for moving convex bodies

Abstract: This paper presents the Hierarchical Walk, or H-Walk algorithm, which maintains the distance between two moving convex bodies by exploiting both motion coherence and hierarchical representations. For convex polygons, we prove that H-Walk improves on the classic Lin-Canny and DobkinKirkpatrick algorithms. We have implemented H-Walk for moving convex polyhedra in three dimensions. Experimental results indicate that, unlike previous incremental distance computation algorithms, H-Walk adapts well to variable coherence in the motion.

Efficient exact geometric computation made easy

Abstract: We show that the combination of the CGAL framework for geometric computation and the number type ledareal yields easy-to-write, correct and efficient geometric programs.

The complexity of the union of (a, b)-covered objects

Abstract: An (α, β)-covered object is a simply connected planar region c with the property that for each point p ∈ ∂c there exists a triangle contained in c and having p as a vertex, such that all its angles are at least α and all its edges are at least β · diam(c)long. This notion extends that of fat convex objects. We show that the combinatorial complexity of the union of n (α, β)-covered objects of ‘constant description complexity’ is O(λs+2(n) log 2 n log logn), where s is the maximum number of intersections between the boundaries of any pair of the given objects.

Controlled perturbation for arrangements of polyhedral surfaces with application to swept volumes

Abstract: We describe a perturbation scheme to overcome degeneracies and precision problems for algorithms that manipulate polyhedral surfaces using floating point arithmetic. The perturbation algorithm is simple, easy to program and completely removes all degeneracies. We describe a software package that implements it, and report experimental results. The perturbation requires O(n log3 n+nDK2) expected time and O(nlog n+nK2) working storage, and has O(n) output size, where n is the total number of facets in the surfaces, K is a small constant in the input instances that we have examined, D is a constant greater than K but still small in most inputs, and both might be as large as n in ‘pathological’ inputs. A tradeoff exists between the magnitude of the perturbation and the efficiency of the computation. Our perturbation package can be used by any application that manipulates polyhedral surfaces and needs robust input, such as solid modeling, manufacturing and robotics. We describe an application for the computation of swept volumes, which uses our perturbation package and is therefore robust and does not need to handle degeneracies. Our work is based on [19] which handles the case of spheres, extending the scheme to the more difficult case of polyhedral surfaces perturbation.

Visibility-based pursuit-evasion in a polygonal room with a door

Abstract: Visibility-based pursuit-evasion problems are as follows: given a polygonal region, one or more searchers with visibility, and an unpredictable intruder that is arbitrarily faster than the searcher, plan the motion of the searchers so as to see the intruder. In this paper, we consider several visibility-based pursuit-evasion problems with a single searcher: l Given a simple polygon with a door (i.e., penetrable vertex) d, can a searcher find an intruder within the polygon in such a way that the intruder couldn’t make a dash for the door d? l Given a simple polygon with a door d, can a searcher make no undetected intruder remain in the polygon (that is, find the intruder or evict it from the polygon through d)? l Given a building (represented as a sequence of simple polygons joined by staircases), can the searcher find the intruder within it? For each of the three problems above, we give a characterization of the class of regions that admits a search strategy and present an O(n2)-time algorithm for constructing a search path, if one exists, for an n-sided region. Interestingly, our characterizations imply that each of the above regions searchable by a searcher with omnidirectional vision (i.e., 360’ vision) is also searchable by a searcher with two flashlights (i.e., ray visions). As a by-product, we improves the time complexity of the corridor search problem in [2], by a factor of log n. *This work was partially supported by KOSEF 98-0102-07-01-3. permission lo make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists. requires prior specific permission and/or a fee. SCG’99 Miami Beach Florida Copyright ACM 1999 I-581 13-068-6/99/06...$5.00

Curve reconstruction, the traveling salesman problem and Menger's theorem on length

Abstract: We give necessary and sufficient regularity conditions under which the curve reconstruction problem is solved by a traveling salesman tour or path, respectively. For the proof we have to generalize a theorem of Menger [12], [13] on arc length.

MAPC: a library for efficient and exact manipulation of algebraic points and curves

Abstract: We present MAPC a library for exact representation of geometric objects speci cally points and algebraic curves in the plane Our library makes use of several new algorithms which we present here including methods for nding the sign of a determinant nding intersections between two curves and breaking a curve into monotonic segments These algorithms are used to speed up the underlying computations The library provides C classes that can be used to easily instantiate manipulate and perform queries on points and curves in the plane The point classes can be used to represent points known in a variety of ways e g as exact rational coordinates or algebraic numbers in a uni ed manner The curve class can be used to represent a portion of an algebraic curve We have used MAPC for applications dealing with algebraic points and curves including sorting points along a curve computing arrangement of curves medial axis computations and boundary evaluation of spline primitives As compared to earlier algorithms and implementations utilizing exact arithmetic our library is able to achieve more than an order of magnitude improvement in performance

Mesh collapse compression

Abstract: We present a novel algorithm for encoding the topology of triangular meshes. A sequence of edge contract and divide operations collapses the entire mesh into a single vertex. This implicitly creates a tree with weighted edges. The weights are vertex degrees and capture the topology of the unlabeled mesh. The nodes are vertices and capture the labeling of the mesh. This weighted-edge tree has a very compact encoding.

Generalizing ham sandwich cuts to equitable subdivisions

Abstract: We prove a generalization of the famous Ham Sandwich Theorem for the plane. Given gn red points and gm blue points in the plane in general position, there exists an equitable subdivision of the plane into g disjoint convex polygons, each of which contains n red points and m blue points. For g = 2 this problem is equivalent to the Ham Sandwich Theorem in the plane. We also present an efficient algorithm for constructing an equitable subdivision.

The path of a triangulation

Abstract: For a planar point set S let T be a triangulation of S and 1 a line properly intersecting T. We show that there always exists a unique path in T with certain properties with respect to 1. This path is then generalized to (non triangulated) point sets restricted to the interior of simple polygons. This so-called triangulation path enables us to treat several triangulation problems on planar point sets in a divide & conquer-like manner. For example, we give the first algorithm for counting triangulations of a planar point set which is observed to run in time sublinear in the number of triangulations. Moreover, the triangulation path proves to be useful for the computation of optimal triangulations.

Approximation and exact algorithms for minimum-width annuli and shells

Abstract: and Shells Pankaj K. Agarwaly Boris Aronovz Sariel Har-Peledx Micha Sharir{ Abstract Let S be a set of n points in Rd . The \roundness" of S can be measured by computing the width ! (S) of the thinnest spherical shell (or annulus in R2 ) that contains S. This paper contains four main results related to computing ! (S): (i) For d = 2, we can compute in O(n logn) time an annulus containing S whose width is at most 2! (S). (ii) For d = 2 we can compute, for any given parameter " > 0, an annulus containing S whose width is at most (1 + ")! (S), in time O(n logn+ n="2). (iii) For d 3, given a parameter " > 0, we can compute a shell containing S of width at most (1+")! (S) in time O n "d log diam(S) ! (S)" or O n logn "d 2 + n "d 1 log diam(S) ! (S)" . (iv) For d = 3, we present an O(n3 1=19+")-time algorithm to compute a minimum-width shell containing S. Work by P.A. was supported by Army Research O ce MURI grant DAAH04-96-1-0013, by a Sloan fellowship, by NSF grants EIA{9870724, and CCR{9732787, by an NYI award, and by a grant from the U.S.-Israeli Binational Science Foundation. Work by B.A. was supported by a Sloan Research Fellowship and by a grant from the U.S.-Israeli Binational Science Foundation. Work by M.S. was supported by NSF Grants CCR-97-32101, CCR-94-24398, by grants from the U.S.-Israeli Binational Science Foundation, the G.I.F., the German-Israeli Foundation for Scienti c Research and Development, and the ESPRIT IV LTR project No. 21957 (CGAL), and by the Hermann Minkowski{MINERVA Center for Geometry at Tel Aviv University. Part of the work by P.A. and B.A. on the paper was done when they visited Tel Aviv University in May 1998. yCenter for Geometric Computing, Department of Computer Science, Box 90129, Duke University, Durham, NC 27708-0129, USA. E-mail: pankaj@cs.duke.edu zDepartment of Computer and Information Science, Polytechnic University, Brooklyn, NY 11201-3840, USA. E-mail: aronov@ziggy.poly.edu xSchool of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel. E-mail: sariel@math.tau.ac.il {School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel; and Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA. E-mail: sharir@math.tau.ac.il

Smooth paths in a polygonal channel

Abstract: We show how to efficiently smooth a polygon with an approximating spline that stays to one side of the polygon. We also show how to find a smooth spline path between two polygons that form a channel. Problems of this type arise in many physical motion planning tasks where not only forbidden regions have to be avoided but also a smooth traversal of tbe motion path is required. Both algorithms are based on a new tight and efficiently computable bound on the distance of a spline from its control polygon and employ only standard linear and quadratic ptogmmming techniques. FIGURE 1. The SUPPORT problem: given an input polygon c (grey) find a spline (black) that stays above but close to

The table layout problem

Abstract: In this paper we study a geometric problem arising in typography: the problem of laying out a two dimensional table. Each cell of the table has content associated with it. We may have choices on the geometry of cells (e.g., number of rows to use for the text in a cell.) The problem is to choose configurations for the cells to optimize an objective function such as minimum table height given a fixed width for the table. We formulate a combinatorial version of the table layout problem, where the objective is to choose cell geometry to minimize table size, The table layout problem is NP-complete, even for very restricted instances. One of our main results is an algorithm for computing the convex hull of the set of feasible table configurations, which gives a heuristic algorithm for table layout. We establish a connection between the fractional (LP) solution to the table layout problem and generalized network flow. We also present experimental results comparing the performance of heuristics. Our final result is an algorithm for a general paragraphing problem, where we are interested in generating all minimal height-width configurations of text. We show that the set of all height-width pairs for a paragraph of n words can be determined in O’(n3f “) time.

Edgewise subdivision of a simplex

Abstract: In this paper we introduce the abacus model of a simplex and use it to subdivide a d-simplex into k d d-simplices all of the same volume and shape characteristics. The construction is an extension of the subdivision method of Freudenthal [3] and has been used by Goodman and Peters [4] to design smooth manifolds.

Sharp bounds on geometric permutations of pairwise disjoint balls in Rd

Abstract: We prove that the maximum number of geometric permutations, induced by line transversals to a collection of n pairwise disjoint balls inR d ,i s2.n di1 /. This improves substantially the upper bound of O.n 2di2 / known for general convex sets (9). We show that the maximum number of geometric permutations of a sufficiently large collection of pairwise disjoint unit disks in the plane is two, improving the previous upper bound of three given in (5).

Determining an optimal penetration among weighted regions in two and three dimensions

Abstract: We present efficient algorithms for solving the problem of computing an optimal penetration (a ray or a semi-ray) among weighted regions in 2-D and 3-D spaces. This problem finds applications in several areas, such as radiation therapy, geological exploration, and environmental engineering. Our algorithms are based on a combination of geometric techniques and optimization methods. Our geometric analysis shows that the d-D (d = 2, 3) optimal penetration problem can be reduced to solving O(n2(d−1)) instances of certain special types of non-linear optimization problems, where n is the total number of vertices of the regions. We also give implementation results of our 2-D algorithms.

Geometric pattern matching: a performance study

Abstract: In this paper, we undertake a performance study of some recent algorithms for geometric pattern matching. These algorithms cover two general paradigms for pattern matching; alignment and combinatorial pattern matching. We present analytical and empirical evaluations of these schemes. Our results indicate that a proper implementation of an alignmentbased method outperforms other (often asymptotically better) approaches.

Efficient perspective-accurate silhouette computation

Abstract: Silhouettes of geometric models (see Fig.1) are salient for many applications, including visualization [LE97] and shape analysis [BV95, Cro77]. The video describes a technique for efficient tracking of perspective-accurate silhouettes by reducing the problem to point location queries. Perspective-accurate silhouette computation is known to be difficult [KM96, KM98, KMGL96]. Note that silhouettes formed under perspective projection are significantly different from those under parallel projection and are more difficult to compute. Given a polyhedral model with well-defined normals, the polygons (facets) whose normals point away from the viewpoint are called back-facing. Similarly, normals of front-facing polygons point toward the viewpoint. An edge shared by a back-facing polygon and a front-facing polygon is said to be on the silhouette of the model. The silhouette of a geometric model is among its most significant visual features. It may be used to describe the shape of a model with only very few details of its geometry. It can also be used for computing the visible portions of a model as it forms the boundary between the visible front-facing and the hidden back-facing polygons. Silhouettes are also useful in shadow generation [Cro77], model simplification [LE97], collision detection [BV95], image registration [LSB95], and several other applications. The video presents an incremental algorithm for computing silhouettes of polyhedral models and updating them as the viewpoint changes. Most applications of interest exhibit coherence in silhouettes between successive computations as viewpoints change slowly. Our algorithm exploits this coherence to obtain fast updates of the silhouettes. Assume that we know the silhouette of a model from .a given view point. An edge ceases to be (resp., becomes) a silhouette edge if and only if it is shared by one front-facing and one back-facing polygon (resp., two polygons of the same facing), and exactly one of its adjacent polygons changes orientation. Thus, when the viewpoint is changed we are interested in finding such orientation-change events. We reduce this problem to a point-location query in a planar-map data structure.

Efficient algorithms for line and curve segment intersection using restricted predicates

Abstract: We consider whether restricted sets of geometric predicates support efficient algorithms to solve line and curve segment intersection problems in the plane. Our restrictions are based on the notion of algebraic degree, proposed by Preparata and others as a way to guide the search for efficient algorithms that can be implemented in more realistic computational models than the Real RAM. Suppose that n (pseudo-)segments have k intersections at which they cross. We show that intersection algorithms for monotone curves that use only comparisons and above/below tests for endpoints, and intersection tests, must take at least Ω(n k ) time. There are optimal O(nlogn+k) algorithms that use a higher-degree test comparing x coordinates of an endpoint and intersection point; for line segments we show that this test can be simulated using CCW () tests with a logarithmic loss of efficiency. We also give an optimal O(nlogn+k) algorithms for red/blue line and pseudo-segment intersection, in which the segments are colored red and blue so that there are no red/red or blue/blue crossings.

Efficient algorithms for maximum regression depth

Abstract: We investigate algorithmic questions that arise in the statistical problem of computing lines or hyperplanes of maximum regression depth among a set of n points. We work primarily with a dual representation and find points of maximum undirected depth in an arrangement of lines or hyperplanes. An O(nd) time and O(nd−1) space algorithm computes undirected depth of all points in d dimensions. Properties of undirected depth lead to an O(nlog 2n) time and O(n) space algorithm for computing a point of maximum depth in two dimensions, which has been improved to an O(nlog n) time algorithm by Langerman and Steiger (Discrete Comput. Geom. 30(2):299–309, [2003]). Furthermore, we describe the structure of depth in the plane and higher dimensions, leading to various other geometric and algorithmic results.