# Canadian Conference on Computational Geometry

Frank Fiala
Evangelos Kranakis
Jörg-Rüdiger Sack
Copyright Date: 1996
https://www.jstor.org/stable/j.ctt7zt13s

## Table of Contents

1. Front Matter
(pp. ii-v)
2. Table of Contents
(pp. vi-x)
3. Preface
(pp. xi-xi)
Frank Fiala, Evangelos Kranakis and Jörg-Rüdiger Sack
4. Monday, August 12
• Session M1: Invited Lecture by Roberto Tamassia
• Robust Proximity Queries in Implicit Voronoi Diagrams
(pp. 1-1)
Roberto Tamassia, G. Liotta and F. P. Preparata
• Session M2
• ${\mathcal {O}}$-Convexity: Computing Hulls, Approximations, and Orientation Sets
(pp. 2-7)
V. Martynchik, N. Metelski and D. Wood

Let${\mathcal {O}}$be a set of unit vectors (orientations) in${{\mathcal {R}}^{2}}$. A line (segment, ray) in the plane is called an${\mathcal {O}}$-line (${\mathcal {O}}$-segment,${\mathcal {O}}$-ray) if its orientation vector is collinear to a vector of${\mathcal {O}}$. Güting [3] introduced, essentially, the notion of an${\mathcal {O}}$-oriented polygon; that is, a polygon whose edges are${\mathcal {O}}$-segments.

A planar point setXis${\mathcal {O}}$-convex if the intersection ofXwith any${\mathcal {O}}$-line is empty or connected. The${\mathcal {O}}$-convex hull of a planar set is defined to be the smallest${\mathcal {O}}$-convex set that contains...

• Efficient Algorithms for Counting and Reporting Pairwise Intersections between Convex Polygons
(pp. 8-13)
Prosenjit Gupta, Ravi Janardan and Michiel Smid

Let${\mathcal {S}}$be a set ofrbounded, convex polygons in the plane with a total ofnvertices. By apolygon, we mean the region consisting of the boundary as well as the interior. PolygonsPandQare said tointersectif they share a point; in particular, they intersect if one is completely contained inside the other or if their boundaries intersect.

We consider efficient algorithms for reporting output-sensitively (resp. counting) all intersecting pairs of polygons in${\mathcal {S}}$. Byoutput sizewe mean the number of intersecting pairs of polygons in${\mathcal {S}}$; we denote this...

• Convex hulls of bounded curvature
(pp. 14-19)
Jean-Daniel Boissonnat and Sylvain Lazard

The convex hull of a set of points in the plane is defined as the smallest set, or equivalently, the set of smallest perimeter that contains all the points. We consider in this paper convex hulls of bounded curvature. A curve is said of bounded curvature if it isC1and if its curvature is upper bounded by 1 everywhere it is defined. We define a convex hull of bounded curvature of a set${\mathcal {S}}$of points in the plane as a set containing${\mathcal {S}}$and whose boundary is a curve of bounded curvature of minimal length.

Convex sets of...

• Enclosing k points in the smallest axis parallel rectangle
(pp. 20-25)
Michael Segal and Klara Kedem

Given a setSofnpoints in the plane, and given an integerk,$\frac{n}{2}\le k\le n$, we want to find the smallest axis parallel rectangle (smallest perimeter or smallest area) that encloses exactlykpoints ofS. Aggarwal et al. [2] present an algorithm for this problem for anykn, which runs in timeO(k2nlogn) and usesO(kn) space. Eppstein et al. [4] and Datta et al. [3] show that this problem can be solved inO(k2n) time; the algorithm in [4] usesO(kn) space, while the algorithm in [3] uses linear space. These algorithms...

• Session M3
• FINDING THE SET OF ALL MINIMAL NESTED CONVEX POLYGONS
(pp. 26-31)
J. Bhadury and R. Chandrasekaran

The Minimal Nested Polygon problem is defined as the following: given two simple polygons Pinand Pout, with Pincompletely contained in Pout(i.e. Pin⊂ Pout), find a nested polygon (i.e. one that is contained in the annulus between Pinand Poutand contains the inner polygon Pin) P*, that has the fewest number of edges. This problem has applications in robotics, collision avoidance, stock cutting etc. and is hence extensively studied - see [1],[2],[3],[5]. When P* is known to be non-convex, anO(n) time algorithm is given in [3] to find it, wherenis the total number...

• Optimizing a Corridor between Two Polygons with an Application to Polyhedral Interpolation
(pp. 32-37)
Gill Barequet and Barbara Wolfers

The problem of reconstructing the boundary of a solid object from a series of parallel planar cross-sections has attracted much attention in the literature during the past two decades. The main motivations for this problem come from medical imaging applications and from geographic information systems.

The input usually consists of a series of parallel planar slices, each consisting of a collection of simple non-crossing polygons (output of an edge-detection process applied to the raw data). The goal is a polyhedral solid model whose cross-sections along the given planes coincide with the input slices. A natural simplification of the problem is...

• Heuristics for the Generation of Random Polygons
(pp. 38-43)
Thomas Auer and Martin Held

In this paper we deal with the random generation of simple polygons on a given set of points¹: Ideally, given a set${\mathcal S}=\{{{s}_{1}},\ldots ,{{s}_{n}}\}$ofnpoints, we would like to generate a simple polygon${\mathcal {P}}$at random with a uniform distribution. In this context, a uniformly random polygon on${\mathcal {S}}$is a polygon which is generated with probability${\frac{1}{k}}$if there existksimple polygons on${\mathcal {S}}$in total. Since no polynomial-time solution is known for the uniformly random generation of simple polygons, we focus on heuristics that offer a good time complexity and still generate a rich variety of...

• Orthogonal Polygon Reconstruction
(pp. 44-49)
L. Jackson and S. K. Wismath

In the field of visibility research, theReconstruction problemrebuilds an (unknown) object from (primarily) visibility information. In the literature, various objects have been considered (e.g., points, line segments, polygons) and a variety of visibility models suggested [Eve90] [O’R87] [Wis85]. In general, visibility information alone is not sufficient to reconstruct an object and frequently extra information is provided [CL91] [Wis94].

This paper presents an algorithm that reconstructs an (unknown) orthogonal polygon inO(nlogn) time from pure visibility information - the “stab visibilities” of the vertices of the polygon. Each vertexvof a simple polygonPhas two...

• Session M4
• CORRECTING TOPOLOGICAL DEFECTS OF TESSELLATIONS
(pp. 50-55)
Dong Wang and John A. Goldak

Geometric modeling technology, which has developed rapidly since 1970, plays a central role in industrial CAD [5]. Solid modelers have complex data structures that are usually proprietary to the computer code used to generate them and therefore are secret. There complexity and secrecy makes it difficult to exchange data between different solid modelers. One of the easiest ways to overcome the difficulties of data exchange between solid modelers is to create a simple boundary representation (BRep) of a tessellation that covers the surface of the solid with planar triangles. In this paper, an STL tessellation of a BRep approximates a...

• Generating Rooted Triangulations with Minimum Degree Four
(pp. 56-61)
David Avis and Chiu Ming Kong

A graph is a triangulation if it is planar and every face is a triangle. A triangulation is rooted if the external triangular face is labelled. Two rooted triangulations with the same external face labels are isomorphic if their internal vertices can be labelled so that both triangulations have identical edge lists.

In this article, we show that in the set of rooted triangulations onnpoints with minimum degree four, there exists a target triangulation${E_{n}^{*}}$such that any other triangulation${{E}_{n}}\ne E_{n}^{*}$in the set can be transformed to${E_{n}^{*}}$via a finite sequence of single and double diagonal...

• On Stable Line Segments in Triangulations
(pp. 62-67)
Andranik Mirzaian, Cao An Wang and Yin-Feng Xu

LetSbe a set ofnpoints in the plane andEdenote the set of all the line segments with endpoints inS. A line segment${\overline{pq}}$withp, qSis called a stable line segment of all triangulations ofS, if no line segment inEproperly intersects${\overline{pq}}$. The intersection of all possible triangulations ofSthen is the set of all stable line segments inS, denoted bySL(S).

As a combinatorial problem, various properties of stable line segments of a set of planar points have been investigated in [Xu92]. It is...

• DIAMONDS ARE NOT A MINIMUM WEIGHT TRIANGULATION’S BEST FRIEND
(pp. 68-73)
Prosenjit Bose, Luc Devroye and William Evans

LetSbe a set ofnpoints in the plane. A triangulationT(S) ofSis a maximal set of non-intersecting edges connecting points inS(that is, the addition of one more edge would create an intersection). The weight of an edge inT(S) is the Euclidean distance between its endpoints. The weight ofT(S) is the sum of the weight of its edges. Computing the minimum weight triangulationMWT(S) of a point setSis an old open problem. In fact, the complexity of this problem remains unresolved. It is not known to be NP-complete or polynomial-time...

• Session M5
• Stabbing Information of a Simple Polygon
(pp. 74-79)
Hazel Everett, Ferran Hurtado and Marc Noy

A simple polygon in the plane is typically represented by means of an ordered list of the real-valued coordinates of its vertices. Although such a representation completely describes the polygon it may not be convenient to store and manipulate. Alternatively then, we may wish to compute and store only that combinatorial information about the polygon required for a particular application. Three well known combinatorial objects defined for simple polygons are the convex hull, the internal visibility graph and the external visibility graph.

Now suppose that we are given a combinatorial object. We ask whether there is a simple polygon which...

• K-transversals of parallel convex sets
(pp. 80-86)
Nina Amenta

The following easy reduction is well known. LetCbe a finite set of parallel line segments in ℝd. We want to find a (d− 1)-transversalforC, that is, a hyperplane intersecting every segment inC. Such a hyperplane has to pass below the upper endpoint of each segment and above the lower endpoint. In the dual, the endpoints correspond to linear halfspaces, and the intersection of these halfspaces corresponds to the set of hyperplane transversals of the parallel segments in the primal. So the problem is solved by linear programming in dimensiond, in linear time if...

• Fast Stabbing of Boxes in High Dimensions
(pp. 87-92)
Franck Nielsen

Let${\mathcal {S}}$be a set ofn d-dimensional geometric objects. We say that${\mathcal {S}}$is stabbed bykpoints if there existkpoints so that each object of${\mathcal {S}}$contains at least one of these points. Given a set${\mathcal {S}}$as above, finding the minimumkso that${\mathcal {S}}$can be stabbed bykpoints has been shown to be NP-complete [FPTB1] as soon asd≥ 21. Therefore this problem is intractable even for small values ofn. This problem is also referenced in the literature as thecovering set problem(or dually as thehitting set problem)...

• Shooter Location Problems
(pp. 93-98)
Subhas C. Nandy, Krishnendu Mukhopadhyaya and Bhargab B. Bhattacharya

The transversal or stabbing problem is well-studied in computational geometry and has applications in image processing, robotics, hidden surface removal in graphics, etc. Given a number of objects distributed on a two- or three-dimensional region, the objective is to check whether a common transversal i.e., a straight line segment palssing through all the objects exists, and if so, determine the same. Edelsbrunner, Overmars and Wood [5] developed a method for computing the transversal inE2, if it exists, inO(n2logn) time andO(n) space, wherenis the number of objects. However anO(nlogn) time algorithm...

5. Tuesday, August 13
• Session Tu1: Invited Lecture by C. Bajaj
• Computational Geometry for Interrogative Visualization
(pp. 99-100)
Chandrajit L. Bajaj
• Session Tu2
• An optimal algorithm for dynamic post-office problem in ${R_{1}^{2}}$ and related problems
(pp. 101-106)
Sergei N. Bespamyatnikh

In this paper we address to the well-known post-office problem. The post-office problem is stated as fo1lows [18].

The post-office problem: Given a setSofnpqints in Rk, store it in a data structure such that for any query pointp∈ Rk, we can efficiently find itsnearest neighbor, i.e., a pointpSthat is closest top,

d(p, p) = min{d(p, q) :qS}.

In the dynamic version of problem the setpis dynamically changed by insertions and deletions of points. It is an open problem [17, 21] if there exists...

• A Topology-Oriented Algorithm for the Voronoi Diagram of Polygons
(pp. 107-112)
Toshiyuki Imai

The algorithm in this paper belongs to an incremental type. At first, we introduce some definitions and notations. In this paper, polygon is considered as a simple closed chain of line segments.

Letgibe a point, a line segment, a segment chain or a polygon for eachi(1 ≤iN). Let us assume that eachgiis simple and anygi,gj(ij) do not intersect.

We also assume that the set$\{{{g}_{1}},\cdots, {{g}_{n}}\}$is a plane graph in which the degree of any vertex is at most 2. We call the set${{G}=\{{g}_{1}},\cdots, {{g}_{N}}\}$a...

• On non-smooth convex distance functions
(pp. 113-118)
Ngọc-Minh Lê

Voronoi diagrams are an important construct in computational geometry. In high-dimensional Euclidean spaces, there are few studies of Voronoi diagrams under convex distance functions other than the Euclidean one. In this paper, we continue the study of general convex distance functions initiated in [4, 6], where only smooth distance functions are considered. The reasons for this study are: 1. To generalize the abstract Voronoi diagram from 2-space [5] to 3-space, we need to know structural properties of bisectors, as well as of their intersections. The work of Boissonnatet al[2] indicates that a direct approach to computing Voronoi diagrams...

• Time-Optimal Proximity Graph Computations on Enhanced Meshes
(pp. 119-124)
Stephan Olariu, Ivan Stojmenović and Albert Y. Zomaya

Recently, an elegant and powerful architecture has been obtained by adding one bus to every row and to every column in the mesh, as illustrated in Figure 1. In [7] an abstraction of such a system is referred to asmesh with multiple broadcasting. The mesh with multiple broadcasting has been implemented in VLSI and is commercially available in the DAP family of multicomputers [7, 12]. In turn, due to its commercial availability, the mesh with multiple broadcasting has attracted a great deal of attention. Applications ranging from image processing [7, 12], to computer graphics and robotics [3, 11], to...

• Session Tu3
• Deforming Curves in the Plane for Tethered-Robot Motion Planning
(pp. 125-130)
Susan Hert and Vladimir Lumelsky

We consider the following problem. A set of 2ndistinct points,Si, Ti, i= 1, …, n, in the plane is given. The pointsS= {Si} lie on the boundary of a convex polygonPwith verticesV; the pointsT= {Ti} lie in its interior. Each pair of points (Si, Ti) is connected by a polygonal line (polyline, for short)PLithat begins atSi(theinitial vertex) and ends atTi(thefinal vertex). In addition, each polyline may have any number ofinternal vertices. These vertices are members of the setTand are...

• Heuristic Motion Planning with Movable Obstacles
(pp. 131-136)
Thomas Chadzelek, Jens Eckstein and Elmar Schömer

Objects in geometric path planning problems are usually divided intomoving objectsandfixedones calledobstacles. In a problem description the moving objects and the obstacles are given - usually as polygons or polyhedra - together with their positions and orientations. For the moving objects goal configurations are specified additionally to complete the problem description.

Much work has been done to solve this kind of problems (→ classical path planning algorithm CPPA). In early papers it has been shown that the problem is decidable for an arbitrary number of moving objects [10]. In subsequent work the focus of interest...

• Viewing a Set of Spheres while Moving on a Linear Flightpath
(pp. 137-142)
Frank Follert

In this paper we investigate a three-dimensional hidden surface removal problem with respect to a moving point of view. Consider a situation where you want to compute the perspective view of a set of fixed nonintersecting spheres in 3-space from a sequence of viewpoints lying on a linear fiightpath. Such a situation arises for example in molecular graphics. Here, a molecule is often modeled as the union of atoms, each represented by a sphere of its van der Waals radius. The possibility of viewing such molecules from a prespecified linear flightpath in 3-space can provide valuable insights into their spatial...

• Approximating shortest paths in arrangements of lines
(pp. 143-148)
Prosenjit Bose, William Evans, David Kirkpatrick, Michael McAllister and Jack Snoeyink

Arrangements, such as the decomposition of the plane into regions by a set ofnlines, are central to much of the research on algorithms in computational geometry. They express a large number of geometric features that have a highly regular structure, such as the$\left( _{2}^{n} \right)$intersections between pairs of lines. Powerful techniques, such as ∊-nets, random sampling, lower envelopes, and implicit representation, have been developed that use the regularity to look at less than the full structure of the arrangements in solving problems on arrangements [3, 4]. These techniques are most applicable to problems of a combinatorial nature; they...

• Session Tu4
• Velocity planning for a robot moving along the shortest straight line path among moving obstacles
(pp. 149-154)
K. Krithivasan, A. Rema, Stefan Schirra and P.I. Vijaykumar

The problem of motion planning is to determine the existence of a path for a moving object from a given source point to a desired destination point, avoiding collision with any obstacles in the environment. The static domain, where the obstacles are stationary, has been studied extensively in [12], [6] and [4]. More specifically, the problem of computing the Euclidean shortest length path in the plane amidst stationary obstacles has been studied in [8], [10], [5] and [9]. Tools such as the visibility graph and the shortest path map have been used to map the connectivity of free space and...

• Lower Bounds for Computing Geometric Spanners and Approximate Shortest Paths
(pp. 155-160)
Danny Z. Chen, Gautam Das and Michiel Smid

Geometric spanners are data structures that approximate the complete graph on a set of points in thed-dimensional space ℝd, in the sense that the shortest path (based on such a spanner) between any pair of given points is not more than a factor oftlonger than the distance between the points in ℝd.

Letτbe a constant such that 1 ≤τ≤ ∞. We measure distances between points in ℝdwith theLτ-metric, whered≥ 1 is a constant. LetSbe a set ofnpoints in ℝd. We consider graphsG= (V,...

• On the Reachable Regions of Chains
(pp. 161-166)
Naixun Pei and Sue Whitesides

Ann-link chainis a sequence ofnrigid rods consecutively connected together at their endjoints, about which the rods may rotate freely. This paper considers the reachability properties of the endjoints of chains confined inside convex polygons.

Figure 1 illustrates ann-link chain Γ with jointsA0, …,An. Herelidenotes the length of linkLi= [Ai−1,Ai]. JointsA0andAnare calledendjointsand the others are calledintermediate joints.

We denote max1≤in{li} bylmaxand say that Γ isboundedbyb, denoted by Γ ≺b, iflmax<b.

We propose a strong notion...

• Heuristic Motion Planning with Many Degrees of Freedom
(pp. 167-172)
Thomas Chadzelek, Jens Eckstein and Elmar Schömer

This work is based on a simple but general path search strategy for spaces of arbitrary dimension. It is a heuristic algorithm for the generalized movers’ problem that doesnot explicitlycompute or represent configuration space but rather utilizes a collision detection subroutine for inquiries about possible paths. The approach uses divide-and-conquer and can be applied to many concrete situations, e.g. to motion planning for a single rigid body moving freely, a jointed robot arm, or even several objects moving concurrently. The fundamental idea is described in [Sch92] and will be investigated and refined here; we show how to handle...

• Session Tu5
• Computing Largest Circles Separating Two Sets of Segments
(pp. 173-178)
Jean-Daniel Boissonnat, Jurek Czyzowicz, Olivier Devillers, Jorge Urrutia and Mariette Yvinec

Let${\mathcal {C}}$denote a family of Jordan curves in the plane. Two setsPandQin the plane are${\mathcal {C}}$-separable if there exists$\xi \in {\mathcal {C}}$, such that every point of one of these sets lies in the closed region insideξ, and every point of the other set lies in the closed region outsideξ. In this paper we restrict our consideration to elements of${\mathcal {C}}$being circles. A circleC(X,r), with centerXand radiusr, separatingPfromQis said to be a largest separating circle if there is a neighbourhoodBofX...

• On the Permutations Generated by Rotational Sweeps of Planar Point Sets
(pp. 179-184)
Hanspeter Bieri and Peter-Michael Schmidt

Thesweep techniquehas proved to be one of the most powerful paradigms in Computational and Combinatorial Geometry, especially when dealing with problems in the 2-dimensional Euclidean plane. In most cases the plane is swept by a straight line whose normal vector never changes its direction. Such aplane sweepis calledtranslational. In some other cases it is more appropriate to perform arotationalplane sweep, i.e. the plane is swept by rotating a straight line or halfline (ray) around a point.

Starting from previous works by J.E. Goodman and R. Pollack ([GoPo80], [GoPo93]) one of the authors studied...

• MAXIMAL LENGTH COMMON NON-INTERSECTING PATHS
(pp. 185-189)
Jurek Czyzowicz, Evangelos Kranakis, Danny Krizanc and Jorge Urrutia

LetPn= {P1, … ,Pn} be a set ofnpoints on the plane. We say thatPnsupportsa planar graphG(V, E) if there is a plane embedding ofG(V, E) on the plane in such a way that its vertices are mapped to the elements ofPnand its edges to straight line segments connecting pairs of adjacent vertices. Given two point setsPnandQn, the problem of finding graphs supported by both of them has received attention recently. For example, Aronov, Seidel and Souvaine [1] and Kranakis and Urrutia [5] studied the problem of...

6. Wednesday, August 14
• Session W1
• Maintaining Multiple Levels of Detail in the Overlay of Hierarchical Subdivisions
(pp. 190-195)
Paola Magillo and Leila De Floriani

In this paper, we consider the overlay of plane subdivisions (a classical problem in computational geometry) and extend it to the case of hierarchically represented subdivisions.

The problem, in this new setting, has relevance in geographic information systems, where plane subdivisions are used to represent maps, and the combination (overlay) of two maps, describing different characteristics of the same spatial domain, is a fundamental operation. The need for multiresolution arises, because huge amounts of data are available, while not all application tasks require the same level of detail. Hierarchical subdivisions compactly encode decompositions of a plane domain at multiple resolutions,...

• Distance-Based Subdivision for Translational LP Containment
(pp. 196-201)
Karen Daniels and Victor J. Milenkovic

A number of industries generate new parts by cutting them from stock material: cloth, leather (hides), sheet metal,etc. These industries require good solutions to containment problems.Containmentis the question of whether a given set of part shapes can be fit, without overlapping, into a given container. The shapes and container are represented as polygons and may be nonconvex. Material such as cloth has a grain, and it sometimes has a “nap” (e.g.velvet or corduroy) or a colored pattern (e.g.stripes or plaid). A part cut out of such material has only one, two, four, or possibly eight...

• Variable Resolution Terrain Surfaces
(pp. 202-210)
E. Puppo

Multiresolution geometric models can be used in several application fields to support the representation and processing of geometric entities at different levels of resolution. The case of topographic surfaces is especially attractive for its impact on applications like geographic information systems, and virtual reality contexts. For instance, visualization in flight simulation can be made faster by rendering portions of terrain close to the observer at high resolution, while far portions are rendered at lower resolution.

In this paper we consider polyhedral terrains defined by a triangulation of a plane domain, where each vertex has an elevation value, and each triangle...

• Session W2
• Generalizing Halfspaces
(pp. 211-216)
Eugene Fink and Derick Wood

The study of convex sets is a branch of geometry that has applications in optimization, statistics, geometric number theory, and combinatorics [8], as well as in more practical areas, such as VLSI design, computer graphics, architectural databases, geographic databases, and motion planning. Researchers have explored a number of nonstandard notions of convexity, driven by application areas. Some examples are: orthogonal convexity [6, 7], NESW convexity [5, 13], finitely oriented convexity [4, 10], and link convexity [1, 12].

Rawlins introduced the notion ofrestricted-orientation convexity, also calledO-convexity, in his doctoral thesis, as a generalization of standard convexity [9]. Rawlins, Wood,...

• Efficient Algorithms for Guarding or Illuminating the Surface of a Polyhedral Terrain
(pp. 217-222)
Prosenjit Bose, David Kirkpatrick and Zaiqing Li

Victor Klee originally posed the problem of determining the minimum number of guards sufficient to cover the interior of ann-sided art gallery (polygon) in 1973. Chvátal showed that ⌊n/3⌋ guards are sufficient and sometimes necessary to cover the interior of ann-sided art gallery using a lengthy combinatorial argument [4]. Subsequently, Fisk [8] gave a concise and elegant proof using the fact that the vertices of a triangulated polygon may be three-colored. Since then, this problem has evolved into what is virtually a new field of study with numerous variations of the original question. The reader is referred to...

• The Surveillance of the Walls of an Art Gallery
(pp. 223-233)
Aldo Laurentini

With the word polygon we refer to a closed set which includes interior and boundary points. If the line segment connecting two points of a polygonPlies entirely inP, we say that they arevisiblefrom each other. A polygonPiscoveredby a set of viewpoints, orguards, lying inp, if each point inPis visible from at least one guard.

The research in this area was triggered in 1975 by Chvatal’s ”Art Gallery” Theorem[3]. He proved that at mostg(n)=⌊n/3⌋ guards are required for covering a simple polygonPwithnedges. The...

• Session W3
• On Rectangle Visibility Graphs. III. External Visibility and Complexity
(pp. 234-239)
Thomas C. Shermer

Let${\mathcal {R}}=\{{{R}_{i}}\}$be a collection of pairwise disjoint closed rectangles in the plane. Two rectanglesRiandRjwill be calledVisibleif there is a closed nondegenerate rectangular regionBij(called aband of visibility) such that one side ofBijis contained in a side ofRi, the opposite side ofBijis contained in a side ofRj, andBijdoes not intersect the interior of any rectangle in${\mathcal {R}}$. This type of visibility is equivalent to what Tamassia and Tollis have called ∊-visibility [8]. Thevisibility graph of${\mathcal {R}}$is the graph of the visibility...

• Maintaining Visibility of a Polygon with a Moving Point of View
(pp. 240-245)
Danny Z. Chen and Ovidiu Daescu

In this paper, we study the following problem: Given a scene with ann-vertex simple polygonPand a trajectory path in the plane, report the perspective view from each of a sequence of successive points on the trajectory. We present conceptually simple and optimal algorithms for the cases of this problem in which the trajectory path consists of several line segments or of a conic curve that contains the polygon. Intuitively, the polygonPcould represent the “opaque” walls of a building.

The general problem of computing visibility information of a geometric scene along a trajectory arises in several...

• Visibility graph of a set of line segments: A dynamic sequential algorithm and its parallel version
(pp. 246-251)
Yosser ATASSI

The visibility graph of a set of nonintersecting line segments G in the plane is a graph whose vertices are the endpoints of the segments and whose edges are the pairs of endpoints (u,v) such that the open line segment between u and v does not intersect any of the line segments of G.

Asano et al. [AGHI86] and Welzl [We185] presented optimal algorithms for constructing the visibility graph fornline segments inO(n2) time. Hershberger [Her89] studied the visibility graph of triangulated simple polygon withnsides. He described an algorithm that finds the visibility graph inO(m)...

• Dynamic algorithms for approximate neighbor searching
(pp. 252-257)
Sergei N. Bespamyatnikh

We consider the dynamic problems of computing

approximate nearest neighbor,

approximatek-nearest neighbor,

approximate range searching,

approximate furthest neighbor and

approximate diameter.

The nearest neighbor searching is one of the fundamental problems in computational geometry. We are given a setSofnpoints in Rd,d≥ 2, and a distance metricLt, 1 ≤t≤ ∞. It is assumed that the dimensiondis a constant independent ofn. Each pointpis given as ad-tuple of real numbers (p1, … ,pd). Letdist(p,q) denote the distance between pointspandq.

Definition 1.1 Given...

• Session W4
• Three-Dimensional Restricted-Orientation Convexity
(pp. 258-263)
Eugene Fink and Derick Wood

The study of convex sets is a branch of geometry that has numerous connections with other areas of mathematics, including analysis, linear algebra, statistics, and combinatorics [4]. Its importance stems from the fact that convex sets arise in many areas of mathematics and are often amenable to rather elementary reasoning. The concept of convexity serves to unify a wide range of mathematical phenomena.

The application of convexity theory to practical problems led to the exploration of nontraditional notions of convexity, such as orthogonal convexity [6], finitely oriented convexity [5], and link convexity [1, 9]. These nontraditional convexities are used in...

• Efficient Algorithms for the Smallest Enclosing Cylinder Problem
(pp. 264-269)
Elmar Schömer, Jürgen Sellen, Marek Teichmann and Chee Yap

A major topic of geometric optimization is to approximate point sets by simple geometric figures. This includes extensively studied planar problems such as smallest enclosing circles, the minimum width annulus, and the minimum width slab. In higher dimensions, there are few non-trivial complexity results for geometric figures beyond hyperplanes or spheres. In this paper, we consider the following:

Smallest Cylinder Problem (P1): LetIbe a given set ofnpoints in 3-space. Find a line ℓ which minimizes max{d(ℓ,c) :cI}.

Here,d(ℓ,c) denotes the minimum Euclidean distance betweencandapoint of...

• On the Ω(n4/3) Weak Lower Bounds for Some 3D Geometric Problems
(pp. 270-275)
Binhai Zhu

Proving the lower bounds of problems is one of the central part in algorithm theory. Established lower bound for a specific problem usually convinces people not to try obtaining better algorithms unless under a different model of computation or when some extra primitives are allowed. In computational geometry (as well as in the general algorithm design area) lower bound is usually obtained via problem reduction, probablistic argument, combinatorial counting. However, in general the lower bound results are very sparse compared with the vast upper bound results in computational geometry [Cha94].

We briefly mention three techniques for proving lower bounds in...

7. Thursday, August 15
• Session Th1: Invited Lecture by P. Raghavan
• Computational Geometry Impact Potential: A Business and Industrial Perspective
(pp. 276-276)
Prabhakar Raghavan
• Session Th2
• Probabilistic algorithms for efficient grasping and fixturing
(pp. 277-282)
Marek Teichmann

Consider an idealized robot hand, consisting of several independently movable force-sensing fingers; this hand is used to grasp a rigid objectB. Each finger contacts the object only at one point onBand can apply a positive force. We assume that at that point the normal toBis unique, and that the contact is frictionless. We wish to find agrasp: a set of points on the boundary ofB. The fingers will then apply forces at these points to grasp the object. In general, we want the number of fingers to be small. Another desirable characteristic of...

• On a problem of immobilizing polygons.
(pp. 283-288)
Jurek CZYZOWICZ, Ivan STOJMENOVIC and Tomasz SZYMACHA

Let P be a polygon and I a set of n points (pins) in the plane. We say that I immobilizes P if any rigid motion of P in the plane forces at least one of the pins of I to penetrate the interior of P. As for small motion only pins from the boundary of P may penetrate it, we will suppose that all pins actually belong to the boundary of the polygon. A rigid motion of polygon P in the plane is a mapping M (different from identity) from the set t×P (t represents time) to the plane,...

• Finding an o(n2logn) algorithm is sometimes hard
(pp. 289-294)
Antonio Hernández Barrera

In the process of designing efficient algorithms, we are faced with the problem of trying to lower known time-bounds, or finding what the real lower bounds are for a given problem. It might be useful to know that a problem we are trying to solve is, at least, as difficult as another well-studied,verydifficult one. Some classes of difficult problems have been found, where the NP-complete class is, maybe, the best known. Inside computational geometry, a large number of problems have been identified for which it is impossible to obtain subquadratic algorithms, unless one manages to improve the complexity...

• Improved orthogonal drawings of 3-graphs
(pp. 295-299)
Therese C. Biedl

Orthogonal graph drawings are an important tool for graph layout, since the minimum angle of 90° makes the drawings easily readable. Specific uses include Data Flow Diagrams and Entity Relationships Diagrams. The precise definition is as follows:

Anorthogonal drawingof a graph is an embedding in the plane such that vertices are drawn as points and edges are drawn as sequences of horizontal and vertical line segments. A point where the drawing of an edge changes its direction is called abendof this edge. We assume that all vertices and bends are placed on points with integer coordinates....

• Session Th3
• Algorithms on Polygonal Embeddings of Graphs
(pp. 300-305)
Leizhen Cai

Anembeddingof a graphG= (V,E) is a mapping of the vertices ofGonto different points in the plane and the edges ofGonto line segments that preserves the incidence relation between edges and vertices. Therefore an edgeuvwill be mapped onto the line segment linking the corresponding points ofuandv. Furthermore, the point corresponding to a vertexvwill not lie on any line segment that does not correspond to an edge incident withvinG. Figure 1 gives three different embeddings of a path with five vertices. We refer...

• Optimal orthogonal drawings of connected plane graphs
(pp. 306-311)
Therese C. Biedl

A (2-dimensional) orthogonal drawing of a graph is a drawing such that every vertex is drawn as a point in the plane, and every edge is drawn as a sequence of horizontal and vertical lines. Orthogonal graph drawings are an important layout tool, for example to display Data Flow Diagrams and Entity Relationships Diagrams. Two of the most important measurements of the quality of a drawing are the grid-size and the number of bends. Orthogonal drawings exist only if every vertex in the graph has at most four incident edges, such a graph is called a4-graph. On the other...

• Straight Line Embeddings of Planar Graphs on Point Sets
(pp. 312-318)
Netzahualcoyotl Castañeda and Jorge Urrutia

LetPnbe a set of points on the plane in general position, andGa graph withnvertices. We say thatPnsupportsGif there is an embedding ofGon the plane in such a way that the vertices ofGare mapped to the elements ofPn, and its edges to non-intersecting open straight line segments joining pairs of elements ofPnwhich correspond to pairs of adjacent vertices inG. We call any such embedding a straight-line embedding ofGonPn.

In 1990, Perles introduced the problem of embedding rooted trees on point...

• Extending Rectangular Range Reporting with Query Sensitive Analysis
(pp. 319-324)
Robin Y. Flatland and Charles V. Stewart

We present an algorithm for reporting the new points incorporated by on-line sequences of nested rectangular range queries. We assume the rectangular regions in a sequence have the same orientation but this orientation may be different for each sequence and is not known in advance (see Figure 1). The problem is formally stated as follows:

Extending Rectangular Range Reporting Problem: Given a set ofNpoints inRdand an on-line sequence of d-dimensional rectangular query regionsQl, … ,QEaligned with some arbitrary orthogonal coordinate system and eachQicompletely containingQi−l, for thei(th) extended query, report...

• Session Th4
• The Complexity of Rivers in Triangulated Terrains
(pp. 325-330)
Mark de Berg, Prosenjit Bose, Katrin Dobrint, Marc van Kreveld, Mark Overmars, Marko de Groot, Thomas Roos, Jack Snoeyink and Sidi Yu

Terrain drainage characteristics provide important information on water resources, possible flood areas, erosion and other natural processes. In natural resource management, for example, the basic management unit is thewatershed, the area around a stream that drains into the stream. Road building, logging, or other activities carried out in a watershed all have the potential to affect the defining stream. Manual quantification of terrain drainage characteristics is a tedious and time consuming job. Fortunately, through spatial analysis of digital representations of surfaces, they can be, by and large, inferred automatically.

In this note, we survey some of the literature on...

• Computing the Angularity Tolerance
(pp. 331-336)
Mark de Berg, Henk Meijer, Mark Overmars and Gordon Wilfong

Manufactured objects are always approximations to some ideal object: parts that are supposed to be flat will not be perfectly flat, round parts will not be perfectly round, and so on. In many situations, however, it is important that the manufactured object is very close to the ideal object. In such cases the specification of an object includes a description of how far the manufactured object is allowed to deviate from the ideal one. The field of dimensional tolerancing [2] provides the language for this. Given a specification, one must test whether the manufactured object meets it, which is the...

• The Complexity of Illuminating Polygons by α-Flood-Lights
(pp. 337-342)
Jay Bagga, Laxmi Gewali and David Glasser

In most illumination problems, visibility from a point is allowed in all directions (360° angular aperture). Recently, several researchers have considered illumination problems that restrict visibility to within a certain angular aperture [3,4,5]. A light source whose illumination angle is restricted toα-degrees is called anα-flood-light (or simplyα-light). A simple polygon may remain unilluminated even if we place a 90º-flood-light at each vertex; and this holds true even if the polygon is restricted to be monotone [4]. Estivill-Castro et. al. [5] presented a surprise on polygon illumination: there are simple polygons (called logarithmic spirals) that can not be...

8. AUTHOR INDEX
(pp. 343-344)