Computational Geometry on Surfaces

Transcription

1 Computational Geometry on Surfaces

2 Computational Geometry on Surfaces Performing Computational Geometry on the Cylinder, the Sphere, the Torus, and the Cone by Clara I. Grima Department 0/ Applied Mathematics (E. u.l. T.A.), University 0/ Seville, Seville, Spain and Alberto Marquez Department 0/ Applied Mathematics (F.I.E.. ), University 0/ Seville, Seville, Spain Springer-Science+Business Media, B.V.

3 A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN ISBN (ebook) DOI / Printed on acid-free paper All Rights Reserved 2001 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in Softcover reprint of the hardcover 1 st edition 2001 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

4 A nuestras familias To our families

5 Contents Preface Acknowledgments 1. PRELIMIN ARlES 1 1. Introduction 1 2. Notations and Terminology The Cylinder The Torus The sphere The cone Orbifolds Point Location and Range Searching Notes and comments EUCLIDEAN POSITION Introduction Euclidean Position Euclidean position on the cylinder and the cone Euclidean position on the torus Euclidean position on the sphere Cylindrical position in the Torus Euclidean position in Orbifolds and in General Surfaces N otes and comments CONVEX RULL Introduction 1.1 Some Extensions of Convexity Ryperconvex Rull Metrically Convex Rull Metrically Convex Rull in the Cylinder Metrically Convex Rull in the Torus Metrically Convex RuH on the Sphere Metrically convex hull on the cone Vll Xl XV

6 VUI COMPUTATIONAL GEOMETRY ON SURFACES 4. Analysis of complexity Minimum enclosing polygon Notes and comments VORONOI DIAGRAMS 61 l. Introduction Voronoi diagrams Voronoi diagrams on the cylinder Voronoi diagrams on the torus Voronoi diagrams on the sphere Voronoi diagrams on the cone Proximity problems and Voronoi diagrams 3.1 Voronoi diagrams and convex hulls Furthest point Voronoi diagram Generalized Voronoi diagrams Voronoi diagrams for a set of points and segments on the cylinder Polar diagram on the cylinder Notes and Comments RADI! 85 l. Introduction The Width of a Convex Set on the Sphere Alternative definitions of width on the sphere Algorithm of the width on the sphere Circumradius Diameter Maximum and minimum distances Notes and remarks VISIBILITY 107 l. Introduction Stabbing line segments Transversal helices Stabbing segments Visibility in the presence of obstacles Notes and comments TRIANGULATIONS 127 l. Introduction Triangulations on the cylinder Maximizing the Smallest Angle Graph of triangulations Triangulations on the sphere and on the torus Triangulations on the sphere Triangulations on the torus The graph of triangulations on non-planar surfaces 168

7 Contents IX 5. Notes and Comments References Topic Index 185 A uthor Index 189

8 Preface In the last thirty years Computational Geometry has emerged as a new discipline from the field of design and analysis of algorithms. That discipline studies geometric problems from a computational point of view, and it has attracted enormous research interest. But that interest is mostly concerned with Euclidean Geometry (mainly the plane or Euclidean 3-dimensional space). Of course, there are some important reasons for this occurrence since the first applieations and the bases of all developments are in the plane or in 3-dimensional space. But, we can find also some exceptions, and so Voronoi diagrams on the sphere, cylinder, the cone, and the torus have been considered previously, and there are many works on triangulations on the sphere and other surfaces. The exceptions mentioned in the last paragraph have appeared to try to answer some quest ions which arise in the growing list of areas in which the results of Computational Geometry are applicable, since, in practiee, many situations in those areas lead to problems of Computational Geometry on surfaces (probably the sphere and the cylinder are the most common examples). We can mention here some specific areas in which these situations happen as engineering, computer aided design, manufacturing, geographie information systems, operations research, roboties, computer graphics, solid modeling, etc. For instance, in geographic information systems and in operations research it is possible to consider worldwide questions which lead to problems in the sphere, in engineering or solid modeling are very common to deal with cases modeled by torus, cylinder or sphere. The cylinder is in general useful when we meet phenomena in which the same configuration appears in cycles. Finally, the arm of a robot does not describe, in general, an Euclidean space but a more complex algebraic surface, which in the simplest cases used to be one of the surfaces considered here. Xl

9 xii COMPUTATIONAL GEOMETRY ON SURFACES As its title declares, this book is about Computational Geometry on Surfaces, but as its subtitle specifies, the material of this book is restricted to four very specific surfaces, the sphere, the cylinder, the cone, and the torus (in fact, this is not exactly true, in so far as we study some questions concerning more general surfaces, but we can say that more than ninety per cent of the book is devoted to the four surfaces mentioned). There are two main reasons for considering those surfaces. On one hand, they are the easiest surfaces after the plane, so naturally they must be the first to be considered when we try to travel beyond the plane. And, on the other hand, we think that restricting the material to those surfaces allows us to reach in an easier way the objective that we had in mind when we decided to start this work. So it is the intention of this book to demonstrate that classical problems of Computational Geometry can be solved when the input and output data are on surfaces other than the plane, but that planar techniques cannot be always adapted successfully and new techniques must be considered. In other words, we try to show here the flavor of Computational Geometry on surfaces. Basically this book is conceived as a graduate text (in fact, its core is C.I. Grima's doctoral dissertation, although a lot of new material has been included), but we think that it can be useful to the professional in the applied fields mentioned above as weil. Finally, it can be a guide for the researeher interested in Computational Geometry 'out of the plane', he or she can find here a sort of catalog of techniques in his/her discipline adapted to the surfaces considered here. In addition, some of the techniques and methods expound here can be adapted to other spaces that have not been treated directly but that share some common characteristics with the surfaces that we consider. EquaIly, we have tried to show not only how to obtain some results, but how it is impossible to obtain those results; in other words, which planar methods are not adaptable to our surfaces. However, it must be pointed out that, as is common in this class of books, this book is not exactly a catalog of readily usable algorithms, but we focus mainly on the keys of the adaptation of planar algorithms to our surfaces. Contents of the book The three fundamental structures in Computational Geometry will be covered: convex huils, Voronoi diagrams, and triangulations. These structures will be considered in three different surfaces, each one of them

10 Preface Xlll with so me special characteristics: the sphere; the cylinder; and the torus (and occasionally the cone). In addition, some other eiassical problems will be studied: width; diameter; stabbing line; visibility; etc. We will start with a first chapter that shows some notations and preliminary concepts, especially those regarding the metric of our surfaces as well as some distinguished elements. After that chapter we will focus on one of the key concepts of the book, Euclidean position. It is easy to imagine that performing Computational Geometry on surfaces will be different from the plane only when the input of an algorithm is a set that is not very small compared to the curvature of the surface. In fact, all surfaces considered here have eiosed geodesics, and, roughly speaking, we will see throughout this book that if the diameter of the set is smaller than half of the length of those geodesics, then to compute any invariant on that set coincides with the computations needed if the input is on the plane, in this case, we will say that the set is in Euclidean position. So, throughout the book, we will center our study on the cases of sets in non-euclidean position. In the third chapter we will study the convex hull of a set on a surface. In particular, two different extensions of convexity to our surfaces will be considered, and we will construct their convex hulls in optimal time. Both extensions will be based in the set of geodesics of each surface. Moreover, we will analyze the expected time of our algorithms and we will see that they run in linear time, hut the bad news is that in most cases the convex hub is too big to he useful as a preprocessing for many problems (width, diameter, etc.), thus other tools must be constructed. In some sense this is one of the main and surprising characteristics of Computational Geometry on Surfaces, since in the plane, convex hull computations appear almost everywhere in order to solve a huge variety of problems, and we leam that, generally, in surfaces, that computation is useless for most of those problems. The fourth chapter is devoted to Voronoi diagrams. As we have said above, these structures have been considered previously by other authors. So in that chapter we will summarize the known results on the eiosest point and the farthest point Voronoi diagrams. In addition, we will complete some of those known results, studying some extensions and giving valid methods for computing those diagrams in the cases which have not been considered previously. So far all structures (convex hulls and Voronoi diagrams) studied in this work have the same complexity as in the plane, but in this chapter we will present a specific generalized Voronoi diagram (the polar diagram ) that is more complex in the cylinder than in the plane.

11 XIV COMPUTATIONAL GEOMETRY ON SURFACES The next two chapters are devoted to solving some practical problems that shape weil the methodology which must be applied when doing Computational Geometry on Surfaces. Thus in Chapter four we will consider the computation of some functionals covered by the general name of radii, such as the width of a convex set or the diameter of a point set. And in the fifth chapter we treat the stabbing line of a segment set, and some visibility problems. In all cases we will show that the solutions given in the plane are not valid anymore (for instance, as we have mentioned above, the convex hull is not a useful tool for computing the diameter of a point set), but, in spite of that, optimal solutions can be found. This is an important part because it shows, probably better than any other part, the flavor of the field. The last chapter intro duces triangulations either of a point set or of a polygon. As in other structures studied before, some important differences appear in this subject. For instance, we can give two possible definitions of triangulations, which are equivalent in the planar case (a maximal subdivision and a triangular subdivision), and it is not c1ear whether both definitions agree outside the plane (we will see that both are equivalent in the case of the cylinder or the sphere but not in the case of the torus). The other three problems studied in this chapter are: what is the domain defined by a set of sites when we perform a triangulation?; is it possible to go from a triangulation to another using diagonal flips?; and, how can we obtain optimal triangulations? we will especially study the connectivity of the graph of triangulations of polygons on surfaces, seeing that, in general, but with some very remarkable exceptions, that graph is not connected.

12 Acknowledgments It would be impossible to thank individually all our colleagues who have contributed to this book. We are grateful to all of them, even if we cannot list all their names here. However, we would like to explicitly thank (in alphabeticalorder) Jose Caceres, Javier Cobos, Carmen Cortes, Juan C. Dana, Angeles Garrido, Ferran Hurtado, Felipe Mateos, Atsuhiro Nakamoto, Lidia Ortega, Joserra Portillo, Francisco Santos and Jesus Valenzuela. Without their contributions this book would never appear at the present time. We are also grateful to Dr. Liesbeth Mol and all the staff from Kluwer Academic Publisher for their support in all stages of the creation of this book. Last, but not least, our thanks go to our families for their support and love during the hours that have led to this finished product. xv