Table Of ContentComputational Geometry on Surfaces
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.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-5908-6 ISBN 978-94-015-9809-5 (eBook)
DOI 10.1007/978-94-015-9809-5
Printed on acid-free paper
All Rights Reserved
© 2001 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 2001.
Softcover reprint of the hardcover 1s t 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.
A nuestras familias
To our families
Contents
Preface
Xl
Acknowledgments
XV
1. PRELIMIN ARlES 1
1. Introduction 1
2. Notations and Terminology 2
2.1 The Cylinder 2
2.2 The Torus 5
2.3 The sphere 8
2.4 The cone 10
3. Orbifolds 12
4. Point Location and Range Searching 14
5. Notes and comments 17
2. EUCLIDEAN POSITION 19
1. Introduction 19
2. Euclidean Position 20
2.1 Euclidean position on the cylinder and the cone 20
2.2 Euclidean position on the torus 24
2.3 Euclidean position on the sphere 25
3. Cylindrical position in the Torus 26
4. Euclidean position in Orbifolds and in General Surfaces 27
5. N otes and comments 28
3. CONVEX RULL 31
1. Introduction 31
1.1 Some Extensions of Convexity 32
2. Ryperconvex Rull 32
3. Metrically Convex Rull 36
3.1 Metrically Convex Rull in the Cylinder 37
3.2 Metrically Convex Rull in the Torus 42
3.3 Metrically Convex RuH on the Sphere 47
3.4 Metrically convex hull on the cone 51
Vll
VUI COMPUTATIONAL GEOMETRY ON SURFACES
4. Analysis of complexity 55
5. Minimum enclosing polygon 56
6. Notes and comments 58
4. VORONOI DIAGRAMS 61
l. Introduction 61
2. Voronoi diagrams 62
2.1 Voronoi diagrams on the cylinder 64
2.2 Voronoi diagrams on the torus 66
2.3 Voronoi diagrams on the sphere 67
2.4 Voronoi diagrams on the cone 68
3. Proximity problems and Voronoi diagrams 69
3.1 Voronoi diagrams and convex hulls 72
4. Furthest point Voronoi diagram 72
5. Generalized Voronoi diagrams 77
5.1 Voronoi diagrams for a set of points and segments
on the cylinder 78
5.2 Polar diagram on the cylinder 79
6. Notes and Comments 81
5. RADI! 85
l. Introduction 85
2. The Width of a Convex Set on the Sphere 87
2.1 Alternative definitions of width on the sphere 89
2.2 Algorithm of the width on the sphere 96
3. Circumradius 98
4. Diameter 98
5. Maximum and minimum distances 101
6. Notes and remarks 104
6. VISIBILITY 107
l. Introduction 107
2. Stabbing line segments 108
2.1 Transversal helices 111
2.2 Stabbing segments 120
3. Visibility in the presence of obstacles 121
4. Notes and comments 124
7. TRIANGULATIONS 127
l. Introduction 127
2. Triangulations on the cylinder 129
2.1 Maximizing the Smallest Angle 141
2.2 Graph of triangulations 151
3. Triangulations on the sphere and on the torus 158
3.1 Triangulations on the sphere 158
3.2 Triangulations on the torus 159
4. The graph of triangulations on non-planar surfaces 168
Contents IX
5. Notes and Comments 170
References 173
Topic Index
185
A uthor Index
189
Preface
In the last thirty years Computational Geometry has emerged as a new
discipline from the field of design and analysis of algorithms. That dis
cipline 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 Eu
clidean 3-dimensional space). Of course, there are some important rea
sons 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, cylin
der, 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 Com
putational 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 re
search, 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 mod
eled 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
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 re
stricted 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 sur
faces 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
