B.Sc. Engg. Thesis
Algorithms in Computational Geometry
By
Syed Ishtiaque Ahmed
Student No.: 0305003
Md. Ariful Islam
Student No.: 0305006
Submitted to
Department of Computer Science and Engineering
in partial fulfillment of the requirements for the degree of
Bachelor of Science in Computer Science and Engineering
Department of Computer Science and Engineering
Bangladesh University of Engineering and Technology (BUET)
Dhaka-1000.
Declaration
This is to certify that the work presented in this thesis entitled “Algorithms
in Computational Geometry” is the outcome of the investigation carried out
by us under the supervision of Dr. Masud Hasan, Assistant Professor, Department
of Computer Science and Engineering, Bangladesh University of Engineering and
Technology (BUET), Dhaka. It is also declared that neither this thesis nor any
part thereof has been submitted or is being currently submitted anywhere else for
the award of any degree or diploma.
−−−−−−−−−−−−−
(Supervisor)
Dr. Masud Hasan
Assistant Professor
Department of Computer Science
and Engineering (BUET), Dhaka-1000.
−−−−−−−−−−−−−
(Author)
Syed Ishtiaque Ahmed
Student No.: 0305003
Department of Computer Science
and Engineering (BUET), Dhaka-1000.
−−−−−−−−−−−−−
(Author)
Md. Ariful Islam
Student No.: 0305006
Department of Computer Science
and Engineering (BUET), Dhaka-1000.
Abstract
Two types of problems were studied in this thesis. The first one is cutting
a convex polygon out of a circle and the second one is to find out the center
of a sphere and an ellipsoid under some definite constraints.
The problem of cutting a convex polygon P out of a piece of paper Q
with minimum total cutting length is a well studied problem. Researchers
studied several variations of the problem, such as P and Q are convex or
non-convex polygons and the cuts are line cuts or rays cuts. In this thesis we
first consider the variation of the problem where Q is a circle and P is convex
polygon such that P is bounded by a half circle of Q and all the cuts are
line cuts. We give a simple linear time O(logn)-approximation algorithm for
this problem where n is the number of vertices of the polygon. We also give
an constant factor approximation algorithm for this problem with a running
time of o(n3 ). Finally, we give an constant factor approximation algorithm in
general case i.e. cutting a convex polygon out of a circle in which the convex
polygon is not necessarily cornered with a running time of O(n3 ). For both
of the cases we also give the (1 + ²) Polynomial Time Approximation Scheme
6
(PTAS)which runs in O( ²n12 )-time.
The problem of finding the center of a circle, starting from a point on
the boundary and using a limited number of operations is an interesting
problem. The searching algorithms for circle and ellipse have been given by
the researchers. In this thesis we give efficient strategies of searching the
centers of spheres and ellipsoids which inherently works for circle and ellipse
respectively.
Acknowledgments
First of all we would like to thank our supervisor, Dr. Masud Hasan, for
introducing us to the amazingly interesting world of computational geometry
and teaching us how to perform research work. Without his continuous
supervision, guidance and valuable advice, it would have been impossible
to complete the thesis. We are especially grateful to him for allowing us
greater freedom in choosing the problems to work on, for his encouragement
at times of disappointment, and for his patience with our wildly sporadic
work habits and passion for travelling. We are grateful to all other friends
for their continuous encouragement and for helping us in thesis writing. We
would like to express our gratitude to all our teachers. Their motivation and
encouragement in addition to the education they provided meant a lot to us.
Last but not least, we are grateful to our parents and to our families for their
patience, interest, and support during our studies.
I
Contents
1 Introduction
1.1 Computational Geometry . . . . . . . . .
1.2 Major Research Problems . . . . . . . . .
1.2.1 Static Problems . . . . . . . . . . .
1.2.2 Geometric query problem . . . . .
1.2.3 Dynamic Problems . . . . . . . . .
1.2.4 Variations . . . . . . . . . . . . . .
1.3 Applications of Computational Geometry
1.3.1 Design and manufacturing . . . . .
1.3.2 Graphics and visualization . . . . .
1.3.3 Information systems . . . . . . . .
1.3.4 Medicine and biology . . . . . . .
1.3.5 Physical sciences . . . . . . . . . .
1.3.6 Robotics . . . . . . . . . . . . . . .
1.3.7 Other applications . . . . . . . . .
1.4 Overview . . . . . . . . . . . . . . . . . .
1.4.1 Preliminaries . . . . . . . . . . . .
1.4.2 Previous Works . . . . . . . . . . .
1.4.3 Cutting Polygons . . . . . . . . . .
1.4.4 Searching for the centers of spheres
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and ellipsoids
2 Preliminaries
2.1 Polygon Cutting . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Definition . . . . . . . . . . . . . . . . . . . . . .
2.2 Preliminaries on searching center of sphere and ellipsoid
2.2.1 Definition . . . . . . . . . . . . . . . . . . . . . .
2.3 Preliminaries on Algorithms . . . . . . . . . . . . . . . .
2.3.1 Definition . . . . . . . . . . . . . . . . . . . . . .
II
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3 Previous Works
3.1 Previous Works on Polygon Cutting . . . . . . . . . . . . . . . . .
3.2 Previous Works on Searching for the Center of a Certain Geometric
Shape Under Definite Constraints . . . . . . . . . . . . . . . . . . .
28
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4 Cutting a Convex Polygon Out of a Circle
4.1 The Basic Problem that We Studied . . . . . . . . . . . . . . . . .
4.2 Algorithm for Cornered Convex Polygon . . . . . . . . . . . . . . .
4.2.1 Algorithm 1 . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Algorithm 2 . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Algorithms for General Case . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Separating Minimum Rectangle Bounding P Using Rotating
Calipers . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Algorithm 1 . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.3 Algorithm 2 . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.4 Algorithm 3 . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5 Searching for the Center of Spheres and Ellipsoids
5.1 Searching the Center of Spheres . . . . . . . . . . . . . . . .
5.1.1 Problem Description . . . . . . . . . . . . . . . . . .
5.1.2 Mathematical Description . . . . . . . . . . . . . . .
5.1.3 Proposed Strategy . . . . . . . . . . . . . . . . . . .
5.1.4 How to detect a direction which is inside the sphere
5.1.5 Correctness Proof . . . . . . . . . . . . . . . . . . .
5.2 Competitive Ratio . . . . . . . . . . . . . . . . . . . . . . .
5.3 Detection Stage of Searching the Center of an Ellipsoid . .
5.3.1 Problem Description . . . . . . . . . . . . . . . . . .
5.3.2 Our Work . . . . . . . . . . . . . . . . . . . . . . . .
5.3.3 Proposed Strategy . . . . . . . . . . . . . . . . . . .
5.3.4 Correctness . . . . . . . . . . . . . . . . . . . . . . .
5.4 Correction Stage . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1 Proposed Correction Strategy . . . . . . . . . . . . .
5.4.2 Convergence of the strategy . . . . . . . . . . . . . .
5.4.3 Performance Analysis . . . . . . . . . . . . . . . . .
5.4.4 Worst Case Analysis . . . . . . . . . . . . . . . . . .
5.4.5 For Ellipsoids . . . . . . . . . . . . . . . . . . . . . .
5.5 Use of Volume Knobs . . . . . . . . . . . . . . . . . . . . .
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III
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6 Conclusion and Future Works
6.1 Cutting a cornered polygon out of a circle
6.2 Cutting a centered polygon out of a circle
6.3 Searching for the center of a sphere . . . .
6.4 Searching for the center of an ellipsoid . .
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Index
85
Bibliography
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IV
List of Figures
2.1
2.2
2.3
2.4
(a) Cornered Convex Polygon (b) Centered Convex Polygon.
(a) l1 and l2 are line cuts, (b)l3 and l4 are not line cuts . . .
Ray cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Axis Diameter . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.1
3.2
Problems associated withe the two dimensional search techniques.
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31
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4.1
(a) A convex polygon. (b,c) Two different cutting sequences to cut
P out of Q. The cost of the sequence in (b) is more than that in
the (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Some preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Critical edge and critical vertex. . . . . . . . . . . . . . . . . . . .
Separating P from c by using more than one cut increases the cutting cost. The cutting length is shown in bold lines. . . . . . . . .
Triangle separation. . . . . . . . . . . . . . . . . . . . . . . . . . .
Obtaining obtuse triangle(s) from Ta . . . . . . . . . . . . . . . . . .
Curving phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Lower bound for C ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . .
shortest chord through a point inside a circle is perpendicular o the
line joining the center and that point. . . . . . . . . . . . . . . . .
(a) Rotating Calipers (b) Rotating two orthogonal pairs of rotating
calipers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Upper bound of the cut length of minimum area rectangle bounding
P when P is centered. . . . . . . . . . . . . . . . . . . . . . . . . .
possible shape of bounding box . . . . . . . . . . . . . . . . . . . .
cutting strategy for O(log n) approximation ratio when P is cornered.
cutting out minimum area rectangle with constant cut length.t . .
Determining the portal points. . . . . . . . . . . . . . . . . . . . .
Determining the estimated cost. . . . . . . . . . . . . . . . . . . . .
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
V
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5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
5.15
5.16
5.17
5.18
5.19
5.20
Searching the center of a sphere . . . . . . . . . . . . . . . . . . . .
Proposed strategy. . . . . . . . . . . . . . . . . . . . . . . . . . . .
How to detect a direction which is inside the sphere . . . . . . . .
Cross section of a sphere . . . . . . . . . . . . . . . . . . . . . . . .
The straight line joining the center of the sphere and the center
of the circular plane produced by a cross-section of that sphere is
perpendicular to that plane. . . . . . . . . . . . . . . . . . . . . . .
Perpendicular plane to a circular plane . . . . . . . . . . . . . . . .
Figure for Lemma 5.1.4. . . . . . . . . . . . . . . . . . . . . . . . .
Correctness of the strategy . . . . . . . . . . . . . . . . . . . . . .
Correctness proof . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Competitive ratio calculation . . . . . . . . . . . . . . . . . . . . .
Figure for Lemma 5.3.2 . . . . . . . . . . . . . . . . . . . . . . . .
Figure for Lemma 5.3.3 . . . . . . . . . . . . . . . . . . . . . . . .
Proposed strategy . . . . . . . . . . . . . . . . . . . . . . . . . . .
Electromagnetic field . . . . . . . . . . . . . . . . . . . . . . . . . .
Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Revised elliptical model . . . . . . . . . . . . . . . . . . . . . . . .
Assumed model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dead regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wrong result by proposed strategy . . . . . . . . . . . . . . . . . .
Correction strategy . . . . . . . . . . . . . . . . . . . . . . . . . . .
VI
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Chapter 1
Introduction
1.1
Computational Geometry
Computational geometry is a branch of computer science devoted to the
study of algorithms which can be stated in terms of geometry. Some purely
geometrical problems arise out of the study of computational geometric algorithms, and such problems are also considered to be part of computational
geometry.
According to the Encyclopedia, the definition of the computational geometry is
The study of algorithms for combinatorial, topological, and metric problems concerning sets of points, typically in Euclidean
space. Representative areas of research include geometric search,
convexity, proximity, intersection, and linear programming.
The main impetus for the development of computational geometry as
a discipline was progress in computer graphics, computer-aided design and
manufacturing (CAD/CAM), but many problems in computational geometry
are classical in nature.
1
Other important applications of computational geometry include robotics
(motion planning and visibility problems), geographic information systems
(GIS) (geometrical location and search, route planning), integrated circuit
design (IC geometry design and verification), computer-aided engineering
(CAE) (programming of numerically controlled (NC) machines).
The main branches of computational geometry are:
• Combinatorial computational geometry, also called algorithmic geometry, which deals with geometric objects as discrete entities. A groundlaying book in the subject by Preparata and Shamos dates the first use
of the term ”computational geometry” in this sense by 1975.
• Numerical computational geometry, also called machine geometry, computeraided geometric design (CAGD), or geometric modeling, which deals
primarily with representing real-world objects in forms suitable for computer computations in CAD/CAM systems. This branch may be seen
as a further development of descriptive geometry and is often considered a branch of computer graphics or CAD. The term ”computational
geometry” in this meaning has been in use since 1971
1.2
Major Research Problems
The core problems in computational geometry may be classified in different
ways, according to various criteria. The following general classes may be
distinguished.
1.2.1
Static Problems
In the problems of this category, some input is given and the corresponding
output needs to be constructed or found. Some fundamental problems of this
type are:
2
• Convex Hull:Given a set of points, find the smallest convex polyhedron/polygon containing all the points.
• Line segment intersection:Find the intersections between a given
set of line segments.
• Polygon cutting: Cutting a polygon out of another geometric shape
with minimum total cutting length.
• Delaunay triangulation
• Voronoi diagram:Given a set of points, partition the space according
to which point is closest.
• Linear programming
• Closest pair of points: Given a set of points, find the two with the
smallest distance from each other.
• Euclidean shortest path: Connect two points in a Euclidean space
(with polyhedral obstacles) by a shortest path.
• Polygon triangulation:Given a polygon, partition its interior into
triangles
The computational complexity for this class of problems is estimated by
the time and space (computer memory) required to solve a given problem
instance.
1.2.2
Geometric query problem
In geometric query problems, commonly known as geometric search problems,
the input consists of two parts: the search space part and the query part,
3
which varies over the problem instances. The search space typically needs to
be preprocessed, in a way that multiple queries can be answered efficiently.
Some fundamental geometric query problems are:
• Range searching: Preprocess a set of points, in order to efficiently
count the number of points inside a query region.
• Point location: Given a partitioning of the space into cells, produce a
data structure that efficiently tells in which cell a query point is located.
• Nearest neighbor: Preprocess a set of points, in order to efficiently
find which point is closest to a query point.
• Ray tracing: Given a set of objects in space, produce a data structure
that efficiently tells which object a query ray intersects first.
If the search space is fixed, the computational complexity for this class of
problems is usually estimated by:
• the time and space required to construct the data structure to be
searched in
• the time (and sometimes an extra space) to answer queries.
1.2.3
Dynamic Problems
A yet another major class are the dynamic problems, in which the goal is to
find an efficient algorithm for finding a solution repeatedly after each incremental modification of the input data (addition or deletion input geometric
elements). Algorithms for problems of this type typically involve dynamic
data structures. Any of the computational geometric problems may be converted into a dynamic one. For example, the range searching problem may
4
be converted into the dynamic range searching problem by providing for addition and/or deletion of the points. The dynamic convex hull problem is
to keep track of the convex hull, e.g., for the dynamically changing set of
points, i.e., while the input points are inserted or deleted.
The computational complexity for this class of problems is estimated by:
• the time and space required to construct the data structure to be
searched in
• the time and space to modify the searched data structure after an
incremental change in the search space
• the time (and sometimes an extra space) to answer a query
1.2.4
Variations
Some problems may be treated as belonging to either of the categories, depending on the context. For example, consider the following problem.
• Point in polygon: Decide whether a point is inside or outside a given
polygon.
In many applications this problem is treated as a single-shot one, i.e., belonging to the first class. For example, in many applications of computer graphics
a common problem is to find which area on the screen is clicked by a mouse
cursor. However in some applications the polygon in question is invariant,
while the point represents a query. For example, the input polygon may represent a border of a country and a point is a position of an aircraft, and the
problem is to determine whether the aircraft violated the border. Finally,
in the previously mentioned example of computer graphics, in CAD applications the changing input data are often stored in dynamic data structures,
which may be exploited to speed-up the point-in-polygon queries.
5
In some contexts of query problems there are reasonable expectations on
the sequence of the queries, which may be exploited either for efficient data
structures or for tighter computational complexity estimates. For example,
in some cases it is important to know the worst case for the total time for
the whole sequence of N queries, rather than for a single query. See also
”amortized analysis”.
1.3
Applications of Computational Geometry
Computational geometry emerged from the field of algorithms design and
analysis in the late 1970s. It has grown into a recognized discipline with its
own journals, conferences, and a large community of active researchers. The
success of the field as a research discipline can on the one hand be explained
from the beauty of the problems studied and the solutions obtained, and, on
the other hand, by the many application domains—computer graphics, geographic information systems (GIS), robotics, and others—in which geometric
algorithms play a fundamental role.
For many geometric problems the early algorithmic solutions were either
slow or difficult to understand and implement. In recent years a number of
new algorithmic techniques have been developed that improved and simplified
many of the previous approaches.
1.3.1
Design and manufacturing
Architecture
The influence of computational geometry in architecture is mainly indirect
but multifaceted, via computer graphics for architectural visualization, vir-
6
tual reality for simulated walkthroughs, computer aided design, geographic
information systems for land use planning, and finite element methods for
structural simulation (see [Alexander, Black, and Tsutsui, ”The Mary Rose
Museum”, Oxford 1995] for a nice example of a design informed by this last
consideration). However it seems likely that within each of those areas, special problems arise in dealing with architectural information, and that more
direct connections can also be made.
Assembly planning
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.
Computer aided manufacturing
Computer aided manufacturing concerns the use of algorithms for planning
and controlling fabrication processes. These algorithms are also important
to computer aided design, since it is important to be able to test manufacturability as part of the design process. Specific areas of computer aided
manufacturing listed as separate topics here include grasping and fixturing,
metrology, robot motion planning, and textile layout.
7
Fixturing
Although one version of this problem comes from robotics and the other
comes from manufacturing, the aim of both is the same: to construct a
placement of obstacles (robot fingers or fixtures) preventing the movement
of some part.
Metrology
Metrology is the science of measurement of objects at all scales. The NIST’s
project on ”Computational Geometry and Metrology” includes the use of
geometric techniques including mesh generation in computer simulations to
study the accuracy and precision of coordinate measuring machines (machines which measure the dimensions of parts in a three-dimensional coordinate system). There are also connections with geographic information
systems relating to problems of surveying or otherwise measuring large areas
of land.
One type of problem of particular interest in manufacturing is determining
from a sequence of measurements the tolerance of a part; that is, its departure
from the Platonic ideal form of its design. There has been some related work
in the computational geometry community, on problems such as constructing
the minimum width annulus containing the boundary of an input figure (a
measure of its roundness) however there has been little systematic treatment
of such problems. Chee Yap recently spoke on this subject at the 5th MSI
Worksh. on Computational Geometry; his talk pointed out the possibility
of designing algorithms that take advantage of some known structure in a
sequence of measurements, and of coupling measurement and computation
in an adaptive probing algorithm.
8
Nanotechnology
Connections have been growing recently between the molecular modeling
community and the computational geometry. Many questions in molecular modeling can be understood geometrically in terms of arrangements of
spheres in three dimensions. Problems include computing properties of such
arrangements such as their volume and topology, testing intersections and
collisions between molecules, finding offset surfaces (related to questions of
accessability of molecule subregions to solvents such as water), data structures for computing interatomic forces and performing molecular dynamics
simulations, and computer graphics algorithms for rendering molecular models accurately and efficiently (taking advantage of their special geometric
structure). Classical molecular modeling has dealt with biological molecules
which generally have a tree-like structure, but applications to nanotechnology
require dealing with more complicated diamond-like structures; it is unclear
to what extent this affects the relevant algorithms.
Solid modelling and constructive solid geometry
Geometric questions related to solid modelling include conversion between
different representations including boundary nets, constructive solid geometry (representations involving Boolean combinations of simple base shapes),
and hierarchical decomposition; combination (such as intersection and union)
of shapes; blending surfaces; and data structures for graphical rendering of
models. More could be done to connect the models used in computational geometry (typically polyhedra) with those in computer aided design (typically
involving splines or other higher order curves and surfaces).
9
Textile layout
The problem here is, given a pattern in the form of a collection of shapes
(polygons), to arrange the shapes so they can be cut from a piece of fabric
with as little wasted fabric as possible. Essentially this is a two-dimensional
bin-packing problem with various complications: the width of the fabric is
fixed and one intends to minimize length; the shapes can sometimes be placed
only in a small number of possible orientations so that the fabric pattern lines
up correctly; the shapes can be quite far from convex; and (for leather) the
hides out of which the shapes must be cut can be quite variable in shape
and quality. This area is also related to more tractible geometric problems
such as testing whether one polygon can be translated to fit inside another.
Textile layout has some visibility in the geometry community largely due to
the efforts of Victor Milenkovic, now at the University of Miami.
VLSI design
VLSI circuit design involves many levels of abstraction; the most geometric
of these is the physical design level, in which circuits are represented as
(typically) collections of rectangles representing layers of various materials
on a chip. Most of the layout, routing, and verification problems at this
level involve some amount of computational geometry. At another level of
abstraction, simulation of the electronic and physical properties of a chip
involves interesting questions of mesh generation, in which there may be
some advantage to be taken from the fact that the geometry is rectilinear
and planar or nearly planar.
10
1.3.2
Graphics and visualization
Computer graphics
This area is quite closely connected with computational geometry; for instance ACM TOG often publishes computational geometry papers. Geometric research topics in graphics include data structures for ray tracing,
clipping, and radiosity; hidden surface elimination algorithms; automatic
simplification for distant objects; morphing; clustering for color quantization; converting triangulated surfaces to strips of triangles (some graphics
engines take inputs in this form to save bandwidth); and construction of
low-discrepancy point sets (for oversampling to eliminate Moire effects in
ray tracing). Specialized applications of graphics in which other geometric
ideas are needed include architecture, virtual reality, and video game programming. This area is also related to mesh generation in several ways:
both fields use triangulation algorithms to partition complex surfaces into
simpler pieces, and radiosity calculations in graphics are essentially a special
case of the finite element method.
Graph drawing
Graph drawing can be thought of as a form of data visualization, but unlike
most other types of visualization the information to be visualized is purely
combinatorial, consisting of edges connecting a set of vertices. Applications
of graph drawing include genealogy, cartography (subway maps form one of
the standard examples of a graph drawing), sociology, software engineering
(visualization of connections between program modules), VLSI design, and
visualization of hypertext links. Typical concerns of graph drawing algorithms are the area needed to draw a graph, the types of edges (straight lines
or bent), the number of edge crossings for nonplanar graphs, separation of
vertices and edges so they can be distinguished visually, and preservation of
11
properties such as symmetry and distance. The area has a large literature,
concentrated in the annual Graph Drawing symposia, and I won’t try to link
here to all available research projects and papers on the subject, only those
with some particular historical or application interest.
Virtual reality
This is to a large extent a branch of graphics, with the focus on generation of
real-time three-dimensional views of dynamic scenes. This is similar to video
game programming but with a greater emphasis on accurate rendering and
physical modeling. Problems of interest here include simplification of objects
(to save time by avoiding the rendering of sub-pixel details) and collision
detection. There are also connections with graph drawing for construction
of 3d virtual environments that represent combinatorial structures.
Video game programming
Video game programming is closely related to computer graphics, but has its
own special needs – often speed is much more important than verisimilitude.
Therefore rather than using slow but accurate graphics techniques such as
radiosity, video game engines are typically based on simpler techniques such
as ray casting, using binary space partitions as a primary geometric data
structure. Simulation of non-player-characters in video games often requires
some geometric computation, for instance of short paths around obstacles.
There may also be the possibility of a connection with graph drawing, for
automated layout of adventure game maps.
12
1.3.3
Information systems
Cartography and geographic information systems
A geographic information system (GIS) is simply a database of information
about natural and man-made geographic features such as roads, buildings,
mountains... These systems can be used for making maps, but also for analyzing data e.g. for facility location. There has been some communication
between the geometry and GIS communities (e.g. geographer Michael Goodchild gave an invited lecture on ”Computational Geography” at the 11th
ACM Symp. Comp. Geom.) but more could be done to bring them together. Geographic problems already visible in the geometry community
include interpolation of surfaces from scattered data, overlaying planar subdivisions, hierarchical representations of terrain information, boundary simplification, lossy compression of elevation data (e.g. by using a piecewise
linear approximation with few facets), and map labelling. Other interesting geometric issues include handling of approximate and inconsistent data,
matching similar features from different databases, compression of large geographic databases, cooperation between raster and vector representations,
visibility analysis, and generation of cartograms (maps with area distorted
to represent other information such as population).
A particularly important geometric data structure in geographic analysis is the Voronoi diagram, which has been used for to identify regions of
influence of clans and other population centers, model plant and animal competition, piece together satellite photographs, estimate ore reserves, perform
marketing analysis, and estimate rainfall.
Data mining and multidimensional analysis
Data mining is the process of querying large databases (such as point-of-sale
records) with the aim of distilling from them broad patterns and smaller col13
lections of useful information. There seems to be little work in this area from
the computational geometry perspective, but there are likely good geometric
problems to be found in it. One such problem is coping with high-dimensional
data, by condensing information down to a small number of relevant dimensions and applying geometric clustering techniques. Any algorithm to be
used in this context must be fast, but it is perhaps more important to deal
with amounts of data that do not fit in memory, and keep to a minimum
the total number of I/O operations needed, as has been considered in recent work of Goodrich et al. (”External-memory computational geometry”,
34th FOCS, 1993, 714-723). There are also interesting connections with geographic information systems, which face similar problems of querying large
databases with more explicitly geometric content.
1.3.4
Medicine and biology
Biochemical modelling
Connections have been growing recently between the molecular modeling
community and the computational geometry. Many questions in molecular modeling can be understood geometrically in terms of arrangements of
spheres in three dimensions. Problems include computing properties of such
arrangements such as their volume and topology, testing intersections and
collisions between molecules, finding offset surfaces (related to questions of
accessability of molecule subregions to solvents such as water), data structures for computing interatomic forces and performing molecular dynamics
simulations, and computer graphics algorithms for rendering molecular models accurately and efficiently (taking advantage of their special geometric
structure). Classical molecular modeling has dealt with biological molecules
which generally have a tree-like structure, but applications to nanotechnology
require dealing with more complicated diamond-like structures; it is unclear
14
to what extent this affects the relevant algorithms.
Biology
Computational biology (Lior Pachter): Computational biology is a new discipline whose domain is the quantitative analysis of biological data, the elucidation of biological principles, and the engineering of biological systems. The
central themes of computational biology have been shaped by radical breakthroughs in biotechnology, which have enabled high throughput gathering
of data about DNA, RNA, and protein in cells and systems. Our lectures
will focus on DNA, and we will explain the relevance of algebraic statistical
models to genome sequence analysis.Computational geometry is often used
as a tool for computational biology.
Medical imaging
A key problem in medical computation is reconstruction of shapes (of organs,
bones, tumors etc) from lower dimensional information such as CAT scans
and sonograms. The CAT scan information is originally one-dimensional, but
is transformed into two-dimensional slices by signal processing techniques.
However the reconstruction of three-dimensional shapes from slices becomes a
more geometric problem, which can be abstracted as that of finding a surface
connecting a collection of contour lines or data points. A different problem
relates to compression of medical images for transmission and storage; this
differs from most other applications of image compression in that little or no
loss of information can be tolerated.
15
1.3.5
Physical sciences
Astronomy
Computational geometry problems arise in astronomy in observation planning, shape reconstruction for irregular bodies such as asteroids, clustering
for galaxy distribution analysis, and hierarchical decomposition of point sets
for n-body simulations.
Molecular modelling
Connections have been growing recently between the molecular modelling
community and the computational geometry. Many questions in molecular modelling can be understood geometrically in terms of arrangements of
spheres in three dimensions. Problems include computing properties of such
arrangements such as their volume and topology, testing intersections and
collisions between molecules, finding offset surfaces (related to questions of
accessability of molecule subregions to solvents such as water), data structures for computing interatomic forces and performing molecular dynamics
simulations, and computer graphics algorithms for rendering molecular models accurately and efficiently (taking advantage of their special geometric
structure). Classical molecular modeling has dealt with biological molecules
which generally have a tree-like structure, but applications to nanotechnology
require dealing with more complicated diamond-like structures; it is unclear
to what extent this affects the relevant algorithms.
Scientific computation
Computation and simulation has rapidly become a third branch of science,
standing next to the older branches of theory and experiment. Much scientific
computation is done with the finite element method or related techniques,
requiring a geometric mesh generation stage to set up the computation, and
16
often involving more geometry in finding sparse decompositions of the resulting matrices. Other scientific problems involve more directly the geometry
of polyhedra or local neighbor interactions. We include separate sections for
astronomy, biology, earth sciences, and molecular modelling.
1.3.6
Robotics
Computer vision
Many computer vision papers tend to apply signal processing techniques
rather than discrete geometry, to be very heuristic, or to simply describe
coordinate systems or other geometric modelling tools. However there has
been some work in the computational geometry community on exact solution
of geometric pattern matching problems associated with vision, for instance
by Dan Huttenlocher’s group at Cornell. Several vision projects use Voronoi
diagrams to represent the structure of the image under study.
Grasp planning
Although one version of this problem comes from robotics and the other
comes from manufacturing, the aim of both is the same: to construct a
placement of obstacles (robot fingers or fixtures) preventing the movement
of some part.
Robot motion planning
The study of exact algorithms for robot motion planning forms a major
subarea of computational geometry, with connections also to symbolic and
algebraic computation. Note that motion planning is useful not only for computer control of actual robots; in his invited talk at FOCS 1995, Jean-Claude
Latombe showed impressive videos of applications of the same techniques in
assembly planning and to computer animation.
17
1.3.7
Other applications
Character recognition
Compiler design
Control theory
Signal processing
Sociology
Earth sciences
Metallurgy
Telecommunication
Timber processing
Typography
Windowing systems
1.4
Overview
In this section we give a short overview of the contents of this thesis.
1.4.1
Preliminaries
Computational geometry is an interdisciplinary subject which combines knowledge and techniques of geometry and computer science among other subjects.
To understand the motivation behind the computational problems in this
field, it is necessary for a computer scientist to understand concept of geometry properly. In Chapter 2 we have presented some fundamental concepts of
18
geometry that we hope will provide insight into the geometrical applications
of computational problems discussed in this thesis.
1.4.2
Previous Works
In chapter 3, we will highlight some of the major works so far done in our
research area. We will divide this section into two parts as we have mainly
focused on two fields of computational geometry in our thesis. In first part we
will discuss about the previous works on polygon cutting and finding center
of some geometric shape like circle, ellipse etc in the next section.
1.4.3
Cutting Polygons
In this thesis, we broadly studied how to cut a geometrical shape out of
another. We considered the geometric shape as the polygons, specially convex
polygon. In chapter 4 we will give some algorithms to cut a convex polygon
out of a circle. We will give logarithmic and constant factor approximation
algorithm for this problem.
We solved the problem with some elementary geometry and some techniques such as rotating calipers.
1.4.4
Searching for the centers of spheres and ellipsoids
Finding center of ellipse and sphere is an interesting as well as important
computational geometric problem. Though finding center of a circle has been
approached, finding center of an ellipse and a sphere is yet to be approached.
We are the first to approach this problem and we will develop some algorithm
regarding this problem in chapter 5.
19
Chapter 2
Preliminaries
2.1
Polygon Cutting
In this section we will describe some of preliminaries regarding the polygon
cutting algorithms.
2.1.1
Definition
Convex polygon
A convex polygon is a simple polygon whose interior is a convex set. The
following properties of a simple polygon are all equivalent to convexity:
• Every internal angle is less than 180 degrees.
• Every line segment between two vertices remains inside or on the boundary of the polygon.
A simple polygon is strictly convex if every internal angle is strictly less than
180 degrees. Equivalently, a polygon is strictly convex if every line segment
between two nonadjacent vertices of the polygon is strictly interior to the
polygon except at its endpoints.
20
Every non-degenerate triangle is strictly convex.
Concave polygon
A polygon that is not convex is called concave or reentrant. A concave
polygon will always have an interior angle with a measure that is greater
than 180 degrees.
It is possible to cut a concave polygon into a set of convex polygons. A
polynomial-time algorithm for finding a decomposition into as few convex
polygons as possible is described by Chazelle & Dobkin .
Vertex Cut
A line cut is a vertex cut through a vertex v of P if it is tangent to P at v.
Edge Cut
Similarly, a line cut is an edge cut through an edge e of P if it contains e.
Cornered convex polygon
A cornered convex polygon P inside a circle Q is a convex polygon which is
positioned completely on one side of a diameter of Q.
Centered Convex Polygon
A convex polygon P inside a circle Q is called centered if it contains the
center of Q. See Figure 2.1(b). See Figure 2.1(a). If the center of Q is on
the perimeter of P , we still call P as a centered polygon.
21
(a)
(b)
Figure 2.1: (a) Cornered Convex Polygon (b) Centered Convex Polygon.
Cutting sequence
A cutting sequence is a sequence of cuts such that after the last cut in the
sequence we have P = Q. In this thesis we consider the problem of cutting P
out of Q by line cuts with total cutting cost as small as possible. See Figure
1(b).
Line Cut
line cut is a line that does not run through the polygon P and divides Q into
two pieces. After a cut is made, Q is updated to the piece containing P .
Q
Q
l3
l1
l2
P
P
l4
Figure 2.2: (a) l1 and l2 are line cuts, (b)l3 and l4 are not line cuts
22
Ray cut
A cut is a ray cut if starts from a point on the perimeter of a polygon or a
circle and can or can not be ended at a point inside that polygon or circle.
Figure 2.3: Ray cut
Cut and uncut edges
The edges of P through which an edge cut has passed are called to be cut
edges of P and other edges of P are called to be uncut edges of P.
Visible edge
An edge e of P is visible from c if for every point p of e the line segment cp
does not intersect P in any other point.c is the center of Q.
2.2
Preliminaries on searching center of sphere
and ellipsoid
2.2.1
Definition
In this subsection we will give some definitions related to sphere and ellipsoid.
23
Great Circle
The circle which has the same center and diameter as a sphere is called
the great circle of that sphere. The plane with having a great circle as its
boundary is called great circular plane.
Axis Diameter
The diameter of a sphere which goes through the center of a circular plane
produced by a cross-section of that sphere and perpendicular to it, is called
the Axis Diameter of that sphere for that particular circular plane. In fig:8
Figure 2.4: Axis Diameter
ABC is a circular plane produced by a cross-section to the sphere S. PQ is a
diameter of S which goes through the center of the circular plane ABC and
perpendicular to it. As a result, PQ is the axis-diameter of the sphere S for
the circular plane ABC.
Perpendicular Great Circular Plane
In the same sphere, if G is a great circular plane which is perpendicular to a
circular plane, P produced by a cross-section of that sphere, G is called the
perpendicular great circular plane of P.
24
2.3
Preliminaries on Algorithms
We will give some important terminologies used widely in algorithms.
2.3.1
Definition
In this subsection we give some preliminaries on algorithms.
NP-Complete Problem
In computational complexity theory, the complexity class NP-complete (abbreviated NP-C or NPC, NP standing for Nondeterministic Polynomial time)
is a class of problems having two properties:
• Any given solution to the problem can be verified quickly (in polynomial
time); the set of problems with this property is called NP.
• If the problem can be solved quickly (in polynomial time), then so can
every problem in NP.
Although any given solution to such a problem can be verified quickly, there
is no known efficient way to locate a solution in the first place; indeed, the
most notable characteristic of NP-complete problems is that no fast solution
to them is known. That is, the time required to solve the problem using any
currently known algorithm increases very quickly as the size of the problem
grows.As a result, the time required to solve even moderately large versions
of many of these problems easily reaches into the billions or trillions of years,
using any amount of computing power available today. As a consequence,
determining whether or not it is possible to solve these problems quickly is
one of the principal unsolved problems in computer science today.
25
NP-Hard Problem
NP-hard (nondeterministic polynomial-time hard), in computational complexity theory, is a class of problems informally “at least as hard as the
hardest problems in NP.” A problem H is NP-hard if and only if there is an
NP-complete problem L that is polynomial time Turing-reducible to H, i.e.
L 6T H. In other words, L can be solved in polynomial time by an oracle
machine with an oracle for H. Informally we can think of an algorithm that
can call such an oracle machine as subroutine for solving H, and solves L
in polynomial time if the subroutine call takes only one step to compute.
NP-hard problems may be of any type: decision problems, search problems,
optimization problems.
Optimization Problem
A computational problem in which the object is to find the best of all possible
solutions. More formally, find a solution in the feasible region which has the
minimum (or maximum) value of the objective function.
Optimal Solution
A solution to an optimization problem which minimizes (or maximizes) the
objective function.
Approximation Algorithm
An algorithm to solve an optimization problem that runs in polynomial time
in the length of the input and outputs a solution that is guaranteed to be
close to the optimal solution. ”Close” has some well-defined sense called the
performance guarantee.
26
Approximation Ratio
It is the ratio between the approximation result and the optimal solution
of an optimization problem. If the approximation result of an optimization
problem is C and optimal solution to that problem is C ∗ , then the approximation ratio is,
a = C/C ∗
Constant Factor Approximation Algorithm
The approximation ratio of a constant factor algorithm is constant.
An approximation algorithm is called a c-approximation algorithm for
some constant c if it can be proven that the solution that the algorithm finds
is at most c times worse than the optimal solution. Here, c is called the
approximation ratio. For example, the vertex cover problem and travelling
salesman problem with triangle inequality each have simple 2-approximation
algorithms. In contrast to that it’s proven that the travelling salesman problem with arbitrary edge-lengths can not be approximated with approximation
ratio bounded by a constant as long as the Hamiltonian-path problem can
not be solved in polynomial time.
Polynomial Time Approximation Scheme(PTAS)
A Polynomial time approximation scheme (abbrebiated PTAS) is an algorithm which takes an instance of an optimization problem and a parameter
ε > 0 and, in polynomial time, produces a solution that is within a factor εof
being optimal. For example, for the Euclidean travelling salesman problem,
a PTAS would produce a tour with length at most (1 + ε)L, with L being
the length of the shortest tour.
27
Chapter 3
Previous Works
In this chapter we will briefly discuss about some of the previous works related
to our research topics which motivated us to carry on our research interests.
In the first section, we will discuss about the previous works behind polygon
cutting and in the subsequent section, we will discuss the previous works in
the field of the geometry to find out the center of some geometric shapes like
circle, ellipse etc.
3.1
Previous Works on Polygon Cutting
Overmars et.al were the first to introduce this problem in 1985. In [13]
Overmars et.al. considered the following problem:
Given a polygonal piece of paper Q with a polygon P drawn on
it, cut P out of the piece of paper in the cheapest possible way.
They define the cut as line cut and the cost of the cut is the length of the
intersection of the cutline with the piece of paper.They asked for the cutting
sequence such that the total cost is minimum and defined this sequence as
the optimal cutting sequence. They showed in their paper that when P is
28
convex and the piece paper is convex then there does exist a finite optimal
cutting sequence. They showed the existence of a finite optimal cutting
sequence with O(n) cuts, all touching P .But in the case the piece of paper is
not convex the problem seems much harder.When only edge cuts are allowed
Overmars and Welzl proved an optimal cutting sequence can be computed
in O(m + n3 ) time by a simple dynamic programming algorithm.
The problem seems to be more difficult if the cuts are more general, i.e.,
they are not restricted to touch only the edges of P . In that case Bhaduri
and Chandrasekaran showed that the problem has optimal solutions that
lie in the algebraic extension of the input data field [3] and due to this
algebraic nature of this problem, an approximation scheme is the best that
one can achieve [3]. They also gave an approximation scheme [3] with
pseudo-polynomial running time.
After the indication of Bhaduri and Chandrasekaran [3] on the hardness of
the problem several researchers have given polynomial time approximation algorithms for this problem. Dumitrescu proposed an O(log n)-approximation
algorithm with O(mn + n log n) running time [10, 11]. Then Daescu and
Luo [8] gave the first constant factor approximation algorithm for this problem. Their algorithm has a running time of O(n3 + (n + m) log (n + m)).
The best known constant factor approximation algorithm is due to Tan
with an approximation ratio of 7.9 and running time of O(n3 + m) [16].
In the same paper [16], he also proposed an O(log n)-approximation algorithm with improved running time of O(n + m). As the best known result
so far, very recently, Bereg, Daescu and Jiang gave a simple O(m +
n6
)²12
time (1 + ²)-approximation algorithm for this problem [2]. This problem has
also been studied when Q is a non-convex polygon [13], P is a non-convex
polygon [7, 8, 16], and/or the cuts are ray cuts [7, 8, 16].
29
3.2
Previous Works on Searching for the Center of a Certain Geometric Shape Under
Definite Constraints
Biedl et al.
[4] first posed the question of finding the center of a circle,
starting with a point on the boundary, where the only allowed operations are
an interior and exterior query, marking a point, and computing the line or
midpoint between any two marked points. The problem was motivated by
the problem of searching a skier who is victim of an avalanche. The skiers
bear with them small radios which are known as transceivers. Transceivers
have two modes, transmitting and receiving. Whenever a skier becomes
victim of an avalanche he sets his transmitter on transmitting mode and the
rest of the skiers set their transmitter on receiving mode. Then the mission
is to find out the victim as quickly as possible because the victim cannot
survive for a long under the snow. The transceiver, in its transmitting mode,
continuously transmits signal which has a fixed range. Any transceiver in
receiving mode can detect this signal if it is inside that range. But if the
receiving transceiver goes beyond that range, the signal fades. The skiers
who try to find out the victim follow the signal and try to reach the victim
as quickly as possible. The transmitting range of the transceiver of the
victim was at first thought to be bounded by a circle with the victim at the
center of it and in the Avalanche Handbook, there was a solution for how
to reach the center of that circle to rescue that victim. Biedl et al. showed
a more efficient strategy for searching the center of a circle which is known
as right angle strategy. Later in his paper he claimed that the boundary of
the range of the transmitter can better be modelled as an ellipse and he
posed an open problem for an efficient strategy of searching the center of an
ellipse. Later Michael A. Burr [5] solved that problem with parallel chord
30
method. In both of their works they considered the searching space to be two
dimensional and did not give any clue of what to do in case of searching in
three dimensional spaces. But the avalanche actually occurs on the slides of a
hill or a mountain and the three dimensional model is much more appropriate
in these cases. Moreover helicopters are used for rescuing skiers nowadays
which can easily move in three dimensions. So, an efficient three dimensional
model for rescuing the avalanche victim is will be very much helpful and
useful. Again, if we consider a victim under water or in space, we can not
use the two dimensional model at all. So, the efficient searching technique
should handle three dimensional cases which inherently takes care of the
cases of two dimensions. Here we showed efficient strategies for searching in
three dimensional spaces in two models, first spherical and then ellipsoidal
corresponding to the circular model proposed by the Avalanche handbook
[14] and T. Beidl [4]respectively. Again, the strategy for elliptical model
was not complete as the ellipse has some dead areas where no signal can
reach. So, the elliptical model proposed by Michael A. Burr [5] does not
work perfectly in some cases due to dead areas.
Figure 3.1: Problems associated withe the two dimensional search techniques.
31
T. Beidl et al. [4] proposed the right angle strategy to find out the center
of a circle for this type of robot. His strategy was to take two consecutive
mutually perpendicular moves to reach from one endpoint to another endpoint of a diameter of the circle. The strategy is described in Figure 2. This
strategy fails to detect the center of a sphere.
Michael A. Burr [5] solved the problem of searching the center of al ellipse
using a special property of an ellipse. The property is that the line joining
the midpoints of any two parallel chords of an ellipse is a diameter of that
ellipse and the midpoint of that diameter is the center of that ellipse. For
Figure 3.2:
example in Figure 18 , According to his strategy, if the searcher starts from
the point A and advance along AB chord , reaches B, finds out the midpoint
of AB chord which is M and then goes to the point B again. Then it turns 90
degrees inside the ellipse and moves along BC chord and reaches C. Then the
searcher turns 90 degrees inside the ellipsoid again, moves along CD chord
and finds out its midpoint, N. Now from N the searcher moves along NM
and gets the PQ diameter and gets the midpoint of it which is eventually the
center of the ellipse.
32
Chapter 4
Cutting a Convex Polygon Out
of a Circle
In this thesis we first consider the problem where Q is a circle and P is convex
polygon such that P is bounded by a half circle of Q and all the cuts are line
cuts. We give a simple linear time O(log n)-approximation algorithm for this
problem where n is the number of vertices of the polygon. We also gave an
constant factor approximation algorithm for this problem. Finally, we gave
an constant factor approximation algorithm in general case i.e. cutting a
convex polygon out of a circle in which the convex polygon is not necessarily
cornered. For both of the cases we also determined the Polynomial Time
Approximation Scheme (PTAS).
4.1
The Basic Problem that We Studied
A line cut is a line that does not run through the polygon P and divides Q
into two pieces. After a cut is made, Q is updated to the piece containing P .
The cost of a cut is the length of the intersection of the cut with Q. A cutting
sequence is a sequence of cuts such that after the last cut in the sequence we
33
have P = Q. In this thesis we consider the problem of cutting P out of Q
by line cuts with total cutting cost as small as possible. See Figure 4.1(b).
Q
Q
Q
P
(a)
P
(b)
P
(c)
Figure 4.1: (a) A convex polygon. (b,c) Two different cutting sequences to
cut P out of Q. The cost of the sequence in (b) is more than that in the (c).
The problem of cutting a polygon P out of a piece of paper Q with
minimum total cutting length is a well studied problem in computational
geometry. Researchers studied several variations of the problem, such as P
and Q are convex or non-convex polygons and the cuts are line cuts or ray
cuts. Though the problem is approached considering the piece of paper is
being a convex polygon, we are the first to consider it as a circle. At first
we approach the problem in a restricted case that is the polygon P inside
the circle Q is cornered and give approximation algorithm regarding this
restricted case.Then we solved the problem in more general way where the
polygon P is not necessarily cornered.
4.2
Algorithm for Cornered Convex Polygon
In this section we shall represent the basic ideas of establishing our algorithms by introducing some lemma and theorem. In the first subsection we
shall represent algorithm for the cornered convex polygon with O(log n) ap34
proximation ratio and O(n) runtime, in the subsequent subsection we shall
give the second algorithm which is a constant factor approximation algorithm
with approximation ratio 6.48 and running time O(n3 ).
Let ll0 be a chord of Q. ll0 divides Q into two circular segments, one is
bigger than a half circle and is called the bigger segment created by ll0 , and
the other one is called the smaller segment created by ll0 . Let tt0 be another
chord intersecting ll0 at x such that tx is in the smaller segment of ll0 . See
Figure. Then the following lemma holds.
Lemma 4.2.1. Length of xt is no bigger than length of ll0 .
a
b0
c0
l
b0
c0
b0
t
x
t0
c
b
l0
(a)
(b)
Figure 4.2: Some preliminaries.
Following lemma is also obvious, whose illustration can be found in Figure.
Lemma 4.2.2. Let 4abc be an obtuse triangle with ∠bac being the obtuse
angle. Consider any line segment connecting two points b0 and c0 on ab and
ac, respectively, possibly one of b0 and c0 coinciding with b or c respectively.
Then the angle ∠bb0 c0 and ∠cc0 b0 are obtuse.
35
4.2.1
Algorithm 1
Our algorithm has four phases : (1) D-separation, (2) triangle separation,
(3) obtuse phase and (4) curving phase. In D-separation phase, we cut out
a small portion of Q (less than the half of Q in size) which contains P (and
looks like a “D”). Then in triangle separation phase we reduce the size of
Q even more by two additional cuts and bound P by almost a triangle. In
obtuse phase we assure that all the portions of Q that are not inside P are
inside obtuse triangles. Finally, in curving phase we cut P out of Q by
cutting in rounds.
Let C ∗ be the optimal cutting length of P from Q. Clearly C ∗ ≥ |P |,
where |P | is the length of the perimeter of P .
D-Separation
A D of the circle Q is a circular segment of Q which is smaller than an half
circle of Q. By a D-separation of P from Q we mean a line cut along a chord
of Q that creates a D containing P . In general for a circle Q and a convex
polygon P there may not exist any D-separation of P . But in our case since
P is cornered, there always exists a D-separation of P .
We will first observe that there exists only one minimum-cost D-separation
C1 of P and then we will find that C1 . (For our algorithm, however, showing
the uniqueness of the minimum D-separation is not required).
Lemma 4.2.3. The minimum-cost D-separation C1 for P can be found in
O(n) time.
Proof. Clearly C1 must touch P . So, C1 must be a vertex cut or an edge cut
of P . Observe that any vertex cut or edge cut that is a D-separation must
be through a visible vertex or a visible edge. Let e be a visible edge of P .
Let ll0 be a line cut through e. Let cp be the line segment perpendicular to
ll0 at p. If p is a point on e, then we call e a critical edge of P and ll0 a
36
critical edge cut of P . Observe that since P is convex, it can have at most
one critical edge. Similarly, let v be a vertex of P . Let tt0 be a vertex cut
through v. Let cp be the line segment perpendicular to tt0 at p. If tt0 is such
that p = v, then we call v a critical vertex of P and tt0 a critical vertex cut
of P . Again observe that P can have at most one critical vertex. Moreover,
P has a critical edge if and only if it does not have a critical vertex. See
Figure 4.3. Now, if P has the critical edge e, then C1 is the corresponding
critical edge cut ll0 . C1 is minimum, because any other vertex cut or edge cut
of P is either closer to c (and thus bigger) or does not separate c from P . On
the other hand, if P has the critical vertex v, then C1 is the corresponding
critical vertex cut tt0 of P . Again, C1 is minimum by the same reason.
For the running time note that all visible vertices and visible edges of P
can be found in linear time. Then finding whether an edge of P is critical
takes constant time. Over all edges finding the critical edge, if it exists, takes
O(n) time. Similarly, for each visible vertex v we can check in constant time
whether v is critical or not by comparing the angles of cv with two adjacent
edges of v, which takes constant time. Over all visible vertices, it takes linear
time.
Q
Q
l0
t0
c
c
e
v
P
P
t
l
Figure 4.3: Critical edge and critical vertex.
37
Lemma 4.2.4. Cost of C1 is at most C ∗ .
Proof. Consider any optimal cutting sequence C. C must separate P from
c. However, it may do that by using a single cut or by using more than
one cut. If it uses a single cut, then it is in fact doing a D-separation. By
Lemma 4.3.5, it can not do better than C1 , and therefore, cost of C1 is at
most C ∗ .
Now we assume that C separates P from c by more than one cut. There
are several cases here. We will prove that in any case the cutting cost of
separating P from c is even higher than that for a single cut. Let C be the
first cut in C that separates c from P . C can not be the very first cut of
C, otherwise, we are in the previous case considered above. It implies that
C smaller than a chord of Q. See Figure 2.2. Let a and a0 be the two end
points of C. Let bb0 be the chord of Q that contains aa0 where b is closer to
a than to a0 . Now, at least one of a and a0 is not incident on the boundary
of Q. We first assume that a is not incident to the boundary of Q but a0 is.
Let Cx = xx0 be the cut that was applied before C and intersects C at a.
Since Cx does not separate c from P , the bigger segment of Cx contains P ,
c and a. Now if x0 is an end point of Cx , then by Lemma 4.2.1 ab is smaller
than xx0 . It implies that having Cx in addition to C1 increases the cost of
separating P from c (see Figure 2.2(a)). Similarly, if x0 is not an end point
of Cx , and thus another cut is involved, then by Lemma 4.2.1 the cost will
be even more (see Figure 2.2(b)).
Now consider the case when both a and a0 are not incident on Q. So
at least two cuts are required to separate ab and a0 b0 from bb0 . Let those
two cuts be Cx and Cy respectively. Let xx0 and yy 0 be the two chords of
Q along which Cx and Cy are applied. Now if Cx and Cy do not intersect,
then by applying the above argument for each of them we can say that the
cutting cost is no better than that for a single cut. But handling the case
when Cx and Cy intersect is not obvious. (See Figure 2.2(c,d)). Let z 0 be
38
their intersection point. Again there may be several subcases: x and y may
or may not be end points of Cx and Cy . Assume that both x and y are end
points of Cx and Cy . Remember that none of Cx and Cy separates P from
c. So region bounded by xz 0 y must contain P and c inside of it and in that
case the total length of xz 0 and yz 0 is at least the diameter of Q, which is
bigger than bb0 (see Figure 2.2(c)). For the other cases, where at least one of
x and y is not an end point of Cx and Cy , it can be recursively proven that
the cost is even more (For example, see Figure 2.2(d)).
a0 = b0
a0 = b0 x0
y0
c
b0
z0
a0
c
c
x0
a
x0
b
x
a
b0
z0
c
b
x
(b)
y
a
b
x
a0
y
a
b
(a)
x0
y0
(c)
x
(d)
Figure 4.4: Separating P from c by using more than one cut increases the
cutting cost. The cutting length is shown in bold lines.
Triangle Separation
In this section we apply two more cuts C2 and C3 and “bound” P inside a
“triangle”. From there we achieve three triangles inside which we bound the
remaining uncut edges of P (in the D-separation phase, at most one edge of
P becomes cut).
Let C1 = aa0 be the cut applied during D-separation. We apply C2 = at
such that at is tangent to P . Then we apply the third cut C3 = a0 t0 such
that a0 t0 is also tangent to P . If a0 t0 intersects at inside or on the boundary
of Q, then let z be the intersection point. (See Figure 4.5(a)). If a0 t0 and at
39
do not intersect, then let z be the intersection point of the lines containing
at and a0 t0 . (See Figure 4.5(b)).
We get three resulting triangles Ta , Ta0 , Tz having a, a0 and z, respectively,
as a vertex as follows. We describe only Ta , description of Ta0 , Tz are analogous. If C1 is an edge cut, then let rr0 be the corresponding edge such that
a is closer to r than to r0 . (See Figure 4.5(a)). If C1 is a vertex cut, then let
r be the corresponding vertex of P . Let s be the similar vertex due to C2 .
(See Figure 4.5(b)). Then Ta = 4ars. The polygonal chain of P bounded by
this triangle is the edges from r to s that reside inside Ta .
a0
a0
t0
r0
r
t
r
z
z
s
t0
s
a
t
a
(a)
(b)
Figure 4.5: Triangle separation.
Lemma 4.2.5. Total cost of C2 and C3 to achieve Ta , Ta0 and Tz is at most
2C ∗ . Moreover, C2 and C3 can be found in linear time.
Proof. Whether z is within Q or outside Q, the length of at and a0 t0 can not
be more than twice the length of aa0 , which, by Lemma 4.3.5, is no more
40
than C ∗ . So the total cost of C2 and C3 is at most 2C ∗ .
To find at linearly, we can simply scan the boundary of P starting from
the vertex or edge where aa0 touches P and check in constant time whether
a tangent of P is possible through that vertex or edge. Similarly we can find
a0 t0 within the same time.
Obtuse Phase
Consider the triangle Ta obtained in the previous section. We call the vertex
a of Ta the peak of Ta and the edge rs the base of Ta . Let the polygonal
chain of P that is bounded by Ta be Hrs . Observe that the angle of Ta at a
is acute. Similarly, the angle of Ta0 at its peak a0 is also acute. However, the
angle of Tz at its peak z may be acute or obtuse.
We obtain, by applying a cut, for each of Ta , Ta0 and Tz (if Tz is acute)
one or two triangles such that the angle at their peaks are obtuse and they
combinely bound the polygonal chain of the corresponding triangle. We
describe the construction of the triangle(s) from Ta only, description for the
triangles obtained from Ta0 and Tz are analogous.
Let Pa be the polygonal chain bounded by Ta . Length of ar and as may
not be equal. W.l.o.g. assume that as is not larger than ar. Let s0 be the
point on ar such that length of as0 equals to the length of as. Connect ss0
if s0 is different from r. We will find a line segment inside Ta such that it
is tangent to Pa and is parallel to ss0 . We have two cases: (i) ss0 itself is a
tangent to Pa , and (ii) ss0 itself is a tangent to Pa . Case (i): If ss0 itself is
tangent to Pa (at s), we apply our cut along ss0 . This may be a vertex cut
through s. In that case we get the resulting obtuse triangle 4rss0 and the
polygonal chain bounded by this triangle is same as Pa . (See Figure 4.6(a)).
On the other hand, this may be an edge cut through the edge of Pa that is
incident to s. Let u be the vertex other than s of the edge of the edge cut.
Then our resulting obtuse triangle is 4rus0 and the polygonal chain bounded
41
by this triangle is the edges from r to u. (See Figure 4.6(b)). Case(ii): For
this case, let u, u0 be two points on as and ar, respectively, such that uu0 is
tangent of Pa and is parallel to ss0 . We apply the cut along uu0 . Again this
cut may be a vertex cut or an edge cut. If it is a vertex cut, let g be the
vertex of the cut. Then we get two obtuse triangles 4ru0 g and 4sug and
the polygonal chains bounded by them are the sets of edges from r to g and
from g to s respectively. (See Figure 4.6(c)). If it is an edge cut, then let
gg 0 be the edge of the edge cut with u being closer to g than to g 0 . Then
we get two obtuse 4ru0 g 0 and 4sug and the polygonal chains bounded by
them are the sets of edges from r to g 0 and from g to s respectively. (See
Figure 4.6(d)).
a
u0
s0
s0
s
u
a
a
a
g
s0
g0
g
s0
s
r
r
(a)
(b)
(c)
(d)
Lemma 4.2.6. Total cost of obtaining obtuse triangles from Ta , Ta0 , and Tz
is at most C ∗ . Moreover, they can be found in O(log n) time.
Proof. Consider the construction of the obtuse triangle(s) from Ta . Length
of the cut ss0 or uu0 is at most the length of their base ss0 , which is bounded
by the length of Pa . Over all three triangles Ta , Ta0 and Tz , the total cutting
length is bounded by the length of the perimeter of P . Since C ∗ is at least
the length of the perimeter of P , the first part of the lemma holds.
42
u
s
s
r
r
u0
u
a
u0
g
s0
a
u0
u
g0
g
s0
s
u
s
r
r
(c)
(d)
Figure 4.6: Obtaining obtuse triangle(s) from Ta .
For running time, in Ta , we can find the tangent of Pa in O(log |Pa |) time
by using a binary search. Over all three triangles Ta , Ta0 and Tz , we need a
total of O(log n) time.
Curving Phase
After the obtuse phase the edges of P that are not yet cut are partitioned
and bounded into polygonal chains with at most six obtuse triangles. In this
section we apply the cuts in rounds until all edges of P are cut. Our cutting
procedure is same for all obtuse triangles and we describe for only one.
Let Tu = 4gus with peak u and base gs be an obtuse triangle. See
Figure 4.7. Let the polygonal chain bounded by Tu be Pu . Let the edges of
Pu be e1 , . . . , ek with k ≥ 2. We will apply cuts in rounds and all cuts are
edge cuts. In the first round we apply an edge cut C 0 along the edge ek/2 .
Then we connect g and s with the two end points of ek/2 to get two disjoint
triangles. Since Tu is obtuse, by Lemma 4.2.2 these two new triangles are
also obtuse. So, as a result we cut out one polygonal edge and get two new
obtuse triangles. In the next round we work on each of these two triangles
43
recursively, then in the next next round we work on four triangles and so on.
We continue in this way until all edges of Pu are cut.
u
g
s
Figure 4.7: Curving phase
Lemma 4.2.7. There are at most O(log k) rounds of cuts to cut all edges
of Pu . Moreover, the total cost of these cuts is |Pu | log k, where |Pu | is the
length of Pu , and the total running time is O(|Pu |).
Proof. At each round the number of triangles get doubled and the number
of edges that become cut also get doubled. So after log k rounds all k edges
are cut.
At each round the total length of the bases plus the length of the edges
that are being cut is no more than |Pu |. So, the total cutting cost is |Pu | log k
over all rounds.
For running time, for applying an edge cut we always move to the middle
edge of a polygonal chain. So, finding an edge cut takes constant time.
Moreover, after each cut the corresponding edge is never considered again.
So, over all rounds we can have at most |Pu | cuts, which gives a running time
of O(|Pu |).
Corollary 4.2.8. The total cost of curving phase is C ∗ log n and the running
time is O(n).
44
Combining the results of all four phases, we get the following theorem.
Theorem 4.2.9. Given a circle Q and a cornered convex polygon P of n
edges within Q, P can be cut out of Q by using line cuts in O(n) time with
a cutting cost of O(log n) times the optimal cutting cost.
4.2.2
Algorithm 2
In this section we shall present our second algorithm which cuts P out of Q
with a constant approximation ratio of 6.48 and running time of O(n3 ). This
algorithm has three phases: (1) D-separation, (2) cutting out a minimum
area rectangle that bounds P and (3) cutting P out of that rectangle by only
edge cuts. The D-separation phase is same as that of our first algorithm and
we assume that this phase has been applied.
Cutting a Minimum Area Bounding Rectangle
We will use the technique of rotating calipers, which is a well known concept
in computational geometry and was first introduced by Toussaint [ref]. We
use the method described by Toussaint [ref].
A pair rotating calipers remain in parallel. It rotate along the boundary
of an object with two calipers being tangents to the object. If the object is
a convex polygon P , then one fixed caliper, which we call the base caliper,
is tangent along an edge e of P and the other caliper is tangent to a vertex
or an edge of P . In the next step of the rotation, the base caliper moves to
the next edge adjacent to e and continue. The rotation is complete when the
base caliper has encountered all edges of P .
For our case we use two pairs of rotating calipers, where one pair is
orthogonal to the other. We fixed only one caliper, among the four, as the
base caliper. As we rotate along the boundary of P , we always place the base
caliper along an edge of P and adjust other three calipers as the tangents of
45
P . The four calipers give us a bounding rectangle of P . After the rotation
is complete, we identify the minimum area rectangle among the n bounding
rectangles. For that rectangle we apply one cut along each of its edges that
are not collinear with the chord of the D.
The above technique can be done in O(n) time [ref]. Once the base
calipers is placed along an edge, the other three calipers are also rotated and
adjusted to make them tangent to P . Notice that each caliper “traverses”
an edge or a vertex exactly once.
fig here
Lemma 4.2.10. The cost of cutting a minimum area rectangle out of the D
achieved from the D-separation phase is no more than 2.57C ∗ .
Proof. There can be at most four pieces, other than the one inside the bounding rectangle, resulting from four cuts applied to the D. The length of each
cut is no more than the portion of the perimeter of D that is separated by
that cut. So, as a whole the total cutting cost is no more than the perimeter
of D. Also see ref[fig].
Now the perimeter of D is CD + RΘ, where CD is the length of the chord
of D and Θ is the angle made by the arc of D at the center c. Since CD
is the cost of D-separation, which by ref[lemma] is bounded by C ∗ ,Θ is at
most Π, and C ∗ can not be more than 2R, the maximum perimeter of D is
C ∗ + (C ∗ /2)Π = 2.57C ∗ .
Cutting P out of a Minimum Area Rectangle by Only Edge Cuts
For this phase we simply apply the constant factor approximation algorithm
of Tan [ref]. If P is bounded by a minimum area rectangle, then Tans
algorithm cuts P out of the rectangle by using only edge cuts in O(n3 ) time
√
and with approximation ratio (1.5 + (2)) [ref]. We summerize the result of
our second algorithm in the following theorem.
46
Theorem 4.2.11. Given a circle Q and a cornered convex polygon P of n
edges within Q, P can be cut out of Q by using line cuts in O(n3 ) time with
a cutting cost of 6.48 times the optimal cutting cost.
Proof. We have the cutting cost of C ∗ for D-Separation, 2.57C ∗ for cutting
√
the minimum area rectangle, and (1.5 + (2))C ∗ for cutting P out of the
rectangle, which give a total cost of 6.48C ∗ .
4.3
Algorithms for General Case
In this section, we consider the problem where Q is a circle and P is a convex polygon which may or may not be cornered. We give three algorithms
for this problem. Our first algorithm runs in O(n) time and has an approximation ratio of O(log n). Next, we give a constant factor approximation
algorithm having approximation ratio of 6.74 and 7.9-constant factor when
P is cornered or not respectively, and runs in O(n3 ) time. Finally, we give a
polynomial time approximation scheme (P T AS) which has an approximation
6
ration of (1 + ²) and runs in O( ²n12 ) time.
In this thesis we denote the radius of Q by R, diameter of P by D, the
boundary of P (resp. Q) by ∂P (resp. ∂Q) and length of the perimeter of
P (resp. Q) by |P |(resp. |Q|). We denote an optimal cutting cost of P out of
Q as C ∗ . Let Q0 denote the minimum rectangle containing P . Let Dx (resp.
Dy ) denote the x − distance (resp. y − distance) between two vertical (resp.
√
horizontal) edges of Q0 . Then, Dx = D, Dy = D,and Dx + Dy = |S ∗ |/2
hold.
The following two lemmas are obvious:
Lemma 4.3.1. C ∗ ≥ |P |.
Lemma 4.3.2. C ∗ ≥ 2D.
47
Now we prove an important lemma which would be necessary to compute
the approximation ratio later in this thesis.
Lemma 4.3.3. If P is a centered convex polygon, C ∗ ≥ 2R.
b
z
y
m
c
n
a
d
P
l
x
Figure 4.8: Lower bound for C ∗ .
Proof. by Lemma 4.3.1, C ∗ ≥ |P |.Observe that, any cutting sequence that
separate P from Q and has two disjoint subsequences each start from a point
of Q and ends at a point of P . Let a and m are two such points of P and
b and n are the two corresponding points on Q respectively. See Figure 4.9.
Now, |ac| + |ab| ≥ R, and |cm| + |mn| ≥ R. Again, |ac| + |cm| ≤ |P |. So,
the total cutting cost is at least |P | + |mn| + |ab| ≥ 2R. Hence proved.
For a vertex v of P , we define the chord through v as the intersecting segment of Q with a line through v. Let Cv denote a shortest chord
through a vertex v of P . Let V (P ) be the set of vertices of P , and let
|C| = maxv∈V (P ) |Cv |.
Then, |C| ≤ |C ∗ | holds [10].
A chord through
v ∈ V (P ) is called an ST − chord if it is a shortest chord through v and
tangent to P . We denote it by STv .Let |ST | = minv∈V (P ) |STv |.
Now the following lemma is obvious.
48
Lemma 4.3.4. The ST − chord can be found in O(n) time.
P
Figure 4.9: shortest chord through a point inside a circle is perpendicular o
the line joining the center and that point.
4.3.1
Separating Minimum Rectangle Bounding P Using Rotating Calipers
Almost all of our algorithms cuts a minimum rectangle bounding P at the
first step. We choose the rotating calipers technique for determining this
rectangle. This technique was introduced by G.Toussaint in [17]. Chandrasekaran et. al. used this approach in [7] but in a different way so that
their algorithm becomes dependent on the number of edges of Q. But we
used it in such a way so that the running time is only dependant on n. We
discuss the technique here.
In [17], G. Toussaint proved that the rectangle of minimum area enclosing
a convex polygon has a side collinear with one of the edges of the polygon.
This theorem can be deployed to determine the minimum area rectangle
bounding P . For cutting out the minimum area rectangle , we use two pairs
of orthogonal rotating calipers as the method described by G. Toussaint [17]
in O(n) time. This process involves rotating two orthogonal pairs of calipers
49
around P keeping one side of one calipers always collinear with one of the
edges of P . Hence, we get a minimum area rectangle around P .
P
=⇒
Figure 4.10: (a) Rotating Calipers (b) Rotating two orthogonal pairs of
rotating calipers.
Lemma 4.3.5. The cost of cutting out minimum area rectangle bounding P
√
is no more that 2 2C ∗ where P is centered.
a
a
2R
a
a
Figure 4.11: Upper bound of the cut length of minimum area rectangle
bounding P when P is centered.
Proof. We know that the maximum rectangle enclosed in a circle is a square
whose diagonal is 2R. See Figure 4.11. Let us assume that the side of the
50
square be a. So, 2a2 = (2R)2 or a =
√
2R. As a result, the cost of cutting out
the minimum area rectangle separated by rotating callipers is no more than
√
√
4 2R. So, the total cutting length of this step cannot be greater than 4 2R.
√
As C ∗ ≥ 2R, the total cutting length is less than or equal to 2 2C ∗ . The
bounding box thus got by rotating calipers is not always guaranteed to be
in the full shape of a rectangle. The perimeter of Q may cut some portions
of the rectangle. However, if we estimate the cutting cost considering the
part of the rectangle inside the circle, the estimated cost would always be
√
less than or equal to 2 2C ∗ .
Figure 4.12: possible shape of bounding box
4.3.2
Algorithm 1
In this section we represent an algorithm for this problem which runs in O(n)
time with O(log n) approximation. This algorithm first determines whether
P is cornered or centered in O(n) time. If it is cornered, we follow the
algorithm proposed in [1]. If P is centered, we give here a new algorithm for
cutting it out of Q.
For deciding whether P is cornered or centered, we can use an easy O(n)
algorithm which checks the position of the center with respect to all n edges
of P .
51
If P is cornered then Ahmed et. al gave an algorithm in [1] to cut P out
of Q in O(n) running time and O(log n) approximation ratio. We follow the
same algorithm here. The algorithm has four phases : (1) D-separation, (2)
triangle separation, (3) obtuse phase and (4) curving phase. In D-separation
phase, a small portion of Q (less than the half of Q in size) which contains P
(and looks like a “D”) is cut out. Then in triangle separation phase the size
of Q is reduced even more by two additional cuts and bounds P by almost
a triangle. In obtuse phase, it is assured that all the portions of Q that are
not inside P are inside obtuse triangles. Finally, in curving phase P is cut
out of Q by cutting in rounds.
If P is centered we cut P out of Q in two steps: (1) cutting out a minimum
area rectangle containing P and (2) cutting P out of that rectangle.
For cutting out minimum area rectangle, we use the rotating calipers
technique described in Section 4.3.1. As a result the separation phase runs
√
in O(n) time and the cut length is less than or equal to 2 2C ∗ .
After that we cut P out of the minimum area rectangle by cutting in
rounds in the similar way described in the curving phase of the [1]. We do
not need the obtuse phase here , as all the angles of the rectangle are right
angles. Hence, the curving phase will run in O(n) time and will have an
approximation ration of O(log n).
So, the following theorem holds.
Theorem 4.3.6. Algorithm 1 cuts P out of Q in O(n) time with O(log n)
approximation ratio.
4.3.3
Algorithm 2
In this section we represent an algorithm with constant factor approximation ratio and O(n3 ) running time. The constant factor is 6.74 or 7.9 if P
is centered or cornered respectively. This algorithm actually modifies the
52
a0
a0
Q
Q
t
r
z
s
t
z
t0
e
t0
s
a
c
c
r
l
P
(b)
a
u0
u
s0
a
a
a
s
v
P
a
(a)
s0
g
s0
u0
u
s0
s
s
r
r
t
g0 g
u
s
r
(c)
r
u
g
s
(d)
Figure 4.13: cutting strategy for O(log n) approximation ratio when P is
cornered.
separation phase of Tan’s constant factor algorithm [16].
This algorithm also has two phases: (1) cutting out minimum axis aligned
rectangle and (2) cutting P out of that rectangle with edge cuts chosen
optimally.
If P is centered, for cutting out the minimum axis aligned rectangle,
we follow the same rotating calipers technique of Section 4.3.1. So, it cuts
a minimum area rectangle in O(n) time. As a result, for this case, the
√
maximum cost of this phase is 2 2C ∗ by Lemma 4.3.3.
If P is cornered, we could also use the rotating callipers technique. But
53
lemma 4.3.3 is applicable only when P is centered. So, we use different
strategy in this case. However, using rotating callipers technique yields a
simple algorithm with better approximation ratio when P is centered. In the
case of P being cornered, we first find out the ST − chord. By lemma 4.3.4,
ST − chord can be found in O(n) time. Suppose that a chord is ST and
touches a vertex v of P . Let lv denote the supporting line of the chord ST
. Our first cut is made along lv . See Fig. 4.14, where the cuts are shown
with bold and dotted lines. Let u be the vertex of P whose distance to lv is
maximum, and u0 the point of lv such that the angle ∠uu0 v is π/2. Denote
by Cu a shortest chord through u,and d, e two endpoints of Cu on δQ. Let
lu and le denote the lines parallel to lv and through u and e, respectively.
W.l.o.g., assume that the point e lies in between lu and lv .(The point e may
be on lv .) Let f denote the intersection point of lv with the supporting line
of de. There can be following two cases.
Case 1 The chord Cu is tangent to P . In this case, we make the second
cut along Cu , whose length is at most |S ∗ |. W.l.o.g., assume that the slopes
of the first two cuts are different. The lines through two cuts ST and Cu
partition the plane into four wedges. Define the angle of the wedge containing
P as the enclosing angle of ST and Cu .
Case 1.1 The enclosing angle of ST and Cu is at most π/2. Consider a
line segment ap tangent to P ,with p on lv and a on the semi-line originating
from u0 and going through u.(The point p may not lie in Q.) Let b be the
intersection point of ap with lu . Assume also that b and e lie in different sides
of au0 ( 4.14(a)). Our third cut is made along ap, with the point a giving the
minimum value |ap|min of |ap|.(The point a giving |ap|min may not lie in Q.)
Let us now give an upper bound on |ap|min . Set Y = |au|. Since |au0 | =
|au| + |uu0 | = Y + Dy and |ab| ≤ |au| + |bu| = Y + |bu|, we have |ap| =
|au0 |(|ab|/|au|) = (Dy /Y + 1)(Y + |bu|). Since |bu| = Dx holds, we consider
the function F (Y ) = (Dy /Y + 1)(Y + Dx ). A simple analysis shows that the
54
a
d
d0
a
Q
u
d
Q
lv
u
u
lu
b
P
p
u0
v
lv
lu
(a)
p
v
le
e
u0
f
(b)
lu
a
c
c
0
P
P
c
Q
b=u
d
b
p
lv
e
h
v
(c)
a
d
d
Q
lu
le
g
p
lv
b
u
c
Q
Q
lu
g
b u
g
b
u
e
e
lv
v
c
P
P
e
d
lu
P
(d)
c
i
p
s
(e)
f
lv
v
(f)
f
Figure 4.14: cutting out minimum area rectangle with constant cut length.t
√
minimum value F (Y )min of F (Y ) is achieved when Y = Dx Dy . Hence,
√
√
√
F (Y )min < (Dy / Dx Dy + 1)( Dx Dy + Dx ) ≤ (1 + 2/2)|S ∗ |.
It is clear that |ap|min = F (Y )min . Thus, the length of the cut along ap
√
is less than (1 + 2/2)|S ∗ |. Finally, we make a cut along lu , and two cuts
parallel to au’ and tangent to P . This results in the minimum rectangle Q0
containing P (only part of Q0 may result). See Fig. 4.14(b). The sum of the
cost taken for the cut along the chord ST and the costs for the last three cuts
is less than |Q0 | + |ST | − |T |. Since |ST | = |T |, we have |Q0 | + |ST | − |T | ≤
√
√
2(|P | + |ST | − |T |) < 2|S ∗ |. Hence, the total cost of these six cuts is less
√
than (2 + 3 2)|S ∗ |.
Case 1.2 The enclosing angle of ST and Cu is larger than π/2 ( 4.14(b)).
Consider a line segment ap tangent to P , with the point p on lv and the
55
point a on the semi-line originating from f and going through d, such that
|ap| > |af |. Let b be the intersection point of ap with lu . See Fig. 4.14(c).
Set Y = |au|.Since the point f is contained in the right-angled triangle
4uu0 v in this case, |uf | < |uv| ≤ D holds. Thus, |af | = |au| + |uf | ≤ Y + D.
Since |bu| ≤ Dx holds, we have |ab| < |au| + |bu| ≤ Y + Dx . Hence, |ap| =
|af |(|ab|/|au|) < (D + Y )(Y + Dx )/Y ≤ (D/Y + 1)(Y + Dx ). Also, the
following upper bound on the minimum value |ap|min of |ap| holds.
√
|ap|min < D + 2 DDx + Dx = |S ∗ |/2 + |S ∗ | + |S ∗ |/2 = 2|S ∗ |.
To cut out the rectangle Q0 , we compute the position of a giving |ap|min ,
and make the third cut along ap. Finally, make a cut along lu and a cut
tangent to P and perpendicular to lv . The sum of the costs for the cut along
√
ST and for the last two cuts is at most 2|S ∗ |. The total cost is less than
√
(3 + 2)|S ∗ |.
Case 2 The chord Cu intersects the interior of P . Let c be the point of
lu such that the angle ∠dcu is π/2.
Case 2.1 The supporting line of cd intersects the interior of P (Fig. 1c).
Because of the convexity of Q, we can move a line segment c0 d0 ,with c0 on
lu and d0 on ∂Q, to a place such that the supporting line of c0 d0 is parallel
to cd and tangent to P ,and |c0 d0 | = |cd|. See Fig. 4.14(d). In this case, the
second cut is made along the supporting line of c0 d0 .Denoteby h the other
intersection of this cut with ∂Q. For efficiency, the cost of our second cut is
considered as two parts |c0 h| and |c0 d0 |. Note that |c0 d0 | ≤ |cd| < |ud| < Cu ≤
|S ∗ |.
Consider a line segment ap tangent to P , with p on the line le and a on the
semi-line originating from h and going through d0 .Let b be the intersection
point of the line lu with ap. Then, |bc| = Dx holds. See Fig. 4.14(e). Again,
we compute the position of a that gives the minimum value of |ap|, and make
the third cut along ap. The rectangle Q0 can finally be cut out by making
two cuts along lu and along the other tangent to P which is parallel to d0 h.
56
The sum of |c0 h| and the costs taken for the first cut ST and the last two
√
cuts is at most |Q0 | + ST − T = 2|S ∗ |. As in Case 1.1, the total cost is less
than (2 +
√3 )|S ∗ |.
2
Case 2.2 The supporting line of cd does not intersect the interior of P .
Let g denote the vertex of the rectangle Q0 such that it is on lu and |dg| > |du|.
Case 2.2.1 |gu| = |du|. Suppose first that d and e are to different sides of
the vertex v (as viewed from v). See Fig. 4.14(e). Let dp denote the segment
tangent to P , with p on le , such that |dp| > |de|. The second cut is made
along the supporting line of dp. Let b be the intersection point of the cut with
lu . Since |bu| = |gu|,we have |dp| = |de|(|db|/|du|) < |de|(|bu| + |du|)/|du| ≤
|de|(|gu|/|du| + 1) = 2|de|.
To cut out the rectangle Q0 , we make two cuts tangent to P and perpendicular to lv . Also, the longer cut is considered as two parts: one is between
lv and lu , and the other is between lu and the line segment bd, whose length
is less than |dc| < |du| < Cu < |S ∗ |. See Fig. 4.14(e). Finally, make a cut
along lu . As analyzed above (like Case 1.2), the total cost is then less than
√
(3 + 2)|S ∗ |.
Case 2.2.2 |gu| > |du| (Fig. 4.14(f )). Assume first that d and e are to
the same side of the vertex v. Let ap be a line segment tangent to P , with
p on lv and a on the semi-line originating from f and going through d, such
that |ap| > |af |. Let b and i be the intersection points of ap with lu and with
the line through d and parallel to lv , respectively. See Fig. 4.14(f ).
Set Y = |ad| and H = |df |. Since |uf | < |uv| holds in this case, H =
|du|+|uf | < |gu|+|uv| ≤ Dx +D ≤ |S ∗ |. Since |af | = |ad|+|df | = Y +H and
|ai| < |ad| + |di| = Y + |di|, we have |ap| = |af |(|ai|/|ad|) < (H/Y + 1)(Y +
√
|di|). Since |di| < |bu| = Dx holds, we have |ap|min < H + 2 HDx + Dx =
√
(1.5 + 2)|S ∗ |.
To cut out the rectangle Q, we compute the position of a giving |ap|min ,
and make the third cut along ap. Next, make a cut along di, whose length is
57
less than |bu|. Two cuts, tangent to P and perpendicular to lv , are then made.
Note that the longer cut can be considered as two parts: one is between lv
and lu and the other is between lu and the supporting line of di (whose length
is less than |gu|). Finally, a cut along lu is made. The sum of |gu| and the
cost of the cut along lu is clearly less than 2Dx ≤ |S ∗ |. Consider now the
costs represented by |bu| and taken for the first cut along ST and two cuts
that are tangent to P and perpendicular to lv , excluding the part of the
longer cut between lu and the supporting line of di. By noticing the relation
between the segment bu and two considered vertical cuts (Fig.
4.15), we
∗
have that the sum of these costs is less than |P | + |ST | − |T | = |S |. Putting
√
all together, the total cost of our cutting sequence is less than (3.5 + 2)|S ∗ |.
For the case that d and e are to different sides of v, we can also show that
√
the cutting cost is less than (3.5 + 2)|S ∗ |. We leave the detail to readers.
It should be noted that , after finding the ST − chord in O(n) time we can
make the rest cuts in no more than O(n) time.
For cutting P out of that rectangle we follow the method used by Tan [16].
That is, we use the dynamic programming for choosing the optimal sequence
of edge cuts. In his paper Tan proved that this dynamic programming method
√
with all edge cuts has the maximum cutting length of (1.5 + 2)C ∗ . Hence
the over all running time becomes O(n3 ) and the total cutting length is
√
√
less than or equal to (2 2 + 1.5 + 2)C ∗ or 6.74 where P is centered and
√
√
(3.5 + 2 + 1.5 + 2)C ∗ or 7.9 where P is cornered.
Hence , the following theorem holds.
Theorem 4.3.7. Algorithm2 cuts P out of Q in O(n3 ) time with 6.74 and
7.9 approximation ratio where P is centered and cornered respectively.
58
4.3.4
Algorithm 3
In this section we propose a polynomial time approximation scheme. We
modify the separation phase of the algorithm given by Bereg et. al [2]. They
used the separation phase same as the one Tan used. We replace it with
the separation phase of Algorithm2. Hence, we get the minimum bounding
rectangle in O(n) time with constant cut length. Next we follow the same
approach suggested by Bereg et. al which is not actually dependent on the
number of vertices of Q, hence, we can use it for our case.
A0
P
A
Q
Figure 4.15: Determining the portal points.
First of all, we determine whether P is cornered or centered in O(n)
running time. Next, we cut out a minimum area rectangle bounding P . For
this, we use the similar technique as the one we used in algorithm 4.3.3. That
is, we use the rotating calipers technique for P being centered and in P is
cornered we choose different cutting sequences in different cases as described
in 4.3.3. So, we obtain the minimum area rectangle in O(n) running time
with constant cutting cost. Hence we get an estimate EST of the cost OP T
of the optimal cutting sequence such that OP T ≤ EST ≤ cOP T , where c is
the constant approximation factor.
Next we construct a set E of candidate cuts. First put the n edge cuts
59
l1
p
p1
l2
P
p2
q
r
o
Figure 4.16: Determining the estimated cost.
into E, then, for each vertex p of P , and for each portal point v ∈ V , if the
line through both p and v specifies a valid tangent cut anchored at p (that
is, the line does not cut through P ), put the tangent cut into E. We call the
candidate cuts constructed so far the type − 1 cuts.
For every two type−1 cuts l1 and l2 intersecting at a point o and anchored
at two vertices q and r respectively (q, r,and o may coincide), we compute,
for every vertex p of P outside the triangle 4oqr, all the vertex cuts anchored
at p such that the length of the cutlines segment between l1 and l2 is (locally)
minimal. We put these cuts into the candidate set E and called them the
type − 2 cuts. See Figure 4.16. The determination of a type − 2 cut can be
done in constant time [2].
Finally, in the dynamic programming step, we use dynamic programming [13] to find a sequence of candidate cuts from E with the minimum
cost. This cutting sequence is our approximate solution.
We now analyze the running time of our algorithm. For the two rectangles
A and A0 ,we have |A| = O(|P |) = O(OP T )and |A0 | = |A| + 8EST .Since,
EST = Θ(OP T ), we have |A0 | = O(OP T ). It follows that the perimeter
60
of the polygon Q ∩ A0 is O(OP T ). The number of portal points is therefore
O(OP T )
∂EST
= O(1/∂) = O(n/²2 ). The number of type − 1 cuts is O(n/²2 ).
The number of type − 2 cuts is O(n2 /²4 ). The dynamic programming algorithm [13] takes O((n2 /²4 )3) = O(n6 /²12 ) time. Hence, We have the following
theorem
Theorem 4.3.8. Algorithm3 gives a (1 + ²) approximation scheme in polynomial time.
61
Chapter 5
Searching for the Center of
Spheres and Ellipsoids
Biedl et al. [4] first posed the problem of finding the center of a circle, starting
from a point on the boundary and using a limited number of operations.
They presented an open problem for searching the center of an ellipse in
their paper. Later, Michael A. Burr [5] et al. solved that problem. But there
was still a problem in the elliptical model regarding the dead areas which
was not handled by the strategy of Michael A. Burr [5] and this problem
remained unsolved. However, in both of their works they considered the
searching space to be two dimensional. As a result their strategies do well
on a plane. But in reality the searching space is three dimensional in most
of the cases. In those cases we need an efficient strategy to find out the
center of a sphere or in some cases, the center of an ellipsoid. In this thesis
we approached the problem regarding the dead areas, proposed a model
with dead areas, suggested a way to improve the strategy regarding this and
proposed efficient strategies of searching the centers of spheres and ellipsoids
which inherently works for circle and ellipse respectively.
62
5.1
Searching the Center of Spheres
In this section we discuss about the strategy of searching the center of a
sphere efficiently.
5.1.1
Problem Description
Consider the problem of a robot searching for the center of a sphere. The
robot has only a limited number of capabilities. It can detect whether it is
inside the sphere or not, it can mark the point which it occupies, it can move
along a straight line, possibly towards a mark which it had made previously.
It can make 90 degrees and 180 degrees turns. It can detect and move to
the middle of a line segment determined by two marked end-points, if it is
on the line segment. The starting point of the robot is on the surface of the
sphere and tries to reach at the center of the sphere as quickly as possible.
Figure 5.1: Searching the center of a sphere
63
5.1.2
Mathematical Description
Let, in the Figure 5.1, S is the sphere with center located at O. The initial
position of the searching robot is at a point A on the surface of the sphere.
The target is to reach the center of the sphere as quickly as possible. The
robot has the following available operations:
1. It can detect whether it is inside the sphere or not.
2. It can mark a position.
3. It can move in a straight line, possibly toward a mark which it had
made previously.
4. It can make 90 degrees and 180 degrees turns.
5. It can move to the middle of a line segment determined by two
marked points, if it is on the line segment.
Figure 5.2: Proposed strategy.
5.1.3
Proposed Strategy
See Figure 5.2.
64
1. Leave a marker at point A.
2. Move along a straight line inside the sphere, until you reach another
surface. Let this point be B.
3. Turn 90 degrees and move along a straight line inside the sphere.
Move until you reach another surface at point C.
4. Line AB and BC are perpendicular to each other. So, A, B and
C are not co-linear [except the case where AB is the diameter of the circle
ABC , in which case B and C will be superimposed].As a result, A, B and C
lie on a single plane. Move along the perpendicular line of this plane inside
the sphere until you reach another surface at point D.
5. Mark the point D.
6. Move towards the marker of step 1, to the halfway point between two
markers
5.1.4
How to detect a direction which is inside the
sphere
Whenever the robot is told to move in a direction perpendicular to a plane
which is inside the sphere, it needs to choose between two directions. For
example, when the searching robot is at point L on the surface ABL of the
sphere and is told to move along the perpendicular direction inside the sphere
it has to choose between the direction LM and LN which are perpendicular
to the plane ABL on which the robot is standing. Now, if the object moves
along LN it will be out of the range of the transmitter, so it can understand
that the right direction should be LM . When the robot is told to move along
a perpendicular direction to a straight line when it is on the surface of the
sphere it can choose any of the direction which lies on the perpendicular plane
to that straight line as shown in Figure 5.3. The directions the robot can
choose at the point L is shown in the figure. Whether the chosen direction is
65
Figure 5.3: How to detect a direction which is inside the sphere
leading towards the inside of the sphere or not, can be decided by the above
mentioned method.
5.1.5
Correctness Proof
Now we prove that the strategy leads a robot starting from the surface of a
sphere to the center of that sphere.
Lemma 5.1.1. Every cross section of a sphere produces a circular plane.
Proof. We can easily prove this by solving the equations of a sphere and a
plane. In Figure 5.4, we show how a plane cuts a sphere without the loss
of generality. In the same figure, we can see the resultant circular surface
yielded by that cut. Finally in Figure 5.4, we get a top view of that circular
plane.
Lemma 5.1.2. The straight line joining the center of the sphere and the
66
Figure 5.4: Cross section of a sphere
center of the circular plane produced by a cross-section of that sphere is perpendicular to that plane.
Figure 5.5: The straight line joining the center of the sphere and the center of
the circular plane produced by a cross-section of that sphere is perpendicular
to that plane.
Proof. Let, S be a sphere with center at T . ABC is a circular plane produced
by a cross-section of the sphere. P is the center of the circular plane, ABC.
We join T ,P . According to this lemma T P is perpendicular to the plane
67
ABC. This can be easily proved by elementary geometry by proving M
AT P and M DT P equal.
Lemma 5.1.3. Every plane, containing the straight line that joins the center
of a sphere and the center of the circular plane produced by a cross-section
of the sphere, is perpendicular to that circular plane.
Figure 5.6: Perpendicular plane to a circular plane
Proof. This is trivially proved from Lemma 5.1.2 as all of these planes contain
the straight line which is perpendicular to the circular plane produced by a
cross section. In Figure 5.6, we can see a number of perpendicular planes to
the circular plane ABC which contain the straight line joining the center of
the sphere which is donated as T and the center of the circular plane ABC
which is denoted as O.
Lemma 5.1.4. In a sphere S, if G is a perpendicular great circular plane of
a circular plane P , the line of intersection of G and P is a diameter of the
circular plane P .
68
Figure 5.7: Figure for Lemma 5.1.4.
Proof. By the definition of the perpendicular great circular plane G will
contain the axis diameter of the circular plane, P . The axis diameter is
perpendicular to the circular plane, P . Moreover, the center of the sphere
lies on the axis-diameter. Now, from lemma 5.1.1 we know that the straight
line, L (line T C in the Figure 5.7) joining the centers of the sphere and center
of P is perpendicular to P . So, L must be a part of the axis- diameter of P .
Hence the center of P must lie on G. More preciously it will lie on the line of
intersection of the planes P and G, namely AB. Now,∠ACT = ∠BCT = 90
degrees and A, C and B lie on the plane G. so,∠ACT + ∠BCT = 180
degrees. Hence A,C and B are co- linear. Moreover the line segment AB
passes through the center of the circular plane P . So, AB is the diameter of
the circular plane P . Hence, proved.
69
Correctness of the Strategy
According to the right angle strategy described by T. Beidl et al. [4], the
robot can reach from one end to the other end of a diameter of a circle
with two mutually perpendicular moves as shown in the Figure 5.8.Hence, if
we apply this strategy on a circular plane produced by a cross-section of a
given sphere, it would lead us from one end point to another end point of a
diameter of that circular plane. Now we consider the great circular plane G
described by the great circle ACE. G is perpendicular great circular plane
for the circular plane P , described by ABC and produced by a cross section
of the sphere. If the robot moves with right angle strategy on the plane P
Figure 5.8: Correctness of the strategy
starting from the point A, it would lead the robot to C which is the opposite
end point of the diameter AC. Without the loss of generality we assume that
G intersects P along the line AC and it is always true that G will intersect
P with a straight line which is a diameter of P by Lemma 5.1.5. Now, we
consider the circle ACE. To reach the center of the sphere, the robot needs
to reach the center of this great circle. If we think about applying the right
angle strategy on the circle ACE , we find that the robot has just completed
one of first two perpendicular moves which leads the robot from one endpoint
to the another endpoint of a diameter of the circle ACE by moving from A
70
to C. So, now it needs to move along a line perpendicular to AC and our
strategy suggests the same by telling it to move along the straight line which
is perpendicular to the plane, P .
Figure 5.9: Correctness proof
Moving along that direction the robot reaches at E on the surface. Now,
according to the right angle strategy, the robot has to move toward the point,
A. and on the half way it will reach the center of the circle, ACE. And as,
ACE is a great circle for the sphere S, it will eventually reach the center
of the sphere. So, the strategy correctly leads the robot to the center of a
sphere.
5.2
Competitive Ratio
The total length of traversal is [2R sin x(sin y + cos y) + 2R cos x + R] which
gives the competitive ration of at most 3.82. This strategy needs only 4
moves and 2 markers for a single searcher to reach the center.
71
Figure 5.10: Competitive ratio calculation
5.3
Detection Stage of Searching the Center
of an Ellipsoid
5.3.1
Problem Description
This problem is almost similar to the problem of searching the center of a
sphere. The only difference is that we are considering here the searching
space as an ellipsoid. In their paper T. Beidl et al. [4] claimed that the
searching area is not bounded by a circle in reality. It is actually elliptical.
So, strategy for finding out the center of an ellipse was necessary which was
later done by Michael A. Burr [5]. In three dimensional spaces we can think
this model as an ellipsoidal. So, now the mission is to find out the center
of that ellipsoid for that same robot which has the capabilities mentioned
earlier in this thesis.
5.3.2
Our Work
The strategy of Michael A. Burr [5] works well for an ellipse on a plane but it
is not suitable for three dimensional cases, where the searching space should
72
be modelled as an ellipsoid. We have proposed here an efficient strategy for
searching the center of an ellipsoid. Before we propose our strategy, we would
like to discuss some properties of an ellipsoid. These properties along with
the strategy of Michael A. Burr [5] will lead us to the proposed strategy.
Lemma 5.3.1. A plane cuts an ellipse to an ellipsoid.
Figure 5.11: Figure for Lemma 5.3.2
This property can easily be proved by solving the equations of an ellipsoid
and a plane.
Lemma 5.3.2. We can always have a projection of an ellipsoid with two
parallel elliptical planes inside it which are the planes of intersection of the
ellipsoid with two parallel planes, where the ellipsoid is an ellipse and those
parallel planes are two parallel chords of it.
From the physical property of ellipsoid, we know that any projection of
an ellipsoid is an ellipse. So, if we chose to project the ellipsoid in a way that
the parallel planes will be projected as parallel lines, we are done.
Lemma 5.3.3. Two parallel elliptical planes produced by the intersection of
an ellipsoid with two parallel planes have their centers on a diameter of the
ellipsoid.
73
Figure 5.12: Figure for Lemma 5.3.3
Let us consider the Figure 5.13. Here the ellipsoid E is intersected by
two parallel planes resulting two parallel elliptical-shaped planes inside it.
We name them A and B. If we join the centers of the elliptical planes A
and B and extend the line we shall get a chord which is claimed to be a
diameter by this lemma. Hence the midpoint of this chord will be the center
of the ellipsoid. This can be easily proved by using the lemma 5.1.7. On the
projection of the ellipsoid, we get an ellipse with two parallel chords. The
line joining the midpoints of those parallel chords will produce a diameter
of that ellipse. That diameter of the ellipse is actually the projection of a
diameter of the ellipsoid. Hence, we are done.
5.3.3
Proposed Strategy
1.Find the center of the ellipse as told in the thesis of Michael A.
Burr [5] and mark the center.
74
Figure 5.13: Proposed strategy
2.Go to any of the two antipodal points of the last diameter used in
that searching inside that ellipse.
3.Go along the perpendicular line of that plane until the signal fades.
4. Now, re-execute that strategy of Michael A. Burr [5] and find out
the center of that ellipse and mark it.
5.Go toward the previously marked center and keep going until the
signal fades and mark this point.
6.Turn 180 degrees and keep going until the signal fades and mark
that point.
7.Go to the midpoint of the last two marked points.
5.3.4
Correctness
This strategy is the direct consequence of the strategy used for searching the
center of an ellipse. We apply the strategy using the Lemma 5.1.8 we can
prove it instantly. So, this strategy will certainly lead the robot toward the
center of an ellipsoid.
75
5.4
Correction Stage
In the correction stage we handle the problem with dead areas for an ellipse.
We describe the correction stage for both ellipse and ellipsoid. First of all
we concentrate to the case of an ellipse and its dead areas. Then we advance
to the case of an ellipsoid. The concept of elliptical model comes from the
idea that the shape of the electromagnetic field of the transceiver is similar
to that of a magnetic field of a magnet. Michael A. Burr [5] showed a model
in his paper which is shown bellow.
Figure 5.14: Electromagnetic field
But this model is not realistic according to our point of view because the
dead regions are not so narrow near the transceiver in reality. So, we propose
here a hypothetical model assuming the transceiver as a bar magnet. A bar
magnet has two poles, the North Pole and the South Pole. The magnetic
force lines are assumed to be emitted from the North Pole and enters into the
South Pole as showed in the Figure 5.14. So the electromagnetic field takes
the shape of Figure 5.15 which is composed of two elliptical lobes. The center
of the magnet is at C which we have to find out. The two ellipses are assumed
to be symmetric with respect to the axis of the magnet for our purpose. Now,
the boundary of the electromagnetic field looks like Figure 5.16. We assume
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Figure 5.15: Magnetic Field
that this field is bounded by a larger ellipse such a way that the center of the
ellipse is the center of the magnet, C. It is obvious that the signal cannot
reach to every point inside that ellipse. Figure 5.17 explains the idea. In
this figure, E is the large ellipse surrounding the electromagnetic field and C
is the center of the magnet as well as the center of the large ellipse. D1 and
D2 are the dead regions where no signal reaches. The searching agent starts
searching from any point on the dotted lines and the mission is to reach at
C.
Due to these dead regions the parallel chord strategy of Michael A.
Burr [5] may fail to reach at C. Figure 5.18 describes such a case where the
strategy fails to guide the searcher to the center. The strategy ends at point
F where the original center of the ellipse is located at point C.
We use here a hypothetical assumption on this model. We assume that,
if we take two mutually perpendicular chords of the larger ellipse which
have the same midpoint, not both of them will go through the dead regions.
Using this assumption along with this model, we propose here a strategy
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Figure 5.16: Revised elliptical model
Figure 5.17: Assumed model
which will improve the searching accuracy. In the worst case, this scheme
will result the same point resulted by the detection stage. So, it has no sideeffect. Moreover, this strategy can detect the worst case quickly and ends
up immediately. Even if, our hypothetical model is not the one really exists,
still this strategy returns the same point we got in the detection stage and
ends up sharply.
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Figure 5.18: Dead regions
Figure 5.19: Wrong result by proposed strategy
5.4.1
Proposed Correction Strategy
1.Apply the parallel chord strategy reach the point of 1st assumption.
Remember the last diameter used to get the center. Name that diameter
D(k). Mark the center , name it C(k).
2. Turn 90 degrees and get another Diameter D(k+1) and its midpoint
C(k + 1).
3. Compute the distance between C(k) and C(k + 1). If its less than
a predefined fixed value p then C(k + 1) is the refined result. break and end
the process. If the distance is greater than p then let D(k) = D(k + 1) and
C(k) = C(k + 1). Go to step 2.
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5.4.2
Convergence of the strategy
This strategy will certainly converge to a certain point because of the central
tendency of the ellipse. In each step the midpoints will lead the searcher
closer to the center of the ellipse if the diameter is not affected by the dead
regions. No two consecutive diameters will be affected by the dead region by
the hypothesis. So, this strategy will converge to the center of the ellipse.
For example we can apply this strategy for the case stated in Figure 5.18
where the parallel chord method failed to reach the center. In Figure 5.19,
we show how our strategy refines the solution and reaches the center.
Figure 5.20: Correction strategy
5.4.3
Performance Analysis
If our hypothesis matches the case, the center of the ellipse will be reached
very soon. The actual length of the traversed path in correction stage is a
function of the point returned by the detection stage and the last diameter
used in that stage and the predetermined fixed value p. So, there exists a
function F (D, C, p) describing the length of the traversal. The growth of this
function is expected to be reduced at a rate which is a logarithmic function
of A and B , where A and B are the length of the two axes of the large
ellipse and C is the center returned by the detection stage and D is the last
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diameter used in that stage. The upper bound of this function is roughly
estimated as O([A ∗ log(B) + B ∗ log(A)]).
5.4.4
Worst Case Analysis
In the worst case the model with the hypothesis may fail and both of the
first two consecutive chords of correction stage will be affected by the dead
regions. In that case both of the chords are in fact the chords of one of the
smaller elliptical lobes. As a result, first two midpoints will be close enough
to end the strategy. In other words , the strategy will end up with the same
result of the detection stage and it will take only the cost of travelling one
chord and determining its midpoint , more preciously O((3/2) ∗ (A + B)).
5.4.5
For Ellipsoids
For the correction steps in ellipsoids , we do the almost same thing with three
consecutive chords. After reaching the point returned by the detection steps
, we shift the point to the midpoint of the perpendicular chord drawn at that
point. Then we take another chord perpendicular to that plane and shift
the assumed center to the midpoint of that chord. In this way, we gradually
approach the center of the ellipsoids.
5.5
Use of Volume Knobs
If we use the volume knobs of the transceiver, sometimes the searching can
be even easier. The volume button can be used at the detection stage. We
used the strategies of finding the center of circles and spheres to determine
the center of spheres and ellipsoids. So, the technique of using volume in
searching the center of circles or ellipses can be used in those steps.
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Chapter 6
Conclusion and Future Works
In this thesis we have studied the minimization of the length for the cutting
a convex polygon out of a circle and the efficient strategy to search the center
of a sphere or an ellipsoid. In particular we studied four types of problems:
(i) cutting a cornered convex polygon out of a circle, (ii) cutting a centered
polygon out of a circle, (iii) searching for the center of a sphere and (iv)
searching for the center of an ellipsoid. Below we discuss the summary of
our results, their applications and some possible future works associated with
those.
6.1
Cutting a cornered polygon out of a circle
We have studied the problem of cutting a cornered convex polygon out of a
circle using line cuts. We have given an algorithm with runs in linear time
and has an approximation ratio of O(n). We have also given a 6.74 constant
factor approximation algorithm with runs in O(n3 ) time. Before our works
there were no works done to cut a polygon out of a circle. We are the first
to work on this to our knowledge. And comparing to the algorithms so far
given by the researchers for cutting a polygon out of another, our algorithm is
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parallel in running time and approximation ratio. Moreover, we are hopeful
to extend our algorithm to device another useful way to cut out a polygon
out of another in similar fashion which will oversmart the currently used
algorithms.
6.2
Cutting a centered polygon out of a circle
We have studied the problem of cutting a centered polygon out of a circle. We
have given an algorithm which runs in linear time and has an approximation
ration of O(logn). We have also given an constant factor of 7.9 algorithm
which runs in O(n3 ) time.
We have merger the algorithms for cornered and centered convex polygon
to get the general solution of cutting a convex polygon out of a circle. For
determining whether a convex polygon is cornered of centered we have also
given a linear time algorithm.
Finally, we have given a polynomial time approximation scheme(PTAS)
for this problem.
Minimizing the cutting length of a geometric shape out of another one
is particularly important in metal cutting industries. It is also important in
diamond cutting industries. In computer graphics it can be helpful in rendering different geometric shapes to different frames. In some very recent works
we see that robot-surgeons are used to execute critical operations. Cutting
certain portions of human body by a robot can be aided by minimizing the
necessary cutting length. So, this algorithms are of enormous use both in
the theoretical fields and application of industry and medical science.
We have worked only on line cuts. The similar problems on ray cuts
are still to be approached. The approximation ratio can be improved. The
running time of the constant factor algorithm may get improved by using
rotating calipers algorithms. The PTAS may get improved by using better
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heuristics.
6.3
Searching for the center of a sphere
We have studied the problem of searching the center of a sphere. The strategy for searching for the center of a circle was previously given by various
researchers. But no algorithm was given for the three dimensional cases.
The two dimensional searching strategy is not applicable for three dimensional searching spaces. So, we have given an efficient strategy find out the
center of a sphere under certain limitations. We have also shown that our
strategy is correct and efficient. We have also suggested the solutions when
multiple signal level detection nobs are present.
6.4
Searching for the center of an ellipsoid
We have studied the problem when we have to search the center of an ellipsoid
under definite constraints. For a robot with minimum signal detection power
our given algorithm can search the center of an ellipsoid.
The algorithm of searching the center of an ellipse suffers from the problem of dead regions, so does the algorithm for searching the center of an
ellipsoid. In this thesis we have given a strategy which can be helpful in
avoiding the dead region problem. This strategy can be improved by using
different geometric property of ellipsoids.
The application of and efficient searching strategy of the center of an
sphere or an ellipsoid can be helpful in searching lost objects in binary detection sensor networks. It can also be used in monitoring traffic in busy roads
and the detection of lost starts in space. If we can extend the algorithm in
may be useful in determining the shape of black holes, too.
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Index
Approximation algorithm, 26
D-separation, 36
Approximation ratio, 27
Edge Cut, 21
Axis Diameter, 24
Great Circle, 24
Centered Convex Polygon, 21
computational geometry
line cut, 22
definition, 1
NP-Complete problem, 25
dynamic problems, 4
NP-Hard problem, 26
geometric problem, 3
main branches, 2
obtuse phase, 41
research areas, 2
optimal solution, 26
static problem, 2
optimization problem, 26
computational geometry’s application,
6
Perpendicular Great Circular Plane,
24
concave polygon, 21
Constant factor approximation algo-
Polynomial time approximation scheme,
27
rithm, 27
convex polygon, 20
ray cut, 23
cornered convex polygon, 21
curving phase, 43
triangle separation, 39
cut and uncut edges, 23
Vertex Cut, 21
cutting sequence, 22
Visible edges, 23
85
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