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Computational geometry as an area of research in its own right
emerged in the early seventies of this century. Right from the
beginning, it was obvious that strong connections of various kinds
exist to questions studied in the considerably older field of
combinatorial geometry. For example, the combinatorial structure of
a geometric problem usually decides which algorithmic method solves
the problem most efficiently. Furthermore, the analysis of an
algorithm often requires a great deal of combinatorial knowledge.
As it turns out, however, the connection between the two research
areas commonly referred to as computa tional geometry and
combinatorial geometry is not as lop-sided as it appears. Indeed,
the interest in computational issues in geometry gives a new and
con structive direction to the combinatorial study of geometry. It
is the intention of this book to demonstrate that computational and
com binatorial investigations in geometry are doomed to profit from
each other. To reach this goal, I designed this book to consist of
three parts, acorn binatorial part, a computational part, and one
that presents applications of the results of the first two parts.
The choice of the topics covered in this book was guided by my
attempt to describe the most fundamental algorithms in
computational geometry that have an interesting combinatorial
structure. In this early stage geometric transforms played an
important role as they reveal connections between seemingly
unrelated problems and thus help to structure the field."
Computational geometry as an area of research in its own right
emerged in the early seventies of this century. Right from the
beginning, it was obvious that strong connections of various kinds
exist to questions studied in the considerably older field of
combinatorial geometry. For example, the combinatorial structure of
a geometric problem usually decides which algorithmic method solves
the problem most efficiently. Furthermore, the analysis of an
algorithm often requires a great deal of combinatorial knowledge.
As it turns out, however, the connection between the two research
areas commonly referred to as computa tional geometry and
combinatorial geometry is not as lop-sided as it appears. Indeed,
the interest in computational issues in geometry gives a new and
con structive direction to the combinatorial study of geometry. It
is the intention of this book to demonstrate that computational and
com binatorial investigations in geometry are doomed to profit from
each other. To reach this goal, I designed this book to consist of
three parts, acorn binatorial part, a computational part, and one
that presents applications of the results of the first two parts.
The choice of the topics covered in this book was guided by my
attempt to describe the most fundamental algorithms in
computational geometry that have an interesting combinatorial
structure. In this early stage geometric transforms played an
important role as they reveal connections between seemingly
unrelated problems and thus help to structure the field."
This monograph presents a short course in computational geometry
and topology. In the first part the book covers Voronoi diagrams
and Delaunay triangulations, then it presents the theory of alpha
complexes which play a crucial role in biology. The central part of
the book is the homology theory and their computation, including
the theory of persistence which is indispensable for applications,
e.g. shape reconstruction. The target audience comprises
researchers and practitioners in mathematics, biology, neuroscience
and computer science, but the book may also be beneficial to
graduate students of these fields.
This book combines mathematics (geometry and topology), computer science (algorithms), and engineering (mesh generation) in order to solve the conceptual and technical problems in the combining of elements of combinatorial and numerical algorithms. The book develops methods from areas that are amenable to combination and explains recent breakthrough solutions to meshing that fit into this category. It should be an ideal graduate text for courses on mesh generation. The specific material is selected giving preference to topics that are elementary, attractive, lend themselves to teaching, are useful, and interesting.
The book combines topics in mathematics (geometry and topology),
computer science (algorithms), and engineering (mesh generation).
The motivation for these topics is the difficulty, both
conceptually and in the technical execution, of combining elements
of combinatorial and of numerical algorithms. Mesh generation is a
topic where a meaningful combination of these different approaches
to problem solving is inevitable. The book develops methods from
both areas that are amenable to combination, and explains
breakthrough solutions to meshing that fit into this category. This
book emphasizes topics that are elementary, attractive, useful,
interesting, and lend themselves to teaching, making it an ideal
graduate text for courses on mesh generation.
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