In 19 articles presented by leading experts in the field of geometric modelling the state-of-the-art on representing, modeling, and analyzing curves, surfaces as well as other 3-dimensional geometry is given. The range of applications include CAD/CAM-systems, computer graphics, scientific visualization, virtual reality, simulation and medical imaging. The content of this book is based on selected lectures given at a workshop held at IBFI Schloss Dagstuhl, Germany. Topics treated are: - curve and surface modelling - non-manifold modelling in CAD - multiresolution analysis of complex geometric models - surface reconstruction - variational design - computational geometry of curves and surfaces - 3D meshing - geometric modelling for scientific visualization - geometric models for biomedical applications

There has been a resurgence of interest in classical invariant theory driven by several factors: new theoretical developments; a revival of computational methods coupled with powerful new computer algebra packages; and a wealth of new applications, ranging from number theory to geometry, physics to computer vision. This book provides readers with a self-contained introduction to the classical theory as well as modern developments and applications. The text concentrates on the study of binary forms (polynomials) in characteristic zero, and uses analytical as well as algebraic tools to study and classify invariants, symmetry, equivalence and canonical forms. A variety of innovations make this text of interest even to veterans of the subject; these include the use of differential operators and the transform approach to the symbolic method, extension of results to arbitrary functions, graphical methods for computing identities and Hilbert bases, complete systems of rationally and functionally independent covariants, introduction of Lie group and Lie algebra methods, as well as a new geometrical theory of moving frames and applications. Aimed at advanced undergraduate and graduate students the book includes many exercises and historical details, complete proofs of the fundamental theorems, and a lively and provocative exposition.

New variational methods by Aubry, Mather, and Mane, discovered in the last twenty years, gave deep insight into the dynamics of convex Lagrangian systems. This book shows how this Principle of Least Action appears in a variety of settings (billiards, length spectrum, Hofer geometry, modern symplectic geometry). Thus, topics from modern dynamical systems and modern symplectic geometry are linked in a new and sometimes surprising way. The central object is Mather 's minimal action functional. The level is for graduate students onwards, but also for researchers in any of the subjects touched in the book.

Algebraic projective geometry, with its multilinear relations and its embedding into Grassmann-Cayley algebra, has become the basic representation of multiple view geometry, resulting in deep insights into the algebraic structure of geometric relations, as well as in efficient and versatile algorithms for computer vision and image analysis.

This book provides a coherent integration of algebraic projective geometry and spatial reasoning under uncertainty with applications in computer vision. Beyond systematically introducing the theoretical foundations from geometry and statistics and clear rules for performing geometric reasoning under uncertainty, the author provides a collection of detailed algorithms.

The book addresses researchers and advanced students interested in algebraic projective geometry for image analysis, in statistical representation of objects and transformations, or in generic tools for testing and estimating within the context of geometric multiple-view analysis.

In recent years, geometry has played a lesser role in undergraduate courses than it has ever done. Nevertheless, it still plays a leading role in mathematics at a higher level. Its central role in the history of mathematics has never been disputed. It is important, therefore, to introduce some geometry into university syllabuses. There are several ways of doing this, it can be incorporated into existing courses that are primarily devoted to other topics, it can be taught at a first year level or it can be taught in higher level courses devoted to differential geometry or to more classical topics. These notes are intended to fill a rather obvious gap in the literature. It treats the classical topics of Euclidean, projective and hyperbolic geometry but uses the material commonly taught to undergraduates: linear algebra, group theory, metric spaces and complex analysis. The notes are based on a course whose aim was two fold, firstly, to introduce the students to some geometry and secondly to deepen their understanding of topics that they have already met. What is required from the earlier material is a familiarity with the main ideas, specific topics that are used are usually redone.

Among the intuitively appealing aspects of graph theory is its close connection to drawings and geometry. The development of computer technology has become a source of motivation to reconsider these connections, in particular geometric graphs are emerging as a new subfield of graph theory. Arrangements of points and lines are the objects for many challenging problems and surprising solutions in combinatorial geometry. The book is a collection of beautiful and partly very recent results from the intersection of geometry, graph theory and combinatorics.

This book constitutes the thoroughly refereed post-proceedings of the 4th International Workshop on Automated Deduction in Geometry, ADG 2002, held at Hagenberg Castle, Austria in September 2002.

The 13 revised full papers presented were carefully selected during two rounds of reviewing and improvement. Among the issues addressed are theoretical and methodological topics, such as the resolution of singularities, algebraic geometry and computer algebra; various geometric theorem proving systems are explored; and applications of automated deduction in geometry are demonstrated in fields like computer-aided design and robotics.

The Morse-Sard theorem is a rather subtle result and the interplay between the high-order analytic structure of the mappings involved and their geometry rarely becomes apparent. The main reason is that the classical Morse-Sard theorem is basically qualitative. This volume gives a proof and also an "explanation" of the quantitative Morse-Sard theorem and related results, beginning with the study of polynomial (or tame) mappings. The quantitative questions, answered by a combination of the methods of real semialgebraic and tame geometry and integral geometry, turn out to be nontrivial and highly productive. The important advantage of this approach is that it allows the separation of the role of high differentiability and that of algebraic geometry in a smooth setting: all the geometrically relevant phenomena appear already for polynomial mappings. The geometric properties obtained are "stable with respect to approximation," and can be imposed on smooth functions via polynomial approximation.

This text takes a practical, step-by-step approach to algebraic curves and surface interpolation motivated by the understanding of the many practical applications in engineering analysis, approximation, and curve plotting problems. Because of its usefulness for computing, the algebraic approach is the main theme, but a brief discussion of the synthetic approach is also presented as a way of gaining additional insight before proceeding with the algebraic manipulation. The authors start with simple interpolation, including splines, and extend this in an intuitive fashion to the production of conic sections. They then introduce projective co-ordinates as tools for dealing with higher order curves and singular points. They present many applications and concrete examples, including parabolic interpolation, geometric approximation, and the numerical solution of trajectory problems. In the final chapter they apply the basic theory to the construction of finite element basis functions and surface interpolants over non-regular shapes.

Broad appeal to undergraduate teachers, students, and engineers; Concise descriptions of properties of basic planar curves from different perspectives; useful handbook for software engineers; A special chapter---"Geometry on the Web"---will further enhance the usefulness of this book as an informal tutorial resource.; Good mathematical notation, descriptions of properties of lines and curves, and the illustration of geometric concepts facilitate the design of computer graphics tools and computer animation.; Video game designers, for example, will find a clear discussion and illustration of hard-to-understand trajectory design concepts.; Good supplementary text for geometry courses at the undergraduate and advanced high school levels

Spaces of holomorphic functions have been a prominent theme in analysis since early in the twentieth century. Of interest to complex analysts, functional analysts, operator theorists, and systems theorists, their study is now flourishing. This volume, an outgrowth of a 1995 program at the Mathematical Sciences Research Institute, contains expository articles by program participants describing the present state of the art. Here researchers and graduate students will encounter Hardy spaces, Bergman spaces, Dirichlet spaces, Hankel and Toeplitz operators, and a sampling of the role these objects play in modern analysis.

Since it was ?rst organized in 1997, the Japan Conference on Discrete and C- putational Geometry (JCDCG) continues to attract an international audience. The ?rst ?ve conferences of the series were held in Tokyo, the sixth in Manila, Philippines. This volume consists of the refereed papers presented at the seventh conference, JCDCG 2002, held in Tokai University, Tokyo, December 6-9, 2002. An eighth conference is planned to be held in Bandung, Indonesia. The proceedings of JCDCG 1998 and JCDCG 2000 were published by Springer-Verlag as part of the series Lecture Notes in Computer Science: LNCS volumes 1763 and 2098, respectively. The proceedings of JCDCG 2001 were also published by Springer-Verlag as a special issue of the journal Graphs and C- binatorics, Vol. 18, No. 4, 2002. The organizers are grateful to Tokai University for sponsoring the conf- ence. They wish to thank all the people who contributed to the success of the conference, in particular, Chie Nara, who headed the conference secretariat, and the principal speakers: Takao Asano, David Avis, Greg N. Frederickson, Ferran Hurtado, Joseph O'Rourke, J anos Pach, Rom Pinchasi, and Jorge Urrutia."

The new edition of this non-mathematical review of catastrophe theory contains updated results and many new or expanded topics including delayed loss of stability, shock waves, and interior scattering. Three new sections offer the history of singularity and its applications from da Vinci to today, a discussion of perestroika in terms of the theory of metamorphosis, and a list of 93 problems touching on most of the subject matter in the book.

This work on the foundations of Euclidean geometry aims to present the subject from the point of view of mathematics at the end of the 20th century, taking advantage of all the developments since the appearance of Hilbert's classic work. Here real affine space is characterized by a small number of axioms involving points and line segments making the treatment self-contained and thorough, many results being established under weaker hypotheses than usual. The treatment should be accessible for final year undergraduates and graduate students, and can also serve as an introduction to other areas of mathematics such as matroids and antimatroids, combinatorial convexity, the theory of polytopes, projective geometry and functional analysis.

This proceedings volume includes papers presented at DGCI 2003 in Naples, Italy, November 19-21, 2003. DGCI 2003 was the 11th conference in a series of internationalconferencesonDiscreteGeometryforComputerImagery.Thec- ference was organized by the Italian Institute for Philosophical Studies, Naples and the Institute of Cybernetics "E. Caianiello," National Research Council of Italy, Pozzuoli (Naples). DGCI 2003 was sponsored by the International Asso- ation for Pattern Recognition (IAPR). ThisisthesecondtimetheconferencetookplaceoutsideFrance.Thenumber ofresearchersactiveinthe?eldofdiscretegeometryandcomputerimageryis- creasing. Both these factors contribute to the increased international recognition of the conference. The DGCI conferences attract more and more academic and research institutions in di?erent countries. In fact, 68 papers were submitted to DGCI2003.Thecontributionsfocusondiscretegeometryandtopology, surfaces and volumes, morphology, shape representation, and shape analysis. After ca- ful reviewing by an international board of reviewers, 23 papers were selected for oral presentation and 26 for poster presentation. All contributions were sch- uled in plenary sessions. In addition, the program was enriched by three l- tures, presented by internationally well-known invited speakers: Isabelle Bloch (EcoleNationaleSup erieuredesT el ecommunications, France), LonginJanLa- cki(TempleUniversity, USA), andRalphKopperman(CityCollegeofNewYork, USA). In 2002, a technical committee of the IAPR, TC18, was established with the intention to promote interactions and collaboration between researchers wo- ing on discrete geometry. The ?rst TC18 meeting was planned to be held in conjunction with DGCI 2003, to allow the members to discuss the activity of the technical committee. The outcome from this meeting will help the ongoing research and communication for researchers active within the ?eld during the 18 months between the conferences."

Manifolds are the central geometric objects in modern mathematics. An attempt to understand the nature of manifolds leads to many interesting questions. One of the most obvious questions is the following. Let M and N be manifolds: how can we decide whether M and N are ho- topy equivalent or homeomorphic or di?eomorphic (if the manifolds are smooth)? The prototype of a beautiful answer is given by the Poincar e Conjecture. If n N is S, the n-dimensional sphere, and M is an arbitrary closed manifold, then n it is easy to decide whether M is homotopy equivalent to S . Thisisthecaseif and only if M is simply connected (assumingn> 1, the case n = 1 is trivial since 1 every closed connected 1-dimensional manifold is di?eomorphic toS ) and has the n homology of S . The PoincareConjecture states that this is also su?cient for the n existenceof ahomeomorphism fromM toS . For n = 2this followsfromthewe- known classi?cation of surfaces. Forn> 4 this was proved by Smale and Newman in the 1960s, Freedman solved the case in n = 4 in 1982 and recently Perelman announced a proof for n = 3, but this proof has still to be checked thoroughly by the experts. In the smooth category it is not true that manifolds homotopy n equivalent to S are di?eomorphic. The ?rst examples were published by Milnor in 1956 and together with Kervaire he analyzed the situation systematically in the 1960s."

Kodaira is a Fields Medal Prize Winner. (In the absence of a Nobel prize in mathematics, they are regarded as the highest professional honour a mathematician can attain.)

Kodaira is an honorary member of the London Mathematical Society.

Affordable softcover edition of 1986 classic

This 1998 book describes the progress that had been made towards the development of a comprehensive understanding of the formation of complex, disorderly patterns under conditions far from equilibrium. The application of fractal geometry and scaling concepts to the quantitative description and understanding of structure formed under non-equilibrium conditions is described. Self-similar fractals, multi-fractals and scaling methods are discussed, with examples, to facilitate applications in the physical sciences. Computer simulations and experimental studies are emphasised, but the author also includes discussion of theoretical advances in the subject. Much of the book deals with diffusion-limited growth processes and the evolution of rough surfaces, although a broad range of other applications is also included. The techniques and topics will be relevant to graduate students and researchers in physics, chemistry, materials science, engineering and the earth sciences, interested in applying the ideas of fractals and scaling.

Flavors of Geometry is a volume of lectures on four geometrically-influenced fields of mathematics that have experienced great development in recent years. Growing out of a series of introductory lectures given at the Mathematical Sciences Research Institute in January 1995 and January 1996, the book presents chapters by masters in their respective fields on hyperbolic geometry, dynamics in several complex variables, convex geometry, and volume estimation. Each lecture begins with a discussion of elementary concepts, examines the highlights of the field, and concludes with a look at more advanced material. The style and presentation of the chapters are clear and accessible, and most of the lectures are richly illustrated. Bibiliographies and indexes are included to encourage further reading on the topics discussed.

The exposition studies projective models of K3 surfaces whose hyperplane sections are non-Clifford general curves. These models are contained in rational normal scrolls. The exposition supplements standard descriptions of models of general K3 surfaces in projective spaces of low dimension, and leads to a classification of K3 surfaces in projective spaces of dimension at most 10. The authors bring further the ideas in Saint-Donat's classical article from 1974, lifting results from canonical curves to K3 surfaces and incorporating much of the Brill-Noether theory of curves and theory of syzygies developed in the mean time.

This volume presents up-to-date research on finite geometries and designs, a key area in modern applicable mathematics. An introductory chapter discusses topics presented in each of the main chapters, and is followed by articles from leading international figures in this field. These include a discussion of the current state of finite geometry from a group-theoretical viewpoint, and surveys of difference sets and of small embeddings of partial cycle systems into Steiner triple systems. Also presented are successful searches for spreads and packing of designs, rank three geometries with simplicial residues and generalized quadrangles satisfying Veblen's Axiom. In addition, there are articles on new 7-designs, biplanes, various aspects of triple systems, and many other topics. This book will be a useful reference for researchers working in finite geometries, design theory or combinatorics in general.

'The book is well-illustrated, earlier chapters with monochrome portraits of Mandelbrot, his family and those who influenced him, and later ones with striking colour pictures not only of the Mandelbrot set and other computer generated fractals, but also of aEURO~realaEURO (TM) fractals including cloud formations and rural and mountain scenes ... This celebration of MandelbrotaEURO (TM)s scientific life is largely based on interviews that the author had with him when making films on his work ... A challenge for historians of mathematics and science in coming years will be to produce a more broadly contextual and rounded account of the advent of fractals.'London Math SocietyThe time is right, following Benoit Mandelbrot's death in 2010, to publish this landmark book about the life and work of this maverick math genius.This compact book celebrates the life and achievements of Benoit Mandelbrot with the ideas of fractals presented in a way that can be understood by the interested lay-person. Mathematics is largely avoided. Instead, Mandelbrot's ideas and insights are described using a combination of intuition and pictures. The early part of the book is largely biographical, but it portrays well how Mandelbrot's life and ideas developed and led to the fractal notions that are surveyed in the latter parts of the book.

Combinatorics of Finite Geometries is an introductory text on the combinatorial theory of finite geometry. Assuming only a basic knowledge of set theory and analysis, it provides a thorough review of the topic and leads the student to results at the frontiers of research. This book begins with an elementary combinatorial approach to finite geometries based on finite sets of points and lines, and moves into the classical work on affine and projective planes. Later, it addresses polar spaces, partial geometries, and generalized quadrangles. The revised edition contains an entirely new chapter on blocking sets in linear spaces, which highlights some of the most important applications of blocking sets--from the initial game-theoretic setting to their very recent use in cryptography. Extensive exercises at the end of each chapter insure the usefulness of this book for senior undergraduate and beginning graduate students.

Combinatorics of Finite Geometries is an introductory text on the combinatorial theory of finite geometry. Assuming only a basic knowledge of set theory and analysis, it provides a thorough review of the topic and leads the student to results at the frontiers of research. This book begins with an elementary combinatorial approach to finite geometries based on finite sets of points and lines, and moves into the classical work on affine and projective planes. Later, it addresses polar spaces, partial geometries, and generalized quadrangles. The revised edition contains an entirely new chapter on blocking sets in linear spaces, which highlights some of the most important applications of blocking sets--from the initial game-theoretic setting to their very recent use in cryptography. Extensive exercises at the end of each chapter insure the usefulness of this book for senior undergraduate and beginning graduate students.

Proceedings of the Conference on Algebra and Algebraic Geometry with Applications, July 19 – 26, 2000, at Purdue University to honor Professor Shreeram S. Abhyankar on the occasion of his seventieth birthday. Eighty-five of Professor Abhyankar's students, collaborators, and colleagues were invited participants. Sixty participants presented papers related to Professor Abhyankar's broad areas of mathematical interest. Sessions were held on algebraic geometry, singularities, group theory, Galois theory, combinatorics, Drinfield modules, affine geometry, and the Jacobian problem. This volume offers an outstanding collection of papers by expert authors.