The volume reports on interdisciplinary discussions and interactions between theoretical research and practical studies on geometric structures and their applications in architecture, the arts, design, education, engineering, and mathematics. These related fields of research can enrich each other and renew their mutual interest in these topics through networks of shared inspiration, and can ultimately enhance the quality of geometry and graphics education. Particular attention is dedicated to the contributions that women have made to the scientific community and especially mathematics. The book introduces engineers, architects and designers interested in computer applications, graphics and geometry to the latest advances in the field, with a particular focus on science, the arts and mathematics education.

Different Faces of Geometry - edited by the world renowned geometers S. Donaldson, Ya. Eliashberg, and M. Gromov - presents the current state, new results, original ideas and open questions from the following important topics in modern geometry: These apparently diverse topics have a common feature in that they are all areas of exciting current activity. The Editors have attracted an impressive array of leading specialists to author chapters for this volume: G. Mikhalkin (USA-Canada-Russia), V.D. Milman (Israel) and A.A. Giannopoulos (Greece), C. LeBrun (USA), Ko Honda (USA), P. Ozsvath (USA) and Z. Szabo (USA), C. Simpson (France), D. Joyce (UK) and P. Seidel (USA), and S. Bauer (Germany). One can distinguish various themes running through the different contributions. There is some emphasis on invariants defined by elliptic equations and their applications in low-dimensional topology, symplectic and contact geometry (Bauer, Seidel, Ozsvath and Szabo). These ideas enter, more tangentially, in the articles of Joyce, Honda and LeBrun.Here and elsewhere, as well as explaining the rapid advances that have been made, the articles convey a wonderful sense of the vast areas lying beyond our current understanding. Simpson's article emphasizes the need for interesting new constructions (in that case of Kahler and algebraic manifolds), a point which is also made by Bauer in the context of 4-manifolds and the 11/8 conjecture. LeBrun's article gives another perspective on 4-manifold theory, via Riemannian geometry, and the challenging open questions involving the geometry of even well-known 4-manifolds. There are also striking contrasts between the articles. The authors have taken different approaches: for example, the thoughtful essay of Simpson, the new research results of LeBrun and the thorough expositions with homework problems of Honda. One can also ponder the differences in the style of mathematics. In the articles of Honda, Giannopoulos and Milman, and Mikhalkin, the geometry is present in a very vivid and tangible way; combining respectively with topology, analysis and algebra.The papers of Bauer and Seidel, on the other hand, makes the point that algebraic and algebro-topological abstraction (triangulated categories, spectra) can play an important role in very unexpected ways in concrete geometric problems. - From the Preface by the Editors

Human faces are familiar to our visual systems. We easily recognize a person's face in arbitrary lighting conditions and in a variety of poses; detect small appearance changes; and notice subtle expression details. Can computer vision systems process face images as well as human vision systems can? Face image processing has potential applications in surveillance, image and video search, social networking and other domains. A comprehensive guide to this fascinating topic, this book provides a systematic description of modeling face geometry and appearance from images, including information on mathematical tools, physical concepts, image processing and computer vision techniques, and concrete prototype systems. The book will be an excellent reference for researchers and graduate students in computer vision, computer graphics and multimedia, as well as application developers who would like to gain a better understanding of the state of the art.

Forensic art may be defined as 'portrait art minus a tangible subject.' The main objective of this book is to present a series of practical indices interrelating the key features of the human face that will provide a foundation for any exercise in forensic art from composite sketch to post-mortem 're-facing.' These indices are illustrated with a survey of the numerous and often surprising geometric forms that permeate facial design. The various triangles and rectangles, rhomboids and trapezoids, parallelograms and circles that define the human face (the theme) and give it individuality (variations on the theme) are examined. The chapters provide necessary information to define the cephalometric points, planes, areas and lines that demarcate the human face, including the detailed surface anatomy of the eye, nose, mouth and ear. The underlying geometry of the human facial plan is revealed, illustrating a selection of triangles, rectangles, and other polygons. The graphic facial analysis (GFA) of the frontal face is covered, with sixteen indices and triangles defining and illustrating their means and ranges of variation. The GFA details the lateral face by means of eight angles and indices with special attention given to the nose and ear. With 45 illustrations and two tables in this clear and comprehensive text, this book leaves little to the imagination and is truly a unique treatise and source of information.

3D Face Processing: Modeling, Analysis and Synthesis introduces the frontiers of 3D face processing techniques. It reviews existing 3D face processing techniques, including techniques for 3D face geometry modeling; 3D face motion modeling; and 3D face motion tracking and animation. Then it discusses a unified framework for face modeling, analysis and synthesis. In this framework, the authors present new methods for modeling complex natural facial motion, as well as face appearance variations due to illumination and subtle motion. Then the authors apply the framework to face tracking, expression recognition and face avatar for HCI interface. They conclude this book with comments on future work in the 3D face processing framework. 3D Face Processing: Modeling, Analysis and Synthesis will interest those working in face processing for intelligent human computer interaction and video surveillance. It contains a comprehensive survey on existing face processing techniques, which can serve as a reference for students and researchers. It also covers in-depth discussion on face motion analysis and synthesis algorithms, which will benefit more advanced graduate students and researchers.

This book is unique in that it looks at geometry from 4 different viewpoints - Euclid-style axioms, linear algebra, projective geometry, and groups and their invariants Approach makes the subject accessible to readers of all mathematical tastes, from the visual to the algebraic Abundantly supplemented with figures and exercises

This book constitutes the refereed proceedings of the Second International Conference on Intelligence Science, ICIS 2017, held in Shanghai, China, in October 2017. The 38 full papers and 9 short papers presented were carefully reviewed and selected from 82 submissions. They deal with key issues in intelligence science and have been organized in the following topical sections: theory of intelligence science; cognitive computing; big data analysis and machine learning; machine perception; intelligent information processing; and intelligent applications.

Elementary, yet authoritative and scholarly, this book offers an excellent brief introduction to the classical theory of differential geometry. It is aimed at advanced undergraduate and graduate students who will find it not only highly readable but replete with illustrations carefully selected to help stimulate the student's visual understanding of geometry. The text features an abundance of problems, most of which are simple enough for class use, and often convey an interesting geometrical fact. A selection of more difficult problems has been included to challenge the ambitious student. Written by a noted mathematician and historian of mathematics, this volume presents the fundamental conceptions of the theory of curves and surfaces and applies them to a number of examples. Dr. Struik has enhanced the treatment with copious historical, biographical, and bibliographical references that place the theory in context and encourage the student to consult original sources and discover additional important ideas there. For this second edition, Professor Struik made some corrections and added an appendix with a sketch of the application of Cartan's method of Pfaffians to curve and surface theory. The result was to further increase the merit of this stimulating, thought-provoking text — ideal for classroom use, but also perfectly suited for self-study. In this attractive, inexpensive paperback edition, it belongs in the library of any mathematician or student of mathematics interested in differential geometry.

This book is a comprehensive reference on differential geometry. It shows that Maxwell, Dirac and Einstein fields, which were originally considered objects of a very different mathematical nature, have representatives as objects of the same mathematical nature. The book also analyzes some foundational issues of relativistic field theories. All calculation procedures are illustrated by many exercises that are solved in detail.

This text presents a graduate-level introduction to differential geometry for mathematics and physics students. The exposition follows the historical development of the concepts of connection and curvature with the goal of explaining the Chern–Weil theory of characteristic classes on a principal bundle. Along the way we encounter some of the high points in the history of differential geometry, for example, Gauss' Theorema Egregium and the Gauss–Bonnet theorem. Exercises throughout the book test the reader’s understanding of the material and sometimes illustrate extensions of the theory. Initially, the prerequisites for the reader include a passing familiarity with manifolds. After the first chapter, it becomes necessary to understand and manipulate differential forms. A knowledge of de Rham cohomology is required for the last third of the text. Prerequisite material is contained in author's text An Introduction to Manifolds, and can be learned in one semester. For the benefit of the reader and to establish common notations, Appendix A recalls the basics of manifold theory. Additionally, in an attempt to make the exposition more self-contained, sections on algebraic constructions such as the tensor product and the exterior power are included. Differential geometry, as its name implies, is the study of geometry using differential calculus. It dates back to Newton and Leibniz in the seventeenth century, but it was not until the nineteenth century, with the work of Gauss on surfaces and Riemann on the curvature tensor, that differential geometry flourished and its modern foundation was laid. Over the past one hundred years, differential geometry has proven indispensable to an understanding of the physical world, in Einstein's general theory of relativity, in the theory of gravitation, in gauge theory, and now in string theory. Differential geometry is also useful in topology, several complex variables, algebraic geometry, complex manifolds, and dynamical systems, among other fields. The field has even found applications to group theory as in Gromov's work and to probability theory as in Diaconis's work. It is not too far-fetched to argue that differential geometry should be in every mathematician's arsenal.