Reshaping Convex Polyhedra
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Reshaping Convex Polyhedra
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Author : Joseph O'Rourke
language : en
Publisher: Springer Nature
Release Date : 2024-02-28
Reshaping Convex Polyhedra written by Joseph O'Rourke and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-02-28 with Mathematics categories.
^ the="" study="" of="" convex="" polyhedra="" in="" ordinary="" space="" is="" a="" central="" piece="" classical="" and="" modern="" geometry="" that="" has="" had="" significant="" impact="" on="" many="" areas="" mathematics="" also="" computer="" science.="" present="" book="" project="" by="" joseph="" o’rourke="" costin="" vîlcu="" brings="" together="" two="" important="" strands="" subject="" —="" combinatorics="" polyhedra,="" intrinsic="" underlying="" surface.="" this="" leads="" to="" remarkable="" interplay="" concepts="" come="" life="" wide="" range="" very="" attractive="" topics="" concerning="" polyhedra.="" gets="" message="" across="" thetheory="" although="" with="" roots,="" still="" much="" alive="" today="" continues="" be="" inspiration="" basis="" lot="" current="" research="" activity.="" work="" presented="" manuscript="" interesting="" applications="" discrete="" computational="" geometry,="" as="" well="" other="" mathematics.="" treated="" detail="" include="" unfolding="" onto="" surfaces,="" continuous="" flattening="" convexity="" theory="" minimal="" length="" enclosing="" polygons.="" along="" way,="" open="" problems="" suitable="" for="" graduate="" students="" are="" raised,="" both="" a The focus of this monograph is converting—reshaping—one 3D convex polyhedron to another via an operation the authors call “tailoring.” A convex polyhedron is a gem-like shape composed of flat facets, the focus of study since Plato and Euclid. The tailoring operation snips off a corner (a “vertex”) of a polyhedron and sutures closed the hole. This is akin to Johannes Kepler’s “vertex truncation,” but differs in that the hole left by a truncated vertex is filled with new surface, whereas tailoring zips the hole closed. A powerful “gluing” theorem of A.D. Alexandrov from 1950 guarantees that, after closing the hole, the result is a new convex polyhedron. Given two convex polyhedra P, and Q inside P, repeated tailoringallows P to be reshaped to Q. Rescaling any Q to fit inside P, the result is universal: any P can be reshaped to any Q. This is one of the main theorems in Part I, with unexpected theoretical consequences. Part II carries out a systematic study of “vertex-merging,” a technique that can be viewed as a type of inverse operation to tailoring. Here the start is P which is gradually enlarged as much as possible, by inserting new surface along slits. In a sense, repeated vertex-merging reshapes P to be closer to planarity. One endpoint of such a process leads to P being cut up and “pasted” inside a cylinder. Then rolling the cylinder on a plane achieves an unfolding of P. The underlying subtext is a question posed by Geoffrey Shephard in 1975 and already implied by drawings by Albrecht Dürer in the 15th century: whether every convex polyhedron can be unfolded to a planar “net.” Toward this end, the authors initiate an exploration of convexity on convex polyhedra, a topic rarely studiedin the literature but with considerable promise for future development. This monograph uncovers new research directions and reveals connections among several, apparently distant, topics in geometry: Alexandrov’s Gluing Theorem, shortest paths and cut loci, Cauchy’s Arm Lemma, domes, quasigeodesics, convexity, and algorithms throughout. The interplay between these topics and the way the main ideas develop throughout the book could make the “journey” worthwhile for students and researchers in geometry, even if not directly interested in specific topics. Parts of the material will be of interest and accessible even to undergraduates. Although the proof difficulty varies from simple to quite intricate, with some proofs spanning several chapters, many examples and 125 figures help ease the exposition and illustrate the concepts. ^>
Discrete And Computational Geometry 2nd Edition
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Author : Satyan L. Devadoss
language : en
Publisher: Princeton University Press
Release Date : 2025-07-08
Discrete And Computational Geometry 2nd Edition written by Satyan L. Devadoss and has been published by Princeton University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-07-08 with Mathematics categories.
The essential introduction to discrete and computational geometry—now fully updated and expanded Discrete and Computational Geometry bridges the theoretical world of discrete geometry with the applications-driven realm of computational geometry, offering a comprehensive yet accessible introduction to this cutting-edge frontier of mathematics and computer science. Beginning with polygons and ending with polyhedra, it explains how to capture the shape of data given by a set of points, from convex hulls and triangulations to Voronoi diagrams, geometric duality, chains, linkages, and alpha complexes. Connections to real-world applications are made throughout, and algorithms are presented independent of any programming language. Now fully updated and expanded, this richly illustrated textbook is an invaluable learning tool for students in mathematics, computer science, engineering, and physics. Now with new sections on duality and on computational topology Project suggestions at the end of every chapter Covers traditional topics as well as new and advanced material Features numerous full-color illustrations, exercises, and fully updated unsolved problems Uniquely designed for a one-semester class Accessible to college sophomores with minimal background Also suitable for more advanced students Online solutions manual (available to instructors)
Discrete And Computational Geometry 2nd Edition
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Author : Satyan L. Devadoss
language : en
Publisher: Princeton University Press
Release Date : 2025-07-08
Discrete And Computational Geometry 2nd Edition written by Satyan L. Devadoss and has been published by Princeton University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-07-08 with Computers categories.
The essential introduction to discrete and computational geometry—now fully updated and expanded Discrete and Computational Geometry bridges the theoretical world of discrete geometry with the applications-driven realm of computational geometry, offering a comprehensive yet accessible introduction to this cutting-edge frontier of mathematics and computer science. Beginning with polygons and ending with polyhedra, it explains how to capture the shape of data given by a set of points, from convex hulls and triangulations to Voronoi diagrams, geometric duality, chains, linkages, and alpha complexes. Connections to real-world applications are made throughout, and algorithms are presented independent of any programming language. Now fully updated and expanded, this richly illustrated textbook is an invaluable learning tool for students in mathematics, computer science, engineering, and physics. Now with new sections on duality and on computational topology Project suggestions at the end of every chapter Covers traditional topics as well as new and advanced material Features numerous full-color illustrations, exercises, and fully updated unsolved problems Uniquely designed for a one-semester class Accessible to college sophomores with minimal background Also suitable for more advanced students Online solutions manual (available to instructors)
Reshaping College Mathematics
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Author : Mathematical Association of America. Committee on the Undergraduate Program in Mathematics
language : en
Publisher: Mathematical Association of America (MAA)
Release Date : 1989
Reshaping College Mathematics written by Mathematical Association of America. Committee on the Undergraduate Program in Mathematics and has been published by Mathematical Association of America (MAA) this book supported file pdf, txt, epub, kindle and other format this book has been release on 1989 with Mathematics categories.
International Business In A Vuca World
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Author : Rob van Tulder
language : en
Publisher: Emerald Group Publishing
Release Date : 2019-10-21
International Business In A Vuca World written by Rob van Tulder and has been published by Emerald Group Publishing this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-10-21 with Business & Economics categories.
Dedicated to Professor Peter Buckley, OBE, this volume of Progress in International Business Research explores the new challenges for MNEs, SMEs (small and medium sized enterprises) and INVs (International New Ventures) emerging from this changing and increasingly unpredictable political, economic, social and technological VUCA world.
Convex Polyhedra
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Author : A.D. Alexandrov
language : en
Publisher: Springer Science & Business Media
Release Date : 2005-12-08
Convex Polyhedra written by A.D. Alexandrov and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-12-08 with Mathematics categories.
Convex Polyhedra is one of the classics in geometry. There simply is no other book with so many of the aspects of the theory of 3-dimensional convex polyhedra in a comparable way, and in anywhere near its detail and completeness. It is the definitive source of the classical field of convex polyhedra and contains the available answers to the question of the data uniquely determining a convex polyhedron. This question concerns all data pertinent to a polyhedron, e.g. the lengths of edges, areas of faces, etc. This vital and clearly written book includes the basics of convex polyhedra and collects the most general existence theorems for convex polyhedra that are proved by a new and unified method. It is a wonderful source of ideas for students. The English edition includes numerous comments as well as added material and a comprehensive bibliography by V.A. Zalgaller to bring the work up to date. Moreover, related papers by L.A.Shor and Yu.A.Volkov have been added as supplements to this book.
An Introduction To Ray Tracing
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Author : Andrew S. Glassner
language : en
Publisher: Elsevier
Release Date : 1989-06-01
An Introduction To Ray Tracing written by Andrew S. Glassner and has been published by Elsevier this book supported file pdf, txt, epub, kindle and other format this book has been release on 1989-06-01 with Computers categories.
The creation of ever more realistic 3-D images is central to the development of computer graphics. The ray tracing technique has become one of the most popular and powerful means by which photo-realistic images can now be created. The simplicity, elegance and ease of implementation makes ray tracing an essential part of understanding and exploiting state-of-the-art computer graphics.An Introduction to Ray Tracing develops from fundamental principles to advanced applications, providing "how-to" procedures as well as a detailed understanding of the scientific foundations of ray tracing. It is also richly illustrated with four-color and black-and-white plates. This is a book which will be welcomed by all concerned with modern computer graphics, image processing, and computer-aided design. - Provides practical "how-to" information - Contains high quality color plates of images created using ray tracing techniques - Progresses from a basic understanding to the advanced science and application of ray tracing
Kepler S T Bingen
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Author : Charlotte Methuen
language : en
Publisher: Routledge
Release Date : 1998
Kepler S T Bingen written by Charlotte Methuen and has been published by Routledge this book supported file pdf, txt, epub, kindle and other format this book has been release on 1998 with History categories.
Using previously unstudied sources, this interdisciplinary study considers theology and the beginnings of modern science at the University of TÃ1/4bingen in the time of Johannes Kepler (1571-1630). The author casts light upon the origins of modern scientific method by examining the relationship between theology, astronomy and dialectics at the university in the work of Kepler's teachers. Studies of Kepler generally treat him as a precursor of the modern scientist; the influences upon him are identified as Platonist or Pythagorean and his theological interests have often been ignored, or considered as a mystical aberration, unworthy of in-depth treatment. There has been no serious attempt to place Kepler's work in the wider tradition of mainstream 16th-century thought. This study portrays and analyses the influences and ideas which permeated the life of the university in TÃ1/4bingen in the second half of the 16th century and places them in relationship to the theology of Martin Luther and Philip Melanchthon. It pays particular attention to the use of theological concepts, astronomical observations, logical demonstrations and the categories of physics, and to the interplay between them.
Convex Polyhedra
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Author : Aleksandr Danilovich Aleksandrov
language : en
Publisher:
Release Date : 2010
Convex Polyhedra written by Aleksandr Danilovich Aleksandrov and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010 with Convex surfaces categories.
Convex Polyhedra is one of the classics in geometry. There simply is no other book with so many of the aspects of the theory of 3-dimensional convex polyhedra in a comparable way, and in anywhere near its detail and completeness. It is the definitive source of the classical field of convex polyhedra and contains the available answers to the question of the data uniquely determining a convex polyhedron. This question concerns all data pertinent to the poly hedron, e.g. the lengths of edges, areas of faces, etc. This viatal and clearly written book includes the basics of convex polyhedra and collects the most general existence theorems for convex polyhedra that are proved by a new and unified method. It is a wonderful source of ideas for students. The English edition includes numerous comments as well as added material and a comprehensive bibliography by V.A. Zalgaller to bring the work up to date. Moreover, related papers by L.A. Shor and Yu. A. Volkov have been added as supplements to this book.
Shaping Space
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Author : Marjorie Senechal
language : en
Publisher:
Release Date : 1988
Shaping Space written by Marjorie Senechal and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1988 with Mathematics categories.