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The projective, Möbius, Laguerre, and Minkowski planes over the real numbers are just a few examples of a host of fundamental classical topological geometries on surfaces. This book summarizes all known major results and open problems related to these classical point-line geometries and their close (nonclassical) relatives. Topics covered include: classical geometries; methods for constructing nonclassical geometries; classifications and characterizations of geometries. This work is related to many other fields including interpolation theory, convexity, the theory of pseudoline arrangements, topology, the theory of Lie groups, and many more. The authors detail these connections, some of which are well-known, but many much less so. Acting both as a reference for experts and as an accessible introduction for graduate students, this book will interest anyone wishing to know more about point-line geometries and the way they interact.

Geometry, Projective. --- Surfaces. --- Curved surfaces --- Geometry --- Shapes --- Projective geometry --- Geometry, Modern

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Did you know that any straight-line drawing on paper can be folded so that the complete drawing can be cut out with one straight scissors cut? That there is a planar linkage that can trace out any algebraic curve, or even 'sign your name'? Or that a 'Latin cross' unfolding of a cube can be refolded to 23 different convex polyhedra? Over the past decade, there has been a surge of interest in such problems, with applications ranging from robotics to protein folding. With an emphasis on algorithmic or computational aspects, this treatment gives hundreds of results and over 60 unsolved 'open problems' to inspire further research. The authors cover one-dimensional (1D) objects (linkages), 2D objects (paper), and 3D objects (polyhedra). Aimed at advanced undergraduate and graduate students in mathematics or computer science, this lavishly illustrated book will fascinate a broad audience, from school students to researchers.

Polyhedra --- Polyèdres --- Models --- Data processing --- Modèles --- Geometrical models --- Polyhedral figures --- Polyhedrons --- Geometry, Solid --- Shapes --- Data processing. --- Models.

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Many applications in science and engineering require a digital model of a real physical object. Advanced scanning technology has made it possible to scan such objects and generate point samples on their boundaries. This book, first published in 2007, shows how to compute a digital model from this point sample. After developing the basics of sampling theory and its connections to various geometric and topological properties, the author describes a suite of algorithms that have been designed for the reconstruction problem, including algorithms for surface reconstruction from dense samples, from samples that are not adequately dense and from noisy samples. Voronoi- and Delaunay-based techniques, implicit surface-based methods and Morse theory-based methods are covered. Scientists and engineers working in drug design, medical imaging, CAD, GIS, and many other areas will benefit from this first book on the subject.

Curves on surfaces --- Surfaces --- Surfaces, Models of. --- Models of surfaces --- Models and modelmaking --- Curved surfaces --- Geometry --- Shapes --- Surfaces, Curves on --- Mathematical models.

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The publication of the first edition of Lagerungen in der Ebene, auf der Kugel und im Raum in 1953 marked the birth of discrete geometry. Since then, the book has had a profound and lasting influence on the development of the field. It included many open problems and conjectures, often accompanied by suggestions for their resolution. A good number of new results were surveyed by László Fejes Tóth in his Notes to the 2nd edition. The present version of Lagerungen makes this classic monograph available in English for the first time, with updated Notes, completed by extensive surveys of the state of the art. More precisely, this book consists of: a corrected English translation of the original Lagerungen, the revised and updated Notes on the original text, eight self-contained chapters surveying additional topics in detail. The English edition provides a comprehensive update to an enduring classic. Combining the lucid exposition of the original text with extensive new material, it will be a valuable resource for researchers in discrete geometry for decades to come.

Mathematics. --- Math --- Science --- Convex surfaces. --- Polyhedra. --- Sphere. --- Geometry, Solid --- Shapes --- Orbs --- Polyhedral figures --- Polyhedrons --- Convex areas --- Convex domains --- Surfaces --- Superfícies convexes --- Poliedres --- Esfera

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Architecture. --- Shapes. --- Shells (Engineering) --- Structural optimization. --- Coques (Ingénierie) --- Optimisation des structures --- Architecture --- Formes --- Structural optimization --- 624.04 --- Structural design. Graphical and analytical statics for investigation and calculation of structures --- Shells (Engineering). --- 624.04 Structural design. Graphical and analytical statics for investigation and calculation of structures --- Coques (Ingénierie) --- Shapes --- Optimal structural design --- Optimization, Structural --- Optimization of structural systems --- Optimum design of structures --- Optimum structural design --- Optimum structures --- Structures, Optimum design of --- Structural design --- Structural shells --- Elastic plates and shells --- Structural analysis (Engineering) --- Plates (Engineering) --- Forms (Shapes) --- Shape --- Geometry --- Surfaces --- Architecture, Western (Western countries) --- Building design --- Buildings --- Construction --- Western architecture (Western countries) --- Art --- Building --- Design and construction --- Architecture, Primitive