Mathematical Illustrations

A Manual of Geometry and PostScript
Author: Bill Casselman
Publisher: Cambridge University Press
ISBN: 9780521839211
Category: Computers
Page: 318
View: 1993

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This practical introduction to the techniques needed to produce mathematical illustrations of high quality is suitable for anyone with a modest acquaintance with coordinate geometry. The author combines a completely self-contained step-by-step introduction to the graphics programming language PostScript with advice on what goes into good mathematical illustrations, chapters showing how good graphics can be used to explain mathematics, and a treatment of all the mathematics needed to make such illustrations. The many small simple graphics projects can also be used in courses in geometry, graphics, or general mathematics. Code for many of the illustrations is included, and can be downloaded from the book's web site:; scientists, engineers, and even graphic designers seeking help in creating technical illustrations need look no further.

The Four Pillars of Geometry

Author: John Stillwell
Publisher: Springer Science & Business Media
ISBN: 0387255303
Category: Mathematics
Page: 228
View: 9409

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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

Practical Linear Algebra

A Geometry Toolbox, Third Edition
Author: Gerald Farin,Dianne Hansford
Publisher: CRC Press
ISBN: 1498759777
Category: Computers
Page: 514
View: 5987

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Through many examples and real-world applications, Practical Linear Algebra: A Geometry Toolbox, Third Edition teaches undergraduate-level linear algebra in a comprehensive, geometric, and algorithmic way. Designed for a one-semester linear algebra course at the undergraduate level, the book gives instructors the option of tailoring the course for the primary interests: math, engineering, science, computer graphics, and geometric modeling. New to the Third Edition More exercises and applications Coverage of singular value decomposition and its application to the pseudoinverse, principal components analysis, and image compression More attention to eigen-analysis, including eigenfunctions and the Google matrix Greater emphasis on orthogonal projections and matrix decompositions, which are tied to repeated themes such as the concept of least squares To help students better visualize and understand the material, the authors introduce the fundamental concepts of linear algebra first in a two-dimensional setting and then revisit these concepts and others in a three-dimensional setting. They also discuss higher dimensions in various real-life applications. Triangles, polygons, conics, and curves are introduced as central applications of linear algebra. Instead of using the standard theorem-proof approach, the text presents many examples and instructional illustrations to help students develop a robust, intuitive understanding of the underlying concepts. The authors’ website also offers the illustrations for download and includes Mathematica® code and other ancillary materials.

A Topological Picturebook

Author: George K. Francis
Publisher: Springer Science & Business Media
ISBN: 0387681205
Category: Mathematics
Page: 194
View: 4358

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Praise for George Francis's A Topological Picturebook: Bravo to Springer for reissuing this unique and beautiful book! It not only reminds the older generation of the pleasures of doing mathematics by hand, but also shows the new generation what ``hands on'' really means. - John Stillwell, University of San Francisco The Topological Picturebook has taught a whole generation of mathematicians to draw, to see, and to think. - Tony Robbin, artist and author of Shadows of Reality: The Fourth Dimension in Relativity, Cubism, and Modern Thought The classic reference for how to present topological information visually, full of amazing hand-drawn pictures of complicated surfaces. - John Sullivan, Technische Universitat Berlin A Topological Picturebook lets students see topology as the original discoverers conceived it: concrete and visual, free of the formalism that burdens conventional textbooks. - Jeffrey Weeks, author of The Shape of Space A Topological Picturebook is a visual feast for anyone concerned with mathematical images. Francis provides exquisite examples to build one's "visualization muscles". At the same time, he explains the underlying principles and design techniques for readers to create their own lucid drawings. - George W. Hart, Stony Brook University In this collection of narrative gems and intriguing hand-drawn pictures, George Francis demonstrates the chicken-and-egg relationship, in mathematics, of image and text. Since the book was first published, the case for pictures in mathematics has been won, and now it is time to reflect on their meaning. A Topological Picturebook remains indispensable. - Marjorie Senechal, Smith College and co-editor of the Mathematical Intelligencer

Curves and Surfaces in Geometric Modeling

Theory and Algorithms
Author: Jean H. Gallier
Publisher: Morgan Kaufmann
ISBN: 9781558605992
Category: Computers
Page: 491
View: 3987

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"Curves and Surfaces in Geometric Modeling: Theory and Algorithms offers a theoretically unifying understanding of polynomial curves and surfaces as well as an effective approach to implementation that you can apply to your own work as a graduate student, scientist, or practitioner." "The focus here is on blossoming - the process of converting a polynomial to its polar form - as a natural, purely geometric explanation of the behavior of curves and surfaces. This insight is important for more than just its theoretical elegance - the author demonstrates the value of blossoming as a practical algorithmic tool for generating and manipulating curves and surfaces that meet many different criteria. You'll learn to use this and other related techniques drawn from affine geometry for computing and adjusting control points, deriving the continuity conditions for splines, creating subdivision surfaces, and more." "It will be an essential acquisition for readers in many different areas, including computer graphics and animation, robotics, virtual reality, geometric modeling and design, medical imaging, computer vision, and motion planning."--BOOK JACKET.Title Summary field provided by Blackwell North America, Inc. All Rights Reserved

Convex Optimization & Euclidean Distance Geometry

Author: Jon Dattorro
Publisher: Meboo Publishing USA
ISBN: 0976401304
Category: Mathematics
Page: 570
View: 1592

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The study of Euclidean distance matrices (EDMs) fundamentally asks what can be known geometrically given onlydistance information between points in Euclidean space. Each point may represent simply locationor, abstractly, any entity expressible as a vector in finite-dimensional Euclidean space.The answer to the question posed is that very much can be known about the points;the mathematics of this combined study of geometry and optimization is rich and deep.Throughout we cite beacons of historical accomplishment.The application of EDMs has already proven invaluable in discerning biological molecular conformation.The emerging practice of localization in wireless sensor networks, the global positioning system (GPS), and distance-based pattern recognitionwill certainly simplify and benefit from this theory.We study the pervasive convex Euclidean bodies and their various representations.In particular, we make convex polyhedra, cones, and dual cones more visceral through illustration, andwe study the geometric relation of polyhedral cones to nonorthogonal bases biorthogonal expansion.We explain conversion between halfspace- and vertex-descriptions of convex cones,we provide formulae for determining dual cones,and we show how classic alternative systems of linear inequalities or linear matrix inequalities and optimality conditions can be explained by generalized inequalities in terms of convex cones and their duals.The conic analogue to linear independence, called conic independence, is introducedas a new tool in the study of classical cone theory; the logical next step in the progression:linear, affine, conic.Any convex optimization problem has geometric interpretation.This is a powerful attraction: the ability to visualize geometry of an optimization problem.We provide tools to make visualization easier.The concept of faces, extreme points, and extreme directions of convex Euclidean bodiesis explained here, crucial to understanding convex optimization.The convex cone of positive semidefinite matrices, in particular, is studied in depth.We mathematically interpret, for example,its inverse image under affine transformation, and we explainhow higher-rank subsets of its boundary united with its interior are convex.The Chapter on "Geometry of convex functions",observes analogies between convex sets and functions:The set of all vector-valued convex functions is a closed convex cone.Included among the examples in this chapter, we show how the real affinefunction relates to convex functions as the hyperplane relates to convex sets.Here, also, pertinent results formultidimensional convex functions are presented that are largely ignored in the literature;tricks and tips for determining their convexityand discerning their geometry, particularly with regard to matrix calculus which remains largely unsystematizedwhen compared with the traditional practice of ordinary calculus.Consequently, we collect some results of matrix differentiation in the appendices.The Euclidean distance matrix (EDM) is studied,its properties and relationship to both positive semidefinite and Gram matrices.We relate the EDM to the four classical axioms of the Euclidean metric;thereby, observing the existence of an infinity of axioms of the Euclidean metric beyondthe triangle inequality. We proceed byderiving the fifth Euclidean axiom and then explain why furthering this endeavoris inefficient because the ensuing criteria (while describing polyhedra)grow linearly in complexity and number.Some geometrical problems solvable via EDMs,EDM problems posed as convex optimization, and methods of solution arepresented;\eg, we generate a recognizable isotonic map of the United States usingonly comparative distance information (no distance information, only distance inequalities).We offer a new proof of the classic Schoenberg criterion, that determines whether a candidate matrix is an EDM. Our proofrelies on fundamental geometry; assuming, any EDM must correspond to a list of points contained in some polyhedron(possibly at its vertices) and vice versa.It is not widely known that the Schoenberg criterion implies nonnegativity of the EDM entries; proved here.We characterize the eigenvalues of an EDM matrix and then devisea polyhedral cone required for determining membership of a candidate matrix(in Cayley-Menger form) to the convex cone of Euclidean distance matrices (EDM cone); \ie,a candidate is an EDM if and only if its eigenspectrum belongs to a spectral cone for EDM^N.We will see spectral cones are not unique.In the chapter "EDM cone", we explain the geometric relationship betweenthe EDM cone, two positive semidefinite cones, and the elliptope.We illustrate geometric requirements, in particular, for projection of a candidate matrixon a positive semidefinite cone that establish its membership to the EDM cone. The faces of the EDM cone are described,but still open is the question whether all its faces are exposed as they are for the positive semidefinite cone.The classic Schoenberg criterion, relating EDM and positive semidefinite cones, isrevealed to be a discretized membership relation (a generalized inequality, a new Farkas''''''''-like lemma)between the EDM cone and its ordinary dual. A matrix criterion for membership to the dual EDM cone is derived thatis simpler than the Schoenberg criterion.We derive a new concise expression for the EDM cone and its dual involvingtwo subspaces and a positive semidefinite cone."Semidefinite programming" is reviewedwith particular attention to optimality conditionsof prototypical primal and dual conic programs,their interplay, and the perturbation method of rank reduction of optimal solutions(extant but not well-known).We show how to solve a ubiquitous platonic combinatorial optimization problem from linear algebra(the optimal Boolean solution x to Ax=b)via semidefinite program relaxation.A three-dimensional polyhedral analogue for the positive semidefinite cone of 3X3 symmetricmatrices is introduced; a tool for visualizing in 6 dimensions.In "EDM proximity"we explore methods of solution to a few fundamental and prevalentEuclidean distance matrix proximity problems; the problem of finding that Euclidean distance matrix closestto a given matrix in the Euclidean sense.We pay particular attention to the problem when compounded with rank minimization.We offer a new geometrical proof of a famous result discovered by Eckart \& Young in 1936 regarding Euclideanprojection of a point on a subset of the positive semidefinite cone comprising all positive semidefinite matriceshaving rank not exceeding a prescribed limit rho.We explain how this problem is transformed to a convex optimization for any rank rho.

Mathematical Publishing

A Guidebook
Author: Steven George Krantz
Publisher: American Mathematical Soc.
ISBN: 9780821872598
Category: Mathematics
Page: 297
View: 2362

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Mathematicians are expected to publish their work: in journals, conference proceedings, and books. It is vital to advancing their careers. Later, some are asked to become editors. However, most mathematicians are trained to do mathematics, not to publish it. But here, finally, for graduate students and researchers interested in publishing their work, Steven G. Krantz, the respected author of several "how-to" guides in mathematics, shares his experience as an author, editor, editorial board member, and independent publisher. This new volume is an informative, comprehensive guidebook to publishing mathematics. Krantz describes both the general setting of mathematical publishing and the specifics about all the various publishing situations mathematicians may encounter. As with his other books, Krantz's style is engaging and frank. He gives advice on how to get your book published, how to get organized as an editor, what to do when things go wrong, and much more. He describes the people, the language (including a glossary), and the process of publishing both books and journals. Steven G. Krantz is an accomplished mathematician and an award-winning author. He has published more than 130 research articles and 45 books. He has worked as an editor of several book series, research journals, and for the Notices of the AMS. He is also the founder of the Journal of Geometric Analysis. Other titles available from the AMS by Steven G. Krantz are How to Teach Mathematics, A Primer of Mathematical Writing, A Mathematician's Survival Guide, and Techniques of Problem Solving.

The LaTex Graphics Companion

Illustrating Documents with TeX and PostScript
Author: Michel Goossens,S. P. Q. Rahtz,Frank Mittelbach
Publisher: Addison-Wesley Professional
ISBN: 9780201854695
Category: Computers
Page: 554
View: 640

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Complementing The LaTeX Companion, this new graphics companion addresses one of the most common needs among users of the LaTeX typesetting system: the incorporation of graphics into text. It provides the first full description of the standard LaTeX color and graphics packages, and shows how you can combine TeX and PostScript capabilities to produce beautifully illustrated pages. You will learn how to incorporate graphic files into a LaTeX document, program technical diagrams using several different languages, and achieve special effects with fragments of embedded PostScript. Furthermore, you'll find detailed descriptions of important packages like Xy-pic, PSTricks, and METAPOST; the dvips dvi to PostScript driver; and Ghostscript.

Graphs, Surfaces and Homology

An Introduction to Algebraic Topology
Author: P. Giblin
Publisher: Springer Science & Business Media
ISBN: 9400959532
Category: Science
Page: 329
View: 1336

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viii homology groups. A weaker result, sufficient nevertheless for our purposes, is proved in Chapter 5, where the reader will also find some discussion of the need for a more powerful in variance theorem and a summary of the proof of such a theorem. Secondly the emphasis in this book is on low-dimensional examples the graphs and surfaces of the title since it is there that geometrical intuition has its roots. The goal of the book is the investigation in Chapter 9 of the properties of graphs in surfaces; some of the problems studied there are mentioned briefly in the Introduction, which contains an in formal survey of the material of the book. Many of the results of Chapter 9 do indeed generalize to higher dimensions (and the general machinery of simplicial homology theory is avai1able from earlier chapters) but I have confined myself to one example, namely the theorem that non-orientable closed surfaces do not embed in three-dimensional space. One of the principal results of Chapter 9, a version of Lefschetz duality, certainly generalizes, but for an effective presentation such a gener- ization needs cohomology theory. Apart from a brief mention in connexion with Kirchhoff's laws for an electrical network I do not use any cohomology here. Thirdly there are a number of digressions, whose purpose is rather to illuminate the central argument from a slight dis tance, than to contribute materially to its exposition.

A Student's Guide to the Study, Practice, and Tools of Modern Mathematics

Author: Donald Bindner,Martin Erickson
Publisher: CRC Press
ISBN: 1439846073
Category: Mathematics
Page: 280
View: 5287

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A Student’s Guide to the Study, Practice, and Tools of Modern Mathematics provides an accessible introduction to the world of mathematics. It offers tips on how to study and write mathematics as well as how to use various mathematical tools, from LaTeX and Beamer to Mathematica® and MapleTM to MATLAB® and R. Along with a color insert, the text includes exercises and challenges to stimulate creativity and improve problem solving abilities. The first section of the book covers issues pertaining to studying mathematics. The authors explain how to write mathematical proofs and papers, how to perform mathematical research, and how to give mathematical presentations. The second section focuses on the use of mathematical tools for mathematical typesetting, generating data, finding patterns, and much more. The text describes how to compose a LaTeX file, give a presentation using Beamer, create mathematical diagrams, use computer algebra systems, and display ideas on a web page. The authors cover both popular commercial software programs and free and open source software, such as Linux and R. Showing how to use technology to understand mathematics, this guide supports students on their way to becoming professional mathematicians. For beginning mathematics students, it helps them study for tests and write papers. As time progresses, the book aids them in performing advanced activities, such as computer programming, typesetting, and research.

Canadian Sayings 3

1,000 Folk Sayings Used by Canadians
Author: Bill Casselman
Publisher: N.A
ISBN: 9781552784259
Category: Aphorisms and apothegms
Page: 160
View: 1894

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Bill Casselman, Canada's master-gatherer of funny folk sayings, returns with fresh bounty--hundreds and hundreds of new folk sayings not collected in his previous two volumes of knee-slappers and girdle-splitters. Here are Canadian maxims galore, snappy saws and breezy national adages--enough to fill the barn of delight many times over. As always, Bill has divided these hilarious one-liners into dozens of categories redolent of human nature, categories like Stupidity, Sex Canadiana, Weather and Work. And Bill adds his own witty footnotes and explanations to those saying whose meanings may be lost in time. How about his apt squelch for a deeply annoying store clerk? "Miss, by standing behind the counter, you are depriving a village of their idiot." Bill Casselman writes about our words like no other Canadian--he is still being called "A Bluenose among schooners on the sea of popular etymology."

The Cinderella.2 Manual

Working with The Interactive Geometry Software
Author: Jürgen Richter-Gebert,Ulrich H. Kortenkamp
Publisher: Springer Science & Business Media
ISBN: 354034926X
Category: Mathematics
Page: 458
View: 6622

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Cinderella.2, the new version of the well-known interactive geometry software, has become an even more versatile tool than its predecessor. The geometry component extends the functionality to such spectacular objects as dynamic fractals, and the software includes two major new components: physical simulation such as of mechanical objects, virtual electronic devices, and electromagnetic properties. Cinderella.2 Documentation offers complete instruction and techniques for using Cinderella.2.

Euler's Gem

The Polyhedron Formula and the Birth of Topology
Author: David S. Richeson
Publisher: Princeton University Press
ISBN: 0691154570
Category: Mathematics
Page: 317
View: 3593

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Leonhard Euler's polyhedron formula describes the structure of many objects--from soccer balls and gemstones to Buckminster Fuller's buildings and giant all-carbon molecules. Yet Euler's formula is so simple it can be explained to a child. Euler's Gem tells the illuminating story of this indispensable mathematical idea. From ancient Greek geometry to today's cutting-edge research, Euler's Gem celebrates the discovery of Euler's beloved polyhedron formula and its far-reaching impact on topology, the study of shapes. In 1750, Euler observed that any polyhedron composed of V vertices, E edges, and F faces satisfies the equation V-E+F=2. David Richeson tells how the Greeks missed the formula entirely; how Descartes almost discovered it but fell short; how nineteenth-century mathematicians widened the formula's scope in ways that Euler never envisioned by adapting it for use with doughnut shapes, smooth surfaces, and higher dimensional shapes; and how twentieth-century mathematicians discovered that every shape has its own Euler's formula. Using wonderful examples and numerous illustrations, Richeson presents the formula's many elegant and unexpected applications, such as showing why there is always some windless spot on earth, how to measure the acreage of a tree farm by counting trees, and how many crayons are needed to color any map. Filled with a who's who of brilliant mathematicians who questioned, refined, and contributed to a remarkable theorem's development, Euler's Gem will fascinate every mathematics enthusiast.

That's Maths

The Mathematical Magic in Everyday Life
Author: Peter Lynch
Publisher: Gill & Macmillan Ltd
ISBN: 0717169561
Category: Mathematics
Page: 344
View: 2543

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From atom bombs to rebounding slinkies, open your eyes to the mathematical magic in the everyday. Mathematics isn’t just for academics and scientists, a fact meteorologist and blogger Peter Lynch has spent the past several years proving through his Irish Times newspaper column and blog, That’s Maths. Here, he shows how maths is all around us, with chapters on the beautiful equations behind designing a good concert venue, predicting the stock market and modelling the atom bomb, as well as playful meditations on everything from coin-stacking to cartography. If you left school thinking maths was boring, think again!

From Calculus to Cohomology

De Rham Cohomology and Characteristic Classes
Author: Ib H. Madsen,Jxrgen Tornehave
Publisher: Cambridge University Press
ISBN: 9780521589567
Category: Mathematics
Page: 286
View: 5745

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An introductory textbook on cohomology and curvature with emphasis on applications.


A Programming Handbook for Visual Designers and Artists
Author: Casey Reas,Ben Fry
Publisher: MIT Press
ISBN: 026202828X
Category: Art
Page: 672
View: 9944

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The visual arts are rapidly changing as media moves into the web, mobile devices, and architecture. When designers and artists learn the basics of writing software, they develop a new form of literacy that enables them to create new media for the present, and to imagine future media that are beyond the capacities of current software tools. This book introduces this new literacy by teaching computer programming within the context of the visual arts. It offers a comprehensive reference and text for Processing (, an open-source programming language that can be used by students, artists, designers, architects, researchers, and anyone who wants to program images, animation, and interactivity. Written by Processing's cofounders, the book offers a definitive reference for students and professionals. Tutorial chapters make up the bulk of the book; advanced professional projects from such domains as animation, performance, and installation are discussed in interviews with their creators.This second edition has been thoroughly updated. It is the first book to offer in-depth coverage of Processing 2.0 and 3.0, and all examples have been updated for the new syntax. Every chapter has been revised, and new chapters introduce new ways to work with data and geometry. New "synthesis" chapters offer discussion and worked examples of such topics as sketching with code, modularity, and algorithms. New interviews have been added that cover a wider range of projects. "Extension" chapters are now offered online so they can be updated to keep pace with technological developments in such fields as computer vision and electronics.InterviewsSUE.C, Larry Cuba, Mark Hansen, Lynn Hershman Leeson, Jürg Lehni, LettError, Golan Levin and Zachary Lieberman, Benjamin Maus, Manfred Mohr, Ash Nehru, Josh On, Bob Sabiston, Jennifer Steinkamp, Jared Tarbell, Steph Thirion, Robert Winter

Geometric and Engineering Drawing

Author: Ken Morling
Publisher: Routledge
ISBN: 0415536197
Category: Technology & Engineering
Page: 360
View: 5845

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For all students and lecturers of basic engineering and technical drawing The new edition of this successful text describes all the geometric instructions and engineering drawing information, likely to be needed by anyone preparing or interpreting drawings or designs. There are also plenty of exercises to practise these principles.

Mathematica Navigator

Mathematics, Statistics, and Graphics
Author: Heikki Ruskeepää
Publisher: Gulf Professional Publishing
ISBN: 012603642X
Category: Mathematics
Page: 844
View: 4954

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Mathematica Navigator gives you a general introduction to Mathematica. The book emphasizes graphics, methods of applied mathematics and statistics, and programming. Mathematica Navigator can be used both as a tutorial and as a handbook. While no previous experience with Mathematica is required, most chapters also include advanced material, so that the book will be a valuable resource for both beginners and experienced users.

Math Into LaTeX

An Introduction to LaTeX and AMS-LaTex
Author: George A. Gratzer
Publisher: Boston : Birkhäuser
ISBN: 9780817638054
Category: Design
Page: 451
View: 8494

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