Perspectives on Projective Geometry

Author: Jürgen Richter-Gebert
Publisher: Springer Science & Business Media
ISBN: 3642172865
Release Date: 2011-02-04
Genre: Mathematics

Projective geometry is one of the most fundamental and at the same time most beautiful branches of geometry. It can be considered the common foundation of many other geometric disciplines like Euclidean geometry, hyperbolic and elliptic geometry or even relativistic space-time geometry. This book offers a comprehensive introduction to this fascinating field and its applications. In particular, it explains how metric concepts may be best understood in projective terms. One of the major themes that appears throughout this book is the beauty of the interplay between geometry, algebra and combinatorics. This book can especially be used as a guide that explains how geometric objects and operations may be most elegantly expressed in algebraic terms, making it a valuable resource for mathematicians, as well as for computer scientists and physicists. The book is based on the author’s experience in implementing geometric software and includes hundreds of high-quality illustrations.

5000 Jahre Geometrie

Author: Christoph J. Scriba
Publisher: Springer-Verlag
ISBN: 9783662045008
Release Date: 2013-07-02
Genre: Mathematics

Lange bevor die Schrift entwickelt wurde, hat der Mensch geometrische Strukturen wahrgenommen und systematisch verwendet: ob beim Weben oder Flechten einfacher zweidimensionaler Muster oder beim Bauen mit dreidimensionalen Körpern. Das Buch liefert einen faszinierenden Überblick über die geometrischen Vorstellungen und Erkenntnisse der Menschheit von der Urgesellschaft bis hin zu den mathematischen und künstlerischen Ideen des 20. Jahrhunderts.

Geometrie und ihre Anwendungen in Kunst Natur und Technik

Author: Georg Glaeser
Publisher: Springer-Verlag
ISBN: 9783642418525
Release Date: 2014-04-02
Genre: Mathematics

Die „Geometrie und ihre Anwendungen“ ist für Personen geschrieben, die von relativ einfachen Problemen der ebenen Geometrie bis hin zu schwierigeren Aufgaben der Raumgeometrie Interesse an geometrischen Zusammenhängen haben. Ähnlich wie beim „mathematischen Werkzeugkasten“ stehen Anwendungen aus verschiedenen Disziplinen wie dem Ingenieurwesen, der Biologie, Physik, Astronomie, Geografie, Fotografie, Kunstgeschichte, ja sogar der Musik im Vordergrund. Die Anwendungsbeispiele veranschaulichen wichtige Begriffe der Geometrie wie Normalprojektion und Zentralprojektion, Krümmung von Kurven und Flächen, der Geometrie der Bewegung und sogar der Geometrie nichteuklidischer Räume. Stets hat die Raumvorstellung Vorrang. Das Buch kann daher auch von Personen ohne spezielle mathematische Vorbildung gelesen werden. Die 3. Auflage ist um gut 60 Seiten erweitert und enthält zahlreiche neue Anwendungen mit hochwertigen Grafiken.

Projective Geometry

Author: H.S.M. Coxeter
Publisher: Springer Science & Business Media
ISBN: 0387406239
Release Date: 2003-10-09
Genre: Mathematics

In Euclidean geometry, constructions are made with ruler and compass. Projective geometry is simpler: its constructions require only a ruler. In projective geometry one never measures anything, instead, one relates one set of points to another by a projectivity. The first two chapters of this book introduce the important concepts of the subject and provide the logical foundations. The third and fourth chapters introduce the famous theorems of Desargues and Pappus. Chapters 5 and 6 make use of projectivities on a line and plane, respectively. The next three chapters develop a self-contained account of von Staudt's approach to the theory of conics. The modern approach used in that development is exploited in Chapter 10, which deals with the simplest finite geometry that is rich enough to illustrate all the theorems nontrivially. The concluding chapters show the connections among projective, Euclidean, and analytic geometry.

Projective Geometry

Author: Olive Whicher
Publisher:
ISBN: 9781855843790
Release Date: 2013-07-01
Genre: Mathematics

Whicher explores the concepts of polarity and movement in modern projective geometry as a discipline of thought that transcends the limited and rigid space and forms of Euclid, and the corresponding material forces conceived in classical mechanics. Rudolf Steiner underlined the importance of projective geometry as "a method of training the imaginative faculties of thinking, so that they become an instrument of cognition no less conscious and exact than mathematical reasoning." This seminal approach allows for precise scientific understanding of the concept of creative fields of formative (etheric) forces at work in nature--in plants, animals and in the human being.

Projektive Geometrie und Cayley Klein Geometrien der Ebene

Author: Gerhard Kowol
Publisher: Springer-Verlag
ISBN: 9783764399023
Release Date: 2009-08-11
Genre: Mathematics

Der Autor zeigt am Beispiel der ebenen reellen und komplexen projektiven Geometrie und der davon abgeleiteten Cayley-Klein-Geometrien, dass das Mathematisieren eine Bedeutung hat, die weit über das Fach hinausgeht: Zum einen stellt er den erkenntnistheoretischen Aspekt dar, der durch den anschaulich-synthetischen Zugang belegt und durch eine philosophisch-mathematikhistorische Erörterung untermauert wird; zum anderen den Anwendungsaspekt, der auch auf wenig bekannte Anwendungen in der Botanik, Kristallografie, Mechanik und Psychologie bezogen wird.

Geometrie der Lage

Author: Karl Georg Christian von Staudt
Publisher:
ISBN: BSB:BSB10082875
Release Date: 1847
Genre: Curves


Grundlagen der Geometrie

Author: David Hilbert
Publisher: SEVERUS Verlag
ISBN: 9783863479466
Release Date: 2014-03
Genre: Mathematics

Seine Erkenntnisse beeinflussen bis heute die Forschung: David Hilbert baut in seinen „Grundlagen der Geometrie“ auf Euklids Lehre ein Grundsatzsystem auf, von dem ausgehend er wichtige geometrische Sätze ableitet. Die erstmals 1899 erschienene Abhandlung machte Hilbert zu einem der wichtigsten Mathematiker der Neuzeit, der auch den Formalismus entscheidend prägte.

The Beauty of Physics Patterns Principles and Perspectives

Author: A. R. P. Rau
Publisher: OUP Oxford
ISBN: 9780191019890
Release Date: 2014-09-25
Genre: Science

The beauty of physics lies in its coherence in terms of a few fundamental concepts and principles. Even physicists have occasion to marvel at the overarching reach of basic principles and their ability to account for features stretching from the microscopic sub-atomic world to the cosmological expanses of the Universe. While mathematics is its natural language, physics is mostly about patterns, connections, and relations between objects and phenomena, and it is this aspect that is emphasized in this book. Since science tries to connect phenomena that at first sight appear widely different, while boiling them down to a small set of essential principles and laws, metaphor and analogy pervade our subject. Consider the pendulum, its swing from one extreme to the other often invoked in social or economic contexts. In molecular vibrations, such as in the CO2 molecule, the quantum motions of electrons and nuclei are metaphorically the pendulums. In electromagnetic radiation, including the visible light we observe, there are not even any concrete material particles, only electric and magnetic fields executing simple harmonic motion. But, to a physicist, they are all "just a pendulum". The selection of topics reflects the author's own four-decade career in research physics and his resultant perspective on the subject. While aimed primarily at physicists, including junior students, this book also addresses other readers who are willing to think with symbols and simple algebra in understanding the physical world around us. Each chapter, on themes such as dimensions, transformations, symmetries, or maps, begins with simple examples accessible to all while connecting them later to more sophisticated realizations in more advanced topics of physics.