Introduction: The Projective Plane and Central Collineations
This book grew from the idea that much of projective geometry is the elaboration of a simple concept, the central collineation. A central collineation is a construction, carried out with just a straightedge and a device to construct parallel lines, follow
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Collineations and Conic Sections
An Introduction to Projective Geometry in its History
Collineations and Conic Sections
Christopher Baltus
Collineations and Conic Sections An Introduction to Projective Geometry in its History
Christopher Baltus Department of Mathematics State University of New York at Oswego Oswego, NY, USA
ISBN 978-3-030-46286-4 ISBN 978-3-030-46287-1 (eBook) https://doi.org/10.1007/978-3-030-46287-1 Mathematics Subject Classification: 01A05, 51-03, 51A05, 51N15, 01A45, 01A55, 97G50 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Preface
Searching for an entry into the projective geometry section of a college geometry course I was teaching many years ago, I found central collineations in Dan Pedoe’s Introduction to Projective Geometry. In a central collineation, the plane is mapped to itself so lines go to lines and there is a line of points unchanged by the collineation. Here was a simple idea which led to lovely results in proofs that were actually fun and easily motivated by diagrams. The next discovery, for me, was Philippe de la Hire’s projective introduction to the conic sections, from 1673, to which he added some 20 pages that he called Plani-conique. Plani-conique is a set of simple rules to carry out a plane-to-plane projection when one plane is imposed on the other. These are exactly the rules of the central collineation! Moving ahead 140 years, there is the story of Jean-Victor Poncelet. Captured while Napoleon’s army made its wintry retreat from Moscow in late 1812, as a prisoner he rebuilt the geometry he recalled from student days at the École Polytechnique, without the benefit of books. He made great use of what
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