The Universe of Conics: From the ancient Greeks to 21st century developments

496 Pages·2016·12.39 MB·English

by  Georg Glaeser

The Universe of Conics
Georg Glaeser • Hellmuth Stachel
Boris Odehnal
The Universe of Conics
From the ancient Greeks to
21st century developments
Georg Glaeser Boris Odehnal
Department of Geometry Department of Geometry
University of Applied Arts Vienna University of Applied Arts Vienna
Vienna, Austria Vienna, Austria
Hellmuth Stachel
Institute of Discrete Mathematics and Geometry
Vienna University of Technology
Vienna, Austria
ISBN 978-3-662-45449-7 ISBN 978-3-662-45450-3 (eBook)
DOI 10.1007/978-3-662-45450-3
Library of Congress Control Number: 2016932392
Springer Spektrum
© Springer-Verlag Berlin Heidelberg 2016
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the mate-
rial is concerned, speci(cid:191) cally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting,
reproduction on micro(cid:191) lms 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 de-
veloped.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does
not imply, even in the absence of a speci(cid:191) c 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 as-
sume 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, express or implied, with respect to the mate-
rial contained herein or for any errors or omissions that may have been made.
Printed on acid-free paper
This Springer Spektrum imprint is published by Springer Nature
The registered company is Springer-Verlag GmbH Berlin Heidelberg
Preface
This book attempts to cover the most important properties of conics.
These are the simplest and yet so versatile planar curves, having been
fascinating and useful to mathematicians and other scientists ever since
their first description by the ancient Greeks.
The book has been written for people who love geometry and it is mainly
based on figures and synthetic conclusions rather than on pure analytic
calculations. In many proofs, illustrations help to explain ideas and to
support the argumentation, and in a few cases, the picture can display a
theorem at a glance together with its proof.
Largeportionsofthebookcanbeunderstoodbyundergraduate students.
Some sections, however, require a little more mathematical background
and we hope that they will be a treasure trove even for professional ma-
thematicians and geometers.
4 5 7 6
3 1 8 10
2 9
After a short historical summary (1), we explain the classical definitions
of conics (2). Then, differential geometric properties are discussed (3), in-
cluding thenon-trivial fact that theorbits of planets are ellipses. Chapter
4 shows that conics can be planar intersections of surfaces (especially of
quadratic cones). The framework of Projective Geometry is the proper
setting for the study of conics (5 and 6). Still fitting into this context are
polar systems and pencils of conics (7). After that we switch to the affine
properties of conics (8). Chapter 9 deals with special problems. The last
chapter is devoted to conics in various two-dimensional geometries: on
the sphere, in Pseudo-Euclidean planes, and in hyperbolic planes.
vi Preface
When Isaac Newton proved Johannes Kepler’s conjecture that the
paths of the planets are ellipses, he utilized properties of ellipses which
have in the mean-time almost been forgotten. The 23-years-old Carl
Friedrich Gauß gained his reputation by rediscovering the dwarf pla-
net Ceres which had only been observed on a very few occasions. He used
the fact that an ellipse is uniquely determined by a single focal point (the
Sun) and three additional points. Gauß himself considered that this ty-
pe of problem “commended itself to mathematicians by its difficulty and
elegance”. 215 years later, with Ceres still on its elliptic path, NASA’s
Dawn spacecraft has orbited the dwarf planet, sending stunning images
of this icy remnant of our early solar system.
F
C1
C2 C3
A geometrical homage to a famous problem:
Ceres in its icy beauty (©NASA_Dawn). The path of a planet is fully deter-
mined by three locations, due to fact that an ellipse is unambiguously defined
when we know its focal point F and three points C1, C2, and C3.
We wish all readers a pleasant journey into the universe of conics!
Vienna, September 2015
Acknowledgements
We say thanks to Bob Bix, David Brannan, Fritz Manhart, Andreas Rü-
dinger, and Norman Wildberger for assistence, suggestions, and proof
reading.
Table of Contents
1 Introduction 1
2 Euclidean plane 11
2.1 Classical definitions . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Tangent lines of conics . . . . . . . . . . . . . . . . . . . . . . 30
2.3 Mechanisms tracing conics . . . . . . . . . . . . . . . . . . . . 53
3 Differential Geometry 61
3.1 Conics as orbits of planets . . . . . . . . . . . . . . . . . . . . 62
3.2 Conics in planar differential geometry . . . . . . . . . . . . . . 77
3.3 Conics in differential geometry of surfaces . . . . . . . . . . . 120
4 Euclidean 3-space 127
4.1 Planar intersections of cones of revolution . . . . . . . . . . . 128
4.2 Pairs of focal conics, Dupin cyclides . . . . . . . . . . . . . . . 137
4.3 Perspective images of conics . . . . . . . . . . . . . . . . . . . 153
4.4 Spatial interpretation of conic constructions . . . . . . . . . . 162
5 Projective Geometry 177
5.1 Projective planes . . . . . . . . . . . . . . . . . . . . . . . . . . 178
5.2 Perspectivities, projectivities . . . . . . . . . . . . . . . . . . . 188
5.3 Coordinatization . . . . . . . . . . . . . . . . . . . . . . . . . . 194
5.4 Harmonic quadruples . . . . . . . . . . . . . . . . . . . . . . . 206
6 Projective conics 217
6.1 Steiner’s definition of a conic . . . . . . . . . . . . . . . . . . 218
6.2 Pascal, Pappus, Brianchon . . . . . . . . . . . . . . . . . . . . 220
6.3 Equation and parametrization of a conic . . . . . . . . . . . . 228
6.4 There is only one conic in a projective plane . . . . . . . . . . 239
7 Polarities and pencils 259
7.1 Polar system of a conic . . . . . . . . . . . . . . . . . . . . . . 260
7.2 Definition of a conic according to von Staudt . . . . . . . . . 276
7.3 Pencils of conics . . . . . . . . . . . . . . . . . . . . . . . . . . 283
7.4 Desargues’s involution theorem . . . . . . . . . . . . . . . . 309
7.5 Quadratic Cremona transformations . . . . . . . . . . . . . . . 329
viii Table of Contents
8 Affine Geometry 351
8.1 Conjugate diameters of ellipses and hyperbolas . . . . . . . . 353
8.2 Conics are rational quadratic Bézier curves . . . . . . . . . . . 365
8.3 Conics and number theory . . . . . . . . . . . . . . . . . . . . 369
9 Special problems 375
9.1 Conics in triangle geometry. . . . . . . . . . . . . . . . . . . . 376
9.2 Isoptic curves of conics . . . . . . . . . . . . . . . . . . . . . . 394
9.3 Pedal points on conics, Apollonian hyperbola . . . . . . . . . 402
9.4 Pedal curves of conics . . . . . . . . . . . . . . . . . . . . . . . 406
9.5 Poncelet porisms. . . . . . . . . . . . . . . . . . . . . . . . . . 412
10 Other geometries 435
10.1 Spherical conics . . . . . . . . . . . . . . . . . . . . . . . . . . 436
10.2 Conics in non-Euclidean planes . . . . . . . . . . . . . . . . . 468
References
480
Index 483
1 Introduction
Our Universe is full of conics, even if we cannot always see them – like the
orbitsofthe planets.Itneededveryaccurateobservationstodetect(Johannes
Kepler), and a great physicist (Isaac Newton) to prove that.
© Springer-Verlag Berlin Heidelberg 2016
G. Glaeser et al., The Universe of Conics, DOI 10.1007/978-3-662-45450-3_1
2 Chapter 1: Introduction
Conics play a substantial role in the whole universe. On a large scale,
the orbits of planets look like conics. Astronomers have recognized this
fact and physicists have proved it. Many older and newer visualizations
clearly show the ellipses as orbits of planets of our solar system with the
Sun as the common focus.
Atthispoint,weshouldmention theeffortsoftheancientGreeks.Among
them,especiallyApollonius of Perga(262BC–190BC)collectedma-
ny results which were well known at his time. In his famous eight books,
entitled Konika, he showed that all “four types” of conics can be obtained
from the same cone. He even gave the names ellipse, parabola, and hyper-
bola to these curves. The first four books collect many elementary results
which were known at that time.
FIGURE 1.1. A long history and a back and forth between languages: The 5th
book of Apollonius, here in a translation from Arabic into German.
The books five to seven present material which is probably due to Apol-
lonius. Therein, he discussed the problem of normals to a conic and pre-
pared even the construction of the evolute of a conic. However, Apollo-
nius’booksonlysurvivedinformoftranslations:firstfromGreekintoLa-
tin, then into Arabian, and back into Latin or other languages once again
(cf. Figure 1.1, https://ia601408.us.archive.org/1/items/4737397/
4737397.pdf). Unfortunately, the eighth book is lost.