SHADOWS OF THE CIRCLE

Conic Sections, Optimal Figures and Non-Euclidean Geometry

By: HANSEN VAGN LUNDSGAARD

Write A Review

Hardcover | 14 April 1998

At a Glance

Hardcover
120 Pages

Dimensions(cm)
22.86 x 15.24 x 0.79

New Edition

Published: 4th June 2024

Hardcover

RRP $37.99

$34.75

or 4 interest-free payments of $8.69 with

or

Ships in 15 to 25 business days

The aim of this book is to throw light on various facets of geometry through development of four geometrical themes.

The first theme is about the ellipse, the shape of the shadow cast by a circle. The next, a natural continuation of the first, is a study of all three types of conic sections, the ellipse, the parabola and the hyperbola.

The third theme is about certain properties of geometrical figures related to the problem of finding the largest area that can be enclosed by a curve of given length. This problem is called the isoperimetric problem. In itself, this topic contains motivation for major parts of the curriculum in mathematics at college level and sets the stage for more advanced mathematical subjects such as functions of several variables and the calculus of variations.

The emergence of non-Euclidean geometries in the beginning of the nineteenth century represents one of the dramatic episodes in the history of mathematics. In the last theme the non-Euclidean geometry in the Poincaré disc model of the hyperbolic plane is developed.

Shipping

	Standard Shipping	Express Shipping
Metro postcodes:	$9.99	$14.95
Regional postcodes:	$9.99	$14.95
Rural postcodes:	$9.99	$14.95

Orders over $79.00 qualify for free shipping.

How to return your order

At Booktopia, we offer hassle-free returns in accordance with our returns policy. If you wish to return an item, please get in touch with Booktopia Customer Care.

Additional postage charges may be applicable.

Defective items

If there is a problem with any of the items received for your order then the Booktopia Customer Care team is ready to assist you.

For more info please visit our Help Centre.

More in Geometry

Calculus

3rd Edition

Hardcover

RRP $97.95

$81.99

16%
OFF

Grade 4 Geometry and Measurement

Kumon Math Workbooks

Paperback

RRP $16.99

$13.75

19%
OFF

Essential Trigonometry - Isabella Hughes

$19.75

10%
OFF

Geometry

Grades 6 - 8

Paperback

RRP $24.99

$18.99

24%
OFF

Grade 3 Geometry and Measurement

Paperback

RRP $16.99

$13.75

19%
OFF

Intro to Geometry

Grades 6 - 8

Paperback

RRP $24.99

$18.99

24%
OFF

The Golden Ratio

Divine Beauty of Mathematics

Hardcover

RRP $59.99

$42.99

28%
OFF

The Maths Book

Big Ideas Simply Explained

Hardcover

RRP $42.99

$33.99

21%
OFF

Crocheting Adventures with Hyperbolic Planes

Tactile Mathematics, Art and Craft for all to Explore, Second Edition

Paperback

RRP $62.99

$53.75

15%
OFF

Simply Maths

DK Simply

Hardcover

RRP $19.99

$15.99

20%
OFF

Grade 1 Geometry & Measurement

Kumon Math Workbooks

Paperback

RRP $16.99

$13.75

19%
OFF

Saturated Fusion Systems on $p$-Groups of Maximal Class

Paperback

RRP $197.00

$187.75

Steady Gradient Kahler-Ricci Solitons on Crepant Resolutions of Calabi-Yau Cones

Paperback

RRP $197.00

$187.75

$G$-Global Homotopy Theory and Algebraic $K$-Theory

Paperback

RRP $197.00

$187.75

Para-HyperKahler Geometry of the Deformation Space of Maximal Globally Hyperbolic Anti-de Sitter Three-Manifolds

Paperback

RRP $197.00

$187.75

The Dynamics and Geometry of Semi-Hyperbolic Rational Semigroups

Paperback

RRP $197.00

$187.75

Arithmetic Quotients of the Bruhat-Tits Building for Projective GeneralLinear Groups in Positive Characteristic

Paperback

RRP $197.00

$187.75

The Global Solutions to a Cartan's Realization Problem

Paperback

RRP $197.00

$187.75

Introduction to Lie Algebras

Finite and Infinite Dimension

Hardcover

RRP $312.00

$290.99

The Practice of Algebraic Curves

A Second Course in Algebraic Geometry

Hardcover

RRP $312.00

$290.99

Introduction to Lie Algebras

Finite and Infinite Dimension

Paperback

RRP $206.00

$195.75

The Practice of Algebraic Curves

A Second Course in Algebraic Geometry

Paperback

RRP $206.00

$195.75

Preface
An ellipse in the shadow	p. 1
The ellipse as a plane section of a cylinder	p. 3
The equation of the ellipse	p. 5
A parametrization of the ellipse	p. 6
The ellipse as a locus	p. 8
Directrix for the ellipse	p. 10
Geometrical determination of foci and directrices for the ellipse	p. 12
The tangents of the ellipse	p. 13
An application to gear wheel movements	p. 17
Sources for Chapter 1	p. 21
With conic sections in the light	p. 23
The ellipse as a plane section in a cone	p. 23
Geometric determination of foci and directrices for a conic section	p. 27
The parabola	p. 29
The hyperbola	p. 31
Hyperbolic navigational systems	p. 35
Conic sections as algebraic curves	p. 38
Epilogue	p. 42
Sources for Chapter 2	p. 44
Optimal plane figures	p. 45
Isosceles triangles	p. 46
Perron's paradox	p. 48
Some simple geometrical problems without solutions	p. 48
A fundamental property of the real numbers	p. 49
Maxima and minima of real-valued functions	p. 51
The equilateral triangle as optimal figure	p. 54
The square as optimal figure	p. 57
The regular polygons as optimal figures	p. 59
Some limit values for regular polygons	p. 60
The isoperimetric problem	p. 62
Epilogue: Elements of the history of the calculus of variations	p. 67
Sources for Chapter 3	p. 69
The Poincare disc model of non-Euclidean geometry	p. 71
Euclid's Elements	p. 71
The parallel axiom and non-Euclidean geometries	p. 74
Inversion in a circle	p. 76
Inversion as a mapping	p. 79
Orthogonal circles and Euclid's Postulate 1 in the hyperbolic plane	p. 84
The notion of distance in the hyperbolic plane and Euclid's Postulate 2	p. 86
Isometries in the hyperbolic plane	p. 89
Hyperbolic triangles and n-gons	p. 91
The Poincare half-plane	p. 93
Elliptic geometries	p. 94
Sources for Chapter 4	p. 96
Exercises	p. 97
Index	p. 109
Table of Contents provided by Blackwell. All Rights Reserved.

SHADOWS OF THE CIRCLE

Conic Sections, Optimal Figures and Non-Euclidean Geometry

At a Glance

New Edition

Hardcover

Shipping

How to return your order

Defective items

You Can Find This Book In

More in Geometry

Calculus

3rd Edition

Grade 4 Geometry and Measurement

Kumon Math Workbooks

Essential Trigonometry

College Geometry

Sacred Geometry

Philosophy and Practice

Geometry

Grades 6 - 8

Grade 3 Geometry and Measurement

Intro to Geometry

Grades 6 - 8

The Golden Ratio

Divine Beauty of Mathematics

The Maths Book

Big Ideas Simply Explained

Crocheting Adventures with Hyperbolic Planes

Tactile Mathematics, Art and Craft for all to Explore, Second Edition

Simply Maths

DK Simply

Grade 1 Geometry & Measurement

Kumon Math Workbooks

Saturated Fusion Systems on $p$-Groups of Maximal Class

Steady Gradient Kahler-Ricci Solitons on Crepant Resolutions of Calabi-Yau Cones

$G$-Global Homotopy Theory and Algebraic $K$-Theory

Para-HyperKahler Geometry of the Deformation Space of Maximal Globally Hyperbolic Anti-de Sitter Three-Manifolds

The Dynamics and Geometry of Semi-Hyperbolic Rational Semigroups

Arithmetic Quotients of the Bruhat-Tits Building for Projective GeneralLinear Groups in Positive Characteristic

The Global Solutions to a Cartan's Realization Problem

Introduction to Lie Algebras

Finite and Infinite Dimension

The Practice of Algebraic Curves

A Second Course in Algebraic Geometry

Introduction to Lie Algebras

Finite and Infinite Dimension

The Practice of Algebraic Curves

A Second Course in Algebraic Geometry

This product is categorised by