**Anneli Lax New Mathematical Library**

Volume: 44;
2009;
285 pp;
Softcover

Print ISBN: 978-0-88385-648-2

Product Code: NML/44

List Price: $48.00

AMS Member Price: $36.00

MAA Member Price: $36.00

**Electronic ISBN: 978-0-88385-958-2
Product Code: NML/44.E**

List Price: $48.00

AMS Member Price: $36.00

MAA Member Price: $36.00

# Geometric Transformations IV: Circular Transformations

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*I. M. Yaglom*

Translated by A. Shenitzer.

MAA Press: An Imprint of the American Mathematical Society

The familiar plane geometry of high school figures composed of lines and circles takes on a new life when viewed as the study of properties that are preserved by special groups of transformations. No longer is there a single, universal geometry: different sets of transformations of the plane correspond to intriguing, disparate geometries. This book is the concluding Part IV of Geometric Transformations, but it can be studied independently of Parts I, II, and III, which appeared in this series as Volumes 8, 21, and 24. Part I treats the geometry of rigid motions of the plane (isometries); Part II treats the geometry of shape-preserving transformations of the plane (similarities); Part III treats the geometry of transformations of the plane that map lines to lines (affine and projective transformations) and introduces the Klein model of non-Euclidean geometry. The present Part IV develops the geometry of transformations of the plane that map circles to circles (conformal or anallagmatic geometry). The notion of inversion, or reflection in a circle, is the key tool employed. Applications include ruler-and-compass constructions and the Poincar model of hyperbolic geometry. The straightforward, direct presentation assumes only some background in high-school geometry and trigonometry. Numerous exercises lead the reader to a mastery of the methods and concepts. The second half of the book contains detailed solutions of all the problems.

# Table of Contents

## Geometric Transformations IV: Circular Transformations

- Cover cover11
- Title page iii4
- Contents vii8
- 1 Reflection in a circle (inversion) 110
- 2 Application of inversionsto the solution of constructions 3342
- 3 Pencils of circles. The radical axis of two circles 4352
- 4 Inversion (concluding section) 6170
- 5 Axial circular transformations 8190
- Supplement 143152
- Non-Euclidean Geometry of Lobachevski-Bolyai, or Hyperbolic Geometry 143152
- Notes to Supplement 166175
- Solutions 171180
- About the Author 285294