In this post, we will see the book Inversions by I. Ya. Bakel’man. This book is Volume of Popular Lectures In Mathematics series.

# About the book

In this book, I. Ya. Bakel’man introduces inversion transformations in the Euclidean plane and discusses the interrelationships among more general mathematical concepts. The author begins by defining and giving examples of the concept of a transformation in the Euclidean plane, and then explains the “ point of infinity” and the “ stereographic projection” of the sphere onto the plane. With this preparation, the student is capable of applying the theory of inversions to classical construction problems in the plane.

The author also discusses the theory of pencils of circles, and he uses the acquired techniques in a proof of Ptolemy’s theorem. In the final chapter, the idea of a group is introduced with applications of group theory to geometry. The author demonstrates the group-theoretic basis for the distinction between Euclidean and Lobachevskian geometry.

The book was translated from Russian by Joan A. Teller and Susan Williams and was published in 1974.

Credits to original uploader.

You can get the book here.

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# Contents

Preface vii

## 1. Inversions and Pencils of Circles 1

1.1. Elementary Transformations of the Plane 1

1.2. Stereographic Projection: The Point at Infinity of a Plane 6

1.3. Inversions 8

1.4. Properties of Inversions 11

1.5. The Power of a Point with Respect to a Circle: The Radical Axis of Two Circles 19

1.6. Application of Inversions to the Solution of Construction Problems 24

1.7. Pencils of Circles 32

1.8. Structure of an Elliptical Pencil 40

1.9. Structure of a Parabolic Pencil 41

1.10. Structure of a Hyperbolic Pencil 42

1.11. Ptolemy’s Theorem 45

## 2. Complex Numbers and Inversions 48

2.1. Geometric Representation of Complex Numbers and Operations on Them 48

2.2. Linear Functions of a Complex Variable and Elementary Transformations of the Plane 52

2.3. Linear Fractional Functions of a Complex Variable and Related Pointwise Transformations of the Plane 54

## 3. Groups of Transformations: Euclidean and Lobachevskian Geometries 58

3.1. The Geometry of a Group of Transformations 58

3.2. Euclidean Geometry 64

3.3. Lobachevskian Geometry 68