Inversions

I. Ya. Bakel’man, Joan W. Teller, Susan Williams0226034992, 9780226034997

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.>

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Table of contents :
Front cover……Page 1
Back cover……Page 2
Table of Contents……Page 6
Preface……Page 7
1.1. Elementary Transformations of the Plane……Page 9
1.2. Stereographic Projection: The Point at Infinity of a Plane……Page 14
1.3. Inversions……Page 16
1.4. Properties of Inversions……Page 19
1.5. The Power of a Point with Respect to a Circle: The Radical Axis of Two Circles……Page 27
1.6. Application of Inversions to the Solution of Construction Problems……Page 32
1.7. Pencils of Circles……Page 40
1.8. Structure of an Elliptical Pencil……Page 48
1.9. Structure of a Parabolic Pencil……Page 49
1.10. Structure of a Hyperbolic Pencil……Page 50
1.11. Ptolemy’s Theorem……Page 53
2.1. Geometric Representation of Complex Numbers & Operations on Them……Page 56
2.2. Linear Functions of a Complex Variable & Elementary Transformations of the Plane……Page 60
2.3. Linear Fractional Functions of a Complex Variable & Related Pointwise Transformations of the Plane……Page 62
3.1. The Geometry of a Group of Transformations……Page 66
3.2. Euclidean Geometry……Page 72
3.3. Lobachevskian Geometry……Page 76