Euclidean Geometry and Transformations

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ISBN: 9780486434766 | 0486434761
Cover: Paperback
Copyright: 5/18/2004

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This introduction to Euclidean geometry emphasizes both the theory and the practical application of isometries and similarities to geometric transformations. Contents include modern elementary geometry, isometries and similarities in the plane, vectors and complex numbers in geometry, inversion, and isometries in space. Numerous exercises, many with detailed answers. 1972 edition.

Modern Elementary Geometry

The Beginnings of Geometry

(3)

Directed segments and angles

(5)

Ideal points and ratios

(3)

The theorem of Menelaus

(7)

Ceva's theorem

(6)

Some geometry of the triangle

(8)

More geometry of the triangle

(7)

Geometric constructions

(10)

Isometries in the Plane

The Amazing Greeks

(2)

Introduction to translations, rotations, and reflections

(5)

Introduction to isometries

(4)

Transformation theory

(4)

Isometries as products of reflections

(5)

Translations and rotations

(4)

Halfturns

(2)

Products of reflections

(3)

Properties of isometries; a summary

(2)

Applications of isometries to elementary geometry

(4)

Further elementary applications

(4)

Advanced applications

(6)

Analytic representations of direct isometries

(4)

Analytic representations of opposite isometries

(7)

Similarities in the Plane

The rebirth of mathematical thinking

101

(3)

Introduction to similarities

104

(2)

Homothety

106

(4)

Similarity

110

(4)

Applications of similarities to elementary geometry

114

(5)

Further elementary applications

119

(5)

Advanced applications

124

(4)

Analytic representations of similarities

128

(6)

Vectors and Complex Numbers in Geometry

The search for the meaning of complex numbers

131

(3)

Introduction to complex numbers

134

(4)

Vectors

138

(5)

Vector multiplication

143

(7)

Vectors and complex numbers

150

(5)

Triangles in the Gauss plane

155

(6)

Lines in the Gauss plane

161

(4)

The circle

165

(3)

Isometries and similarities in the Gauss plane

168

(3)

Inversion

171

(61)

Matchless modern mathematics

Inversion

175

(5)

Progressions, ratios, and Peaucellier's cell

180

(5)

Inversion and complex geometry

185

(4)

Applications of inversion

189

(12)

Isometries in Space

What next?

196

(5)

Introduction to three dimensions

201

(3)

Reflection in a plane

204

(4)

Basic space isometries

208

(3)

More space isometries

211

(7)

Some applications

218

(4)

Analytic representations

222

(4)

Appendixes

A. A Summary of Book I of Euclid's Elements

226

(2)

B. Basic Ruler and Compass Constructions

228

(4)

Bibliography

232

(3)

Hints for Selected Exercises

235

(13)

Answers

248

(40)

Index

288

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Euclidean Geometry and Transformations

Summary

Table of Contents

Supplemental Materials

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Euclidean Geometry and Transformations

Summary

Table of Contents

Supplemental Materials

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