Transformational Plane Geometry: 1st Edition (Paperback) book cover

Transformational Plane Geometry

1st Edition

By Ronald N. Umble, Zhigang Han

Chapman and Hall/CRC

234 pages | 91 B/W Illus.

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Designed for a one-semester course at the junior undergraduate level, Transformational Plane Geometry takes a hands-on, interactive approach to teaching plane geometry. The book is self-contained, defining basic concepts from linear and abstract algebra gradually as needed.

The text adheres to the National Council of Teachers of Mathematics Principles and Standards for School Mathematics and the Common Core State Standards Initiative Standards for Mathematical Practice. Future teachers will acquire the skills needed to effectively apply these standards in their classrooms.

Following Felix Klein’s Erlangen Program, the book provides students in pure mathematics and students in teacher training programs with a concrete visual alternative to Euclid’s purely axiomatic approach to plane geometry. It enables geometrical visualization in three ways:

  1. Key concepts are motivated with exploratory activities using software specifically designed for performing geometrical constructions, such as Geometer’s Sketchpad.
  2. Each concept is introduced synthetically (without coordinates) and analytically (with coordinates).
  3. Exercises include numerous geometric constructions that use a reflecting instrument, such as a MIRA.

After reviewing the essential principles of classical Euclidean geometry, the book covers general transformations of the plane with particular attention to translations, rotations, reflections, stretches, and their compositions. The authors apply these transformations to study congruence, similarity, and symmetry of plane figures and to classify the isometries and similarities of the plane.


"This book is designed for a one-semester course at the junior undergraduate level and turns especially to future educators in the USA. … The arrangement and clarity of the text meet the most demanding pedagogical and mathematical requirements. Highlights of the book are the classification of isometries and similarities of the Euclidean plane. … a wonderful first step into transformational plane geometry …"

Zentralblatt MATH 1311

Table of Contents

Axioms of Euclidean Plane Geometry

The Existence and Incidence Postulates

The Distance and Ruler Postulates

The Plane Separation Postulate

The Protractor Postulate

The Side-Angle-Side Postulate and the Euclidean Parallel Postulate

Theorems of Euclidean Plane Geometry

The Exterior Angle Theorem

Triangle Congruence Theorems

The Alternate Interior Angles Theorem and the Angle Sum Theorem

Similar Triangles

Introduction to Transformations, Isometries, and Similarities


Isometries and Similarities

Appendix: Proof of Surjectivity

Translations, Rotations, and Reflections




Appendix: Geometer’s Sketchpad Commands Required by Exploratory Activities

Compositions of Translations, Rotations, and Reflections

The Three Points Theorem

Rotations as Compositions of Two Reflections

Translations as Compositions of Two Halfturns or Two Reflections

The Angle Addition Theorem

Glide Reflections

Classification of Isometries

The Fundamental Theorem and Congruence

Classification of Isometries

Orientation and the Isometry Recognition Problem

The Geometry of Conjugation

Symmetry of Plane Figures

Groups of Isometries

Symmetry Type


Frieze Patterns

Wallpaper Patterns


Plane Similarities

Classification of Dilatations

Classification of Similarities and the Similarity Recognition Problem

Conjugation and Similarity Symmetry

Appendix: Hints and Answers to Selected Exercises



About the Authors

Ronald N. Umble is a professor of mathematics at Millersville University of Pennsylvania. He has directed numerous undergraduate research projects in mathematics. He received his Ph.D. in algebraic topology under the supervision of James D. Stasheff from the University of North Carolina at Chapel Hill.

Zhigang Han is an assistant professor of mathematics at Millersville University of Pennsylvania. He earned his Ph.D. in symplectic geometry and topology under the supervision of Dusa McDuff from Stony Brook University.

About the Series

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Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Geometry / General