Euclidean and Affine Transformations - 1st Edition - ISBN: 9781483228013, 9781483261485

Euclidean and Affine Transformations

1st Edition

Geometric Transformations

Authors: P. S. Modenov A. S. Parkhomenko
Editors: Henry Booker D. Allan Bromley Nicholas DeClaris
eBook ISBN: 9781483261485
Imprint: Academic Press
Published Date: 1st January 1965
Page Count: 170
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Description

Geometric Transformations, Volume 1: Euclidean and Affine Transformations focuses on the study of coordinates, trigonometry, transformations, and linear equations.

The publication first takes a look at orthogonal transformations, including orthogonal transformations of the first and second kinds; representations of orthogonal transformations as the products of fundamental orthogonal transformations; and representation of an orthogonal transformation of space as a product of fundamental orthogonal transformations. The text then examines similarity and affine transformations. Topics include properties of affine mappings, Darboux's lemma and its consequences, affine transformations in coordinates, homothetic transformations, similarity transformations of the plane in coordinates, and similarity mapping.

The book takes a look at the representation of a similarity transformation as the product of a homothetic transformation and an orthogonal transformation; application of affine transformations to the investigation of properties of the ellipse; and representation of any affine transformation as a product of affine transformations of the simplest types.

The manuscript is a valuable reference for high school teachers and readers interested in the Euclidean and affine transformations.

Table of Contents


Preface to Volume 1 of the English Edition

Translator's Note

Preface to the Russian Edition

Introduction

Chapter I. General Definitions

1. Sets and Functions

2. Mappings

3. Groups of Transformations

Chapter II. Orthogonal Transformations

4. Orthogonal Mappings

5. Properties of Orthogonal Mappings

6. Orientation

7. Orthogonal Transformations of the First and Second Kinds

8. The Fundamental Types of Orthogonal Transformation (Translation, Reflection, Rotation)

9. Representations of Orthogonal Transformations as Products of the Fundamental Orthogonal Transformations: Translations, Reflections, and Rotations

10. Orthogonal Transformations of the Plane in Coordinates

11. Orthogonal Transformations in Space

12. Representation of an Orthogonal Transformation of Space as a Product of Fundamental Orthogonal Transformations

13. Orthogonal Transformations of Space in Coordinates

Chapter III. Similarity Transformations

14. Similarity Mappings

15. Properties of Similarity Transformations

16. Homothetic Transformations

17. Representation of a Similarity Transformation as the Product of a Homothetic Transformation and an Orthogonal Transformation

18. Similarity Transformations of the Plane in Coordinates

19. Similarity Transformations in Space

Chapter IV. Affine Transformations

20. Definition of Affine Mappings and Transformations of the Plane

21. Examples of Affine Transformations and Mappings of a Plane

22. Properties of Affine Mappings

23. Darboux's Lemma and Its Consequences

24. Invariance of Length Ratios under Affine Mappings

25. Further Properties of Affine Mappings

26. Representation of Any Affine Transformation as a Product of Affine Transformations of the Simplest Types

27. Noninvariance of Lengths of Segments Under Affine Mappings

28. The Change in Area Under an Affine Mapping of One Plane Onto Another

29. An Application of Affine Transformations to the Investigation of Properties of the Ellipse

30. Affine Transformations in Coordinates

31. Affine Classification of Quadratic Curves

32. Affine Transformations of Space

Appendix to Chapter II. Length-Preserving Mappings

Subject Index

Details

No. of pages:
170
Language:
English
Copyright:
© Academic Press 1965
Published:
Imprint:
Academic Press
eBook ISBN:
9781483261485

About the Author

P. S. Modenov

A. S. Parkhomenko

About the Editor

Henry Booker

D. Allan Bromley

Nicholas DeClaris