Euclidean and Affine Transformations

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.

Contenu

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