Modenov / Parkhomenko / Booker Euclidean and Affine Transformations

Geometric Transformations
1. Auflage 2014
ISBN: 978-1-4832-6148-5
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 170 Seiten, Web PDF

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.

Autoren/Hrsg.

Modenov, P. S.

Parkhomenko, A. S.

Booker, Henry

Bromley, D. Allan

Declaris, Nicholas

Inhaltsverzeichnis

1;Front Cover;1
2;Euclidean and Affine Transformations;4
3;Copyright Page;5
4;Table of Contents;10
5;Preface to Volume 1 of the English Edition;6
6;Translator's Note;7
7;Preface to the Russian Edition;8
8;Introduction;12
9;CHAPTER I.
General Definitions;14
9.1;1. Sets and Functions;14
9.2;2. Mappings;17
9.3;3. Groups of Transformations;21
10;CHAPTER II.
Orthogonal Transformations;28
10.1;4. Orthogonal Mappings;29
10.2;5. Properties of Orthogonal Mappings;30
10.3;6. Orientation;36
10.4;7. Orthogonal Transformations of the First and Second
Kinds;42
10.5;8. The Fundamental Types of Orthogonal Transformation (Translation, Reflection, Rotation);45
10.6;9. Representations of Orthogonal Transformations as Products of the Fundamental Orthogonal Transformations: Translations,
Reflections, and Rotations;55
10.7;10. Orthogonal Transformations of the Plane in
Coordinates;60
10.8;11. Orthogonal Transformations in Space;65
10.9;12. Representation of an Orthogonal Transformation of Space as a Product of Fundamental
Orthogonal Transformations;70
10.10;13. Orthogonal Transformations of
Space in Coordinates;79
11;CHAPTER III.
Similarity Transformations;88
11.1;14. Similarity Mappings;88
11.2;15. Properties of Similarity Transformations;90
11.3;16. Homothetic Transformations;92
11.4;17. Representation of a Similarity Transformation as the Product of a Homothetic Transformation and an Orthogonal Transformation;97
11.5;18. Similarity Transformations of the Plane in Coordinates;102
11.6;19. Similarity Transformations in Space;104
12;CHAPTER IV.
Affine Transformations;108
12.1;20. Definition of Affine Mappings and Transformations of the Plane;108
12.2;21. Examples of Affine Transformations and
Mappings of a Plane;109
12.3;22. Properties of Affine Mappings;124
12.4;23. Darboux's Lemma and its Consequences;127
12.5;25. Further Properties of Affine Mappings;131
12.6;26. Representation of any Affine Transformation as a Product of Affine Transformations of the Simplest Types;133
12.7;27. Noninvariance of Lengths of Segments under Affine Mappings;136
12.8;28. The Change in Area under an Affine Mapping of One Plane on to Another;138
12.9;29. An Application of Affine Transformations to the Investigation of Properties of the Ellipse;139
12.10;30. Affine Transformations in Coordinates;146
12.11;31. Affine Classification of Quadratic Curves;149
12.12;32. Affine Transformations of Space;156
13;APPENDIX TO CHAPTER II:
Length-Preserving Mappings;163
14;Subject Index;168