Euclidean and Affine Transformations

Euclidean and Affine Transformations

Geometric Transformations

1st Edition - January 1, 1965

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  • Authors: P. S. Modenov, A. S. Parkhomenko
  • eBook ISBN: 9781483261485

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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


    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

Product details

  • No. of pages: 170
  • Language: English
  • Copyright: © Academic Press 1965
  • Published: January 1, 1965
  • Imprint: Academic Press
  • eBook ISBN: 9781483261485

About the Authors

P. S. Modenov

A. S. Parkhomenko

About the Editors

Henry Booker

D. Allan Bromley

Nicholas DeClaris

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