Euclidean and Affine Transformations

Geometric Transformations

1st Edition - January 1, 1965
Authors: P. S. Modenov, A. S. Parkhomenko
Editors: Henry Booker, D. Allan Bromley, Nicholas DeClaris
Language: English
eBook ISBN:
9 7 8 - 1 - 4 8 3 2 - 6 1 4 8 - 5

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

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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.

Preface to Volume 1 of the English EditionTranslator's NotePreface to the Russian EditionIntroductionChapter I. General Definitions 1. Sets and Functions 2. Mappings 3. Groups of TransformationsChapter 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 CoordinatesChapter 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 SpaceChapter 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 SpaceAppendix to Chapter II. Length-Preserving MappingsSubject Index