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Affine Maps, Euclidean Motions and Quadrics / by Agustí Reventós Tarrida
Resource type: Ressourcentyp: Buch (Online)Buch (Online)Sprache: Englisch Reihen: Springer Undergraduate Mathematics Series | SpringerLink BücherVerlag: London : Springer-Verlag London Limited, 2011Beschreibung: Online-Ressource (XVIII, 458p. 49 illus, digital)ISBN:- 9780857297105
- 510
- QA1-939
- QA477
Inhalte:
Zusammenfassung: Affine Spaces -- Affinities -- Classification of Affinities -- Classification of Affinities in Arbitrary Dimension -- Euclidean Affine Spaces -- Euclidean motions -- Euclidean Motions of the Line, the Plane and of Space -- Affine Classification of Real Quadrics -- Orthogonal Classification of Quadrics -- Appendices.Zusammenfassung: Affine geometry and quadrics are fascinating subjects alone, but they are also important applications of linear algebra. They give a first glimpse into the world of algebraic geometry yet they are equally relevant to a wide range of disciplines such as engineering. This text discusses and classifies affinities and Euclidean motions culminating in classification results for quadrics. A high level of detail and generality is a key feature unmatched by other books available. Such intricacy makes this a particularly accessible teaching resource as it requires no extra time in deconstructing the author’s reasoning. The provision of a large number of exercises with hints will help students to develop their problem solving skills and will also be a useful resource for lecturers when setting work for independent study. Affinities, Euclidean Motions and Quadrics takes rudimentary, and often taken-for-granted, knowledge and presents it in a new, comprehensive form. Standard and non-standard examples are demonstrated throughout and an appendix provides the reader with a summary of advanced linear algebra facts for quick reference to the text. All factors combined, this is a self-contained book ideal for self-study that is not only foundational but unique in its approach.’ This text will be of use to lecturers in linear algebra and its applications to geometry as well as advanced undergraduate and beginning graduate students.PPN: PPN: 1650947240Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
""Affine Maps, Euclidean Motions and Quadrics""; ""Preface""; ""Organization""; ""Acknowledgements""; ""Contents""; ""1. Affine Spaces""; ""1.1 Introduction""; ""1.2 Definition of Affine Space""; ""1.2.1 Observations""; ""1.3 Examples""; ""1.4 The Dimension of an Affine Space""; ""1.5 First Properties""; ""1.6 Linear Varieties""; ""1.6.1 Observations""; ""1.7 Examples of Straight Lines""; ""1.8 Linear Varieties Generated by Points""; ""1.9 The Affine Grassmann Formulas""; ""1.10 Affine Frames""; ""1.10.1 Change of Affine Frame""; ""1.11 Equations of a Linear Variety""
""1.11.1 Equations of Straight Lines""""1.11.2 Equations of Hyperplanes""; ""1.12 Barycenter""; ""1.12.1 Computations in Coordinates""; ""1.13 Simple Ratio""; ""1.13.1 Characterization of the Points of a Line Segment (k=R)""; ""1.13.2 Characteristic Property of the Barycenter of a Triangle""; ""1.13.3 The Real Plane as a Complex Straight Line""; ""1.14 Theorems of Thales, Menelaus and Ceva""; ""2. Affinities""; ""2.1 Introduction""; ""2.2 Definition of Affinity""; ""2.3 First Properties""; ""2.4 The Affine Group""; ""2.5 Affinities and Linear Varieties""; ""2.6 Equations of Affinities""
""2.7 Invariant Varieties""""2.8 Examples of Affinities""; ""2.8.1 Translations""; ""Equations of Translations""; ""2.8.2 Homotheties""; ""Equations of homotheties""; ""2.8.3 Symmetries""; ""Equations of symmetries""; ""2.8.4 Projections""; ""Equations of projections""; ""2.9 Characterization of Affinities of the Line""; ""2.10 The Fundamental Theorem of Affine Geometry""; ""2.10.1 A Note on Automorphisms of Fields""; ""3. Classification of Affinities""; ""3.1 Introduction""; ""3.2 Similar Endomorphisms""; ""3.3 Similar Affinities""; ""3.4 Computations in Coordinates""
""3.5 Invariance Level""""3.5.1 Invariance Level in Dimension 1""; ""3.5.2 Invariance Level in Dimension 2""; ""3.6 Classification of Affinities of the Line""; ""3.7 Classification of Affinities of the Real Plane""; ""3.7.1 Classification of Endomorphisms in Real Dimension Two""; ""3.7.2 List of Canonical Expressions of Endomorphisms""; ""3.7.3 Canonical Expressions of Affinities""; ""3.8 Invariance Level in the Real Plane""; ""3.8.1 Summary of the Classification of Affinities of the Real Plane""; ""3.9 Geometrical Interpretation""; ""3.9.1 Homologies""; ""3.9.2 Elliptic Affinities""
""3.9.3 Parabolic Affinities""""3.9.4 Hyperbolic Affinities""; ""3.10 Decomposition of Affinities in the Real Plane""; ""4. Classification of Affinities in Arbitrary Dimension""; ""4.1 Introduction""; ""4.2 Jordan Matrices""; ""4.3 Similar Endomorphisms and Canonical Decomposition""; ""4.4 Clarifying Examples""; ""4.5 Classification of Affinities in Arbitrary Dimension""; ""5. Euclidean Affine Spaces""; ""5.1 Introduction""; ""5.2 Definition of Euclidean Affine Space. Pythagoras' Theorem""; ""5.3 The Distance Between Two Varieties""; ""Justification of the Method""
""5.4 Common Perpendicular""
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