This book is an introduction to the theory and applications of "modern geometry" ~ roughly speaking, geometry that was developed after Euclid. It covers three major areas of non-Euclidean geometry and their applica tions: spherical geometry (used in navigation and astronomy), projective geometry (used in art), and spacetime geometry (used in the Special The ory of Relativity). In addition it treats some of the more useful topics from Euclidean geometry, focusing on the use of Euclidean motions, and includes a chapter on conics and the orbits of planets. My aim in writing this book was to balance theory with applications. It seems to me that students of geometry, especially prospective mathe matics teachers, need to be aware of how geometry is used as well as how it is derived. Every topic in the book is motivated by an application and many additional applications are given in the exercises. This emphasis on applications is responsible for a somewhat nontraditional choice of top ics: I left out hyperbolic geometry, a traditional topic with practically no applications that are intelligible to undergraduates, and replaced it with the spacetime geometry of Special Relativity, a thoroughly non-Euclidean geometry with striking implications for our own physical universe. The book contains enough material for a one semester course in geometry at the sophomore-to-senior level, as well as many exercises, mostly of a non routine nature (the instructor may want to supplement them with routine exercises of his/her own).
Reihe
Auflage
Sprache
Verlagsort
Zielgruppe
Für höhere Schule und Studium
Für die Erwachsenenbildung
Lower undergraduate
Illustrationen
150
150 s/w Abbildungen
VIII, 204 p. 150 illus.
Maße
Höhe: 233 mm
Breite: 155 mm
Dicke: 12 mm
Gewicht
ISBN-13
978-0-387-94222-3 (9780387942223)
DOI
10.1007/978-1-4612-0855-6
Schweitzer Klassifikation
1 Euclidean Geometry.- 1.1 Euclidean Space.- 1.2 Isometries and Congruence.- 1.3 Reflections in the Plane.- 1.4 Reflections in Space.- 1.5 Translations.- 1.6 Rotations.- 1.7 Applications and Examples.- 1.8 Some Key Results of High School Geometry: The Parallel Postulate, Angles of a Triangle, Similar Triangles, and the Pythagorean Theorem.- 1.9 SSS, ASA, and SAS.- 1.10 The General Isometry.- 1.11 Appendix: The Planimeter.- 2 Spherical Geometry.- 2.1 Geodesics.- 2.2 Geodesics on Spheres.- 2.3 The Six Angles of a Spherical Triangle.- 2.4 The Law of Cosines for Sides.- 2.5 The Dual Spherical Triangle.- 2.6 The Law of Cosines for Angles.- 2.7 The Law of Sines for Spherical Triangles.- 2.8 Navigation Problems.- 2.9 Mapmaking.- 2.10 Applications of Stereographic Projection.- 3 Conics.- 3.1 Conic Sections.- 3.2 Foci of Ellipses and Hyperbolas.- 3.3 Eccentricity and Directrix; the Focus of a Parabola.- 3.4 Tangent Lines.- 3.5 Focusing Properties of Conics.- 3.6 Review Exercises: Standard Equations for Smooth Conics.- 3.7 LORAN Navigation.- 3.8 Kepler's Laws of Planetary Motion.- 3.9 Appendix: Reduction of a Quadratic Equation to Standard Form.- 4 Projective Geometry.- 4.1 Perspective Drawing.- 4.2 Projective Space.- 4.3 Desargues' Theorem.- 4.4 Cross Ratios.- 4.5 Projections in Coordinates.- 4.6 Homogeneous Coordinates and Duality.- 4.7 Homogeneous Polynomials, Algebraic Curves.- 4.8 Tangents.- 4.9 Dual Curves.- 4.10 Pascal's and Brianchon's Theorems.- 5 Special Relativity.- 5.1 Spacetime.- 5.2 Galilean Transformations.- 5.3 The Failure of the Galilean Transformations.- 5.4 Lorentz Transformations.- 5.5 Relativistic Addition of Velocities.- 5.6 Lorentz-FitzGerald Contractions1.- 5.7 Minkowski Geometry.- 5.8 The Slowest Path is a Line.- 5.9 Hyperbolic Angles and theVelocity Addition Formula.- 5.10 Appendix: Circular and Hyperbolic Functions.- References.
