This is a thorough analysis of the fundamentals of plane geometry. The reader is provided with an abundance of geometric facts such as results of Euclidean and spherical geometry, conic sections, projective geometry, and Minkowski geometry to name a few. The book covers unusual material and provides fresh examples and interesting exercises.

Klappentext

This introduction to modern geometry differs from other books in the field due to its emphasis on applications and its discussion of special relativity as a major example of a non-Euclidean geometry. Additionally, it covers the two important areas of non-Euclidean geometry, spherical geometry and projective geometry, as well as emphasising transformations, and conics and planetary orbits. Much emphasis is placed on applications throughout the book, which motivate the topics, and many additional applications are given in the exercises. It makes an excellent introduction for those who need to know how geometry is used in addition to its formal theory.

Inhalt

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 the Velocity Addition Formula.- 5.10 Appendix: Circular and Hyperbolic Functions.- References.