Modern Geometry with ApplicationsSpringer Science & Business Media, 6 de des. 2012 - 204 pàgines 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). |
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AABC algebraic area inside axis circle completes the proof cone conformal map congruent constant coordinate system Corollary cross ratio curve Definition directrix distance drawing dual curve dual triangle ellipse equal Equation Euclidean plane Euclidean space event Example Exercise FIGURE focus follows formula future pointing Galilean geometry Hence Hint homogeneous coordinates hyperbola images inertial coordinate inertial observers infinitesimal intersect isometry Kepler's latitude law of cosines length line at infinity line segment lines in P² Lorentz M₁ measure Minkowski mirror O's coordinates opposite vertex orbit parabola perpendicular point at infinity projectivization Proposition prove Pythagorean theorem Pythagorean triple radial line radius reflection rotation Show side smooth conic spacetime speed of light sphere spherical triangle stereographic projection subtended tangent timelike vanishing point vectors velocity vertices viewplane worldline