Modern Geometry with Applications
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 angle applications assume axis circle completes compute cone conic connecting constant construction contains coordinate system coordinates cross ratio curve Definition Desargues direction distance drawing dual ellipse equal Equation Euclidean event exactly Example Exercise fact Figure focus follows formula four function geodesic geometry given Hence homogeneous hyperbola images inertial intersect isometry length lies light located measure meet Minkowski mirror motion moving objects observer obtain opposite orbit orientation origin parabola parallel passes perpendicular plane positive preserve projection projectivization Proof Proposition prove Pythagorean radial line reflection relative result rotation says segment Show side signal similar smooth solve space spacetime speed sphere spherical triangle surface takes tangent theorem translation traveling triangle vectors velocity vertex vertices worldline