Shadows Of The Circle: Conic Sections, Optimal Figures And Non-Euclidean Geometry

Description

The aim of this book is to throw light on various facets of geometry through development of four geometrical themes. The first theme is about the ellipse, the shape of the shadow cast by a circle. The next, a natural continuation of the first, is a study of all three types of conic sections, the ellipse, the parabola and the hyperbola.The third theme is about certain properties of geometrical figures related to the problem of finding the largest area that can be enclosed by a curve of given length. This problem is called the isoperimetric problem. In itself, this topic contains motivation for major parts of the curriculum in mathematics at college level and sets the stage for more advanced mathematical subjects such as functions of several variables and the calculus of variations.The emergence of non-Euclidean geometries in the beginning of the nineteenth century represents one of the dramatic episodes in the history of mathematics. In the last theme the non-Euclidean geometry in the PoincarГ© disc model of the hyperbolic plane is developed Front Cover Title Page Copyright Page Preface Table of Contents Chapter 1: An ellipse in the shadow The ellipse as a plane section of a cylinder The equation of the ellipse A parametrization of the ellipse The ellipse as a locus Directrix for the ellipse Geometrical determination of foci and directrices for the ellipse The tangents of the ellipse An application to gear wheel movements Sources for Chapter 1 Chapter 2: With conic sections in the light The ellipse as a plane section in a cone Geometric determination of foci and directrices for a conic section The parabola The hyperbola Hyperbolic navigational systems Conic sections as algebraic curves Epilogue Sources for Chapter 2 Chapter 3: Optimal plane figures Isosceles triangles Perrons paradox Some simple geometrical problems without solutions A fundamental property of the real numbers Maxima and minima of real-valued functions The equilateral triangle as optimal figure The square as optimal figure The regular polygons as optimal figures Some limit values for regular polygons The isoperimetric problem Epilogue: Elements of the history of the calculus of variations Sources for Chapter 3 Chapter 4: The Poincare disc model of non-Euclidean geometry Euclids Elements The parallel axiom and non-Euclidean geometries Inversion in a circle Inversion as a mapping Orthogonal circles and Euclids Postulate 1 in the hyperbolic plane The notion of distance in the hyperbolic plane and Euclids Postulate 2 Isometries in the hyperbolic plane Hyperbolic triangles and n-gons The Poincare half-plane Elliptic geometries Sources for Chapter 4 Exercises Index