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
Author(s): Hansen V.L.
Publisher: World Scientific
Year: 1998
Language: English
Pages: 118
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
