Playing With Dynamic Geometry by Donald A. Cole Copyright © 2010 by Donald A. Cole All rights reserved. Cover Design: A three-dimensional image of the curve known as the Lemniscate of Bernoulli and its graph (see Chapter 15). TABLE OF CONTENTS Preface.................................................................................................................. xix Chapter 1 – Background ............................................................ 1-1 1.1 Introduction ............................................................................................................ 1-1 1.2 Equations and Graph .............................................................................................. 1-1 1.3 Analytical and Physical Properties ........................................................................ 1-4 1.3.1 Derivatives of the Curve ................................................................................. 1-4 1.3.2 Metric Properties of the Curve ........................................................................ 1-4 1.3.3 Curvature......................................................................................................... 1-6 1.3.4 Angles ............................................................................................................. 1-6 1.4 Geometric Properties ............................................................................................. 1-7 1.5 Types of Derived Curves ....................................................................................... 1-7 1.5.1 Evolute ............................................................................................................ 1-8 1.5.2 Involute ........................................................................................................... 1-8 1.5.3 Parallel Curves ................................................................................................ 1-8 1.5.4 Inversion ......................................................................................................... 1-9 1.5.5 Pedal Curves ................................................................................................... 1-9 1.5.6 Conchoid ....................................................................................................... 1-10 1.5.7 Strophoid ....................................................................................................... 1-10 1.5.8 Cissoid........................................................................................................... 1-11 1.5.9 Roulette and Glissette ................................................................................... 1-12 1.5.10 Isoptic and Orthoptic................................................................................... 1-12 1.5.11 Caustic......................................................................................................... 1-12 1.6 Special Considerations ......................................................................................... 1-12 1.7 Dynamic Geometry Construction ........................................................................ 1-12 Chapter 2 – The Cissoid of Diocles ........................................... 2-1 2.1 Introduction ............................................................................................................ 2-2 2.2 Equations and Graph of the Cissoid of Diocles ..................................................... 2-3 2.3 Analytical and Physical Properties of the Cissoid of Diocles................................ 2-5 2.3.1 Derivatives of the Cissoid of Diocles ............................................................. 2-5 2.3.2 Metric Properties of the Cissoid of Diocles .................................................... 2-5 2.3.3 Curvature of the Cissoid of Diocles ................................................................ 2-6 2.3.4 Angles for the Cissoid of Diocles ................................................................... 2-6 2.4 Geometric Properties of the Cissoid of Diocles ..................................................... 2-7 2.5 Doubling the Cube ................................................................................................. 2-7 2.6 Dynamic Geometry of the Cissoid of Diocles ....................................................... 2-8 2.6.1 A Construction Based on the Definition of the Cissoid of Diocles ................ 2-8 2.6.2 Diocles’ Method.............................................................................................. 2-8 2.6.3 Newton’s Method............................................................................................ 2-9 2.6.4 A Construction Based on Dividing a Circle’s Diameter ............................... 2-10 2.6.5 A Construction Based on Three Lines .......................................................... 2-10 2.6.6 The Cissoid of Diocles as the Inversion of a Parabola ................................. 2-11 2.6.7 The Cissoid of Diocles as the Pedal Curve of a Parabola ............................. 2-11 2.6.8 The Tangent to the Cissoid of Diocles .......................................................... 2-12 Table of Contents i Playing With Dynamic Geometry 2.6.9 The Osculating Circle of the Cissoid of Diocles .......................................... 2-12 2.6.10 The Generalized Concept of the Cissoid .................................................... 2-13 2.6.11 An Alternate Construction for the Osculating Circle ................................. 2-14 Chapter 3 – The Strophoid ........................................................ 3-1 3.1 Introduction ............................................................................................................ 3-2 3.2 Equations and Graph of the Right Strophoid ......................................................... 3-3 3.3 Analytical and Physical Properties of the Right Strophoid ................................... 3-4 3.3.1 Derivatives of the Right Strophoid ................................................................. 3-4 3.3.2 Metric Properties of the Right Strophoid ........................................................ 3-4 3.3.3 Curvature of the Right Strophoid .................................................................... 3-6 3.3.4 Angles for the Right Strophoid ....................................................................... 3-6 3.4 Geometric Properties of the Right Strophoid......................................................... 3-7 3.5 Dynamic Geometry of the Strophoid ..................................................................... 3-7 3.5.1 A Construction for the General Strophoid ...................................................... 3-7 3.5.2 The General Strophoid as the Pedal of a Parabola .......................................... 3-7 3.5.3 A Construction Based on the Definition of a Right Strophoid ....................... 3-8 3.5.4 Newton’s Carpenter Square Construction of the Right Strophoid.................. 3-9 3.5.5 The Right Strophoid as an Envelope of Circles .............................................. 3-9 3.5.6 The Right Strophoid as the Inverse of a Hyperbola ...................................... 3-10 3.5.7 The Right Strophoid as an Inversion of Itself ............................................... 3-10 3.5.8 The Right Strophoid and Its Tangent ............................................................ 3-11 3.5.9 The Strophoid and Its Osculating Circle ....................................................... 3-11 Chapter 4 – The Witch of Agnesi .............................................. 4-1 4.1 Introduction ............................................................................................................ 4-2 4.2 Equations and Graph of the Witch of Agnesi ........................................................ 4-3 4.3 Analytical and Physical Properties of the Witch of Agnesi ................................... 4-4 4.3.1 Derivatives of the Witch of Agnesi ................................................................ 4-5 4.3.2 Metric Properties of the Witch of Agnesi ....................................................... 4-5 4.3.3 Curvature of the Witch of Agnesi ................................................................... 4-6 4.3.4 Angles for the Witch of Agnesi ...................................................................... 4-6 4.4 Geometric Properties of the Witch of Agnesi ........................................................ 4-6 4.5 Dynamic Geometry of the Witch of Agnesi .......................................................... 4-7 4.5.1 The Witch of Agnesi Based on the Definition ................................................ 4-7 4.5.2 The Tangent to the Witch of Agnesi ............................................................... 4-7 4.5.3 The Pedal Curves of the Witch of Agnesi ...................................................... 4-8 4.5.4 An Alternate Construction for the Tangent to the Witch of Agnesi ............... 4-8 4.5.5 The Osculating Circle for the Witch of Agnesi .............................................. 4-9 Chapter 5 – The Conchoid of Nicomedes ................................. 5-1 5.1 Introduction ............................................................................................................ 5-2 5.2 Equations and Graph of the Conchoid of Nicomedes ............................................ 5-3 5.3 Analytical and Physical Properties of the Conchoid of Nicomedes ...................... 5-5 5.3.1 Derivatives of the Conchoid of Nicomedes ...................................................