## Exploring Analyic Geometry with |
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Getting Started

Introduction

Historical Background

Mathematica

Starting Descarta2D

Outline of the Book

Introduction [Top]

The purpose of this book is to provide a broad introduction to analytic geometry using the Mathematica and Descarta2D computer programs to enhance the numerical, symbolic and graphical nature of the subject. The book has the following objectives:

• To provide a computer-based alternative to a traditional course in analytic geometry.

• To provide a geometric research tool that can be used to explore numerically and symbolically various theorems and relationships of two-dimensional analytic geometry. Due to the nature of the Mathematica environment in which Descarta2D was written, the system can be easily enhanced and extended.

• To provide a reference of geometric formulas from analytic geometry that are not generally provided in more broad-based mathematical textbooks, nor included in mathematical handbooks.

• To provide a large-scale Mathematica programming tutorial that is instructive in the techniques of object oriented programming, modular packaging and good overall system design. By providing the full source code for the Descarta2D system, students and researchers can modify and enhance the system for their own purposes.

Historical Background [Top]

The word geometry is derived from the Greek words for "earth measure." Early geometers considered measurements of line segments, angles and other planar figures. Analytic geometry was introduced by René Descartes in his La Géométrie published in 1637. Accordingly, after his name, analytic or coordinate geometry is often referred to as Cartesian geometry. It is essentially a method of studying geometry by means of algebra. Earlier mathematicians had continued to resort to the conventional methods of geometric reasoning as set forth in great detail by Euclid and his school some 2000 years before. The tremendous advances made in the study of geometry since the time of Descartes are largely due to his introduction of the coordinate system and the associated algebraic or analytic methods.

With the advent of powerful mathematical computer software, such as Mathematica, much of the tedious algebraic manipulation has been removed from the study of analytic geometry, allowing comfortable exploration of the subject even by amateur mathematicians. Mathematica provides a programmable environment, meaning that the user can extend and expand the capabilities of the system including the addition of completely new concepts not covered by the kernel Mathematica system. This notion of expandability serves as the basis for the implementation of the Descarta2D system, which is essentially an extension of the capabilities of Mathematica cast into the world of analytic geometry.

Mathematica [Top]

In this book an assumption is made that you have at least a rudimentary understanding of how to run the Mathematica program, how to enter commands and receive results, and how to arrange files on a computer disk so that programs can locate them. A sufficient introduction to Mathematica would be gained by reading the "Tour of Mathematica" in Stephen Wolfram's book Mathematica: A System for Doing Mathematics by Computer.

The syntax Mathematica uses for mathematical operations differs somewhat from traditional mathematical notation. Since Descarta2D is implemented in the Mathematica programming language it follows all the syntactic conventions of the Mathematica system. See Wolfram's Mathematica book for more detailed descriptions of the syntax. Once you become familiar with Mathematica you will begin to appreciate the consistency and predictability of the system.

Starting Descarta2D [Top]

All of the underlying concepts of analytic geometry presented in this book are implemented in a Mathematica program called Descarta2D. Descarta2D consists of a number of Mathematica programs (called packages) that provide a rich environment for the study of analytic geometry. In order to avoid loading all the packages at one time, a master file of package declarations has been provided. You must load this file at the beginning of any Mathematica session that will make use of the Descarta2D packages. Once the package declarations have been loaded, all of the additional packages will be loaded automatically when they are needed. To load the Descarta2D package declarations from the file init.m use the command

<<Descarta2D`

If this is the first command in the Mathematica session, the Mathematica kernel will be loaded first, and then the declarations will be loaded. Depending on the speed of your computer this may take a few seconds or several minutes. After the initial start-up, packages will load at automatically as new Descarta2D functions are used for the first time. When a package is first loaded, you may notice a delay in computing results; after the package is loaded, results are computed immediately and the time taken depends on the complexity of the computation.

The examples in this book that illustrate the usage of Descarta2D were chosen primarily for their simplicity, rather than to correspond to significant calculations in analytic geometry. At the end of each chapter a section entitled "Explorations" provides more realistic applications of Descarta2D. All of the examples in this book were generated by running an actual copy of Mathematica Version 6. The interactive dialogs of each Mathematica session are provided in the corresponding chapter notebook, so very little typing is required to replicate the output and plots in each chapter. If you choose to enter the commands yourself instead of using the notebook, you should enter the commands exactly as they are printed (including all spaces and line breaks). This will insure that you obtain the same results as printed in the text. Once you become more familiar with Mathematica and Descarta2D, you will learn what deviations from the printed text are acceptable.

Plotting Descarta2D Objects

Graphically rendering (plotting) the geometric objects encountered in a study of analytic geometry greatly enhances the intuitive understanding of the subject. Mathematica provides a wide variety of commands for plotting objects including Graphics, Plot and ParametricPlot. There are also specialized commands such as ImplicitPlot and PolarPlot. Each of these commands has a wide variety of options, giving the user detailed control over the plotted output.

These Mathematica commands can also be used to plot Descarta2D objects, and, in fact, the figures found in this book were generated using the Mathematica plotting commands named above. However, the Descarta2D system provides another command, Sketch2D, for plotting Descarta2D objects. The Sketch2D command has a very simple syntax as illustrated in the following example.

Example: Plot these objects using the Sketch2D command: Point2D[{-1,2}], Line2D[2,-3,1] and Circle2D[{1,0},2]. (The meaning of these geometric objects will be explained in subsequent chapters; for now it is sufficient to understand that we are plotting a point, a line and a circle.)

Solution: The Descarta2D function Sketch2D[objList] plots a list of geometric objects.

Sketch2D[{Point2D[{-1,2}],Line2D[2,-3,1],Circle2D[{1,0},2]}]

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Outline of the Book [Top]

The book is divided into nine sections. The first five sections deal with the subject matter of analytic geometry; the remaining sections provide a reference manual for the use of the Descarta2D computer program and a listing of the source code for the packages that implement Descarta2D, as well as the solutions to the explorations.

Part I of the book serves as an introduction and begins with the material in this chapter aimed at getting started with the subject; the next chapter continues the introduction by providing a high-level tour of Descarta2D. Part II introduces the basic geometric objects studied in analytic geometry, including points and coordinates, equations and graphs, lines, line segments, circles, arcs and triangles. Part III continues by studying second-degree curves, parabolas, ellipses and hyperbolas. In addition, Part III provides a more general study of conic curves by examining general conics, conic arcs and medial curves.

Part IV covers geometric functions including transformations (translation, rotation, scaling and reflection) and the computation of areas and arc lengths. The subject of tangent curves is covered in Part V with specific chapters dedicated to tangent lines, tangent circles and tangent conics. The final chapter in Part V is an overview of biarc circles, which are a special form of tangent circles. The intent of this chapter is to illustrate how new capabilities can be added to Descarta2D.

Generally, the chapters comprising Parts I through V present material in sections with simple examples. The examples are sometimes supplemented with Descarta2D and Mathematica Hints that illustrate the more subtle usages of the commands. Each chapter ends with an "Explorations" section containing several more challenging problems in analytic geometry. The solutions for the explorations are available as Mathematica notebooks, as well as being listed alphabetically in Part VIII.

Parts VI and VII serve as a reference manual for the Descarta2D system. The reference manual includes a description of the geometric objects provided by Descarta2D, a browser for quickly finding command syntax and options, and a listing of the error messages that may be generated. Part VII provides a complete listing, with comments, of all the packages comprising Descarta2D.

Part VIII of the book contains reproductions of the notebooks which provide the solutions to the explorations found at the end of each chapter. The notebooks are listed in alphabetical order by their file names. The exploration notebook files may also be reviewed directly using Mathematica or MathReader.

Part IX contains the instructions for installing Descarta2D on your computer system as well as a Bibliography and a detailed index.

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