Exploring Analyic Geometry with Mathematica®

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Classical Analytic Geometry
Organization of the Book
About the Author

The study of two-dimensional analytic geometry has gone in and out of fashion several times over the past century, however this classic field of mathematics has once again become popular due to the growing power of personal computers and the availability of powerful mathematical software systems, such as Mathematica, that can provide an interactive environment for studying the field. By combining the power of Mathematica with an analytic geometry software system called Descarta2D, the author has succeeded in meshing an ancient field of study with modern computational tools, the result being a simple, yet powerful, approach to studying analytic geometry. Students, engineers and mathematicians alike who are interested in analytic geometry can use this book and software for the study, research or just plain enjoyment of analytic geometry.

Mathematica provides an attractive environment for studying analytic geometry. Mathematica supports both numeric and symbolic computations, meaning that geometry problems can be solved numerically, producing approximate or exact answers, as well as producing general formulas with variables. Mathematica also has good facilities for producing graphical plots which are useful for visualizing the graphs of two-dimensional geometry.

Features [Top]

Exploring Analytic Geometry with Mathematica, Mathematica and Descarta2D provide the following outstanding features:

• The book can serve as classical analytic geometry textbook with in-line Mathematica dialogs to illustrate key concepts.

• A large number of examples with solutions and graphics is keyed to the textual development of each topic.

• Hints are provided for improving the reader's use and understanding of Mathematica and Descarta2D.

• More advanced topics are covered in explorations provided with each chapter, and full solutions are illustrated using Mathematica.

• A detailed reference manual provides complete documentation for Descarta2D, with complete syntax for over 100 new commands.

• Complete source code for Descarta2D is provided in 30 well-documented Mathematica notebooks.

• The complete book is integrated into the Mathematica Help Browser for easy access and reading.

• A complete software system and mathematical reference is packaged as an affordable book.

Classical Analytic Geometry [Top]

Exploring Analytic Geometry with Mathematica begins with a traditional development of analytic geometry that has been modernized with in-line chapter dialogs using Descarta2D and Mathematica to illustrate the underlying concepts. The following topics are covered in 21 chapters:

Coordinates Points Equations Graphs Lines Line Segments Circles Arcs Triangles Parabolas Ellipses Hyperbolas General Conics Conic Arcs Medial Curves Transformations Arc Length Area Tangent Lines Tangent Circles Tangent Conics Biarcs.

Each chapter begins with definitions of underlying mathematical terminology and develops the topic with more detailed derivations and proofs of important concepts.

Explorations [Top]

Each chapter in Exploring Analytic Geometry with Mathematica concludes with more advanced topics in the form of exploration problems to more fully develop the topics presented in each chapter. There are more than 100 of these more challenging explorations, and the full solutions are provided as Mathematica notebooks as well as printed in Part VIII of the book. Sample explorations include some of the more famous theorems from analytic geometry:

Carlyle's Circle Castillon's Problem Euler's Triangle Formula Eyeball Theorem Gergonne's Point Heron's Formula Inversion Monge's Theorem Reciprocal Polars Reflection in a Point Stewart's Theorem plus many more.

Descarta2D [Top]

Descarta2D provides a full-scale Mathematica implementation of the concepts developed in Exploring Analytic Geometry with Mathematica. A reference manual section explains in detail the usage of over 100 new commands that are provided by Descarta2D for creating, manipulating and querying geometric objects in Mathematica. To support the study and enhancement of the Descarta2D algorithms, the complete source code for Descarta2D is provided, both in printed form in the book and as Mathematica notebook files.

Notebooks [Top]

The complete text of the book in Abode Portable Document Format (PDF) for interactive reading. In addition, the following Mathematica notebooks are provided:

• Chapters with Mathematica dialogs, 24 interactive notebooks

• Reference material for Descarta2D, three notebooks

• Complete Descarta2D source code, 30 notebooks

Descarta2D packages, 30 loadable files

• Exploration solutions, 125 notebooks.

These notebooks have been thoroughly tested and are compatible with Mathematica Version 6. Maximum benefit of the book and software is gained by using it in conjunction with Mathematica, but a passive reading and viewing of the book and notebook files can be accomplished without using Mathematica itself.

Organization of the Book [Top]

Exploring Analytic Geometry with Mathematica is a 900-page volume divided into nine parts:

• Introduction (Getting Started and Descarta2D Tour)

• Elementary Geometry (Points, Lines, Circles, Arcs, Triangles)

• Conics (Parabolas, Ellipses, Hyperbolas, Conics, Medial Curves)

• Geometric Functions (Transformations, Arc Length, Area)

• Tangent Curves (Lines, Circles, Conics, Biarcs)

Descarta2D Reference (philosophy and command descriptions)

Descarta2D Packages (complete source code)

• Explorations (solution notebooks)

• Epilogue (Installation Instructions, Bibliography and a detailed index).

About the Author [Top]

Donald L. Vossler is a mechanical engineer and computer software designer with more than 25 years experience in computer aided design and geometric modeling. He has been involved in solid modeling since its inception in the early 1980's and has contributed to the theoretical foundation of the subject through several published papers. He has managed the development of a number of commercial computer aided design systems and holds a US Patent involving the underlying data representations of geometric models.

Copyright © 1999-2007 Donald L. Vossler, Descarta2D Publishing