Exploring Analytic Geometry with Mathematica

883 Pages·1999·2.82 MB·English

Exploring Analytic Geometry
with Mathematica(cid:13)
R
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Exploring Analytic Geometry
with Mathematica(cid:13)
R
Donald L. Vossler
BME, Kettering University, 1978
MM, Aquinas College, 1981
ACADEMIC PRESS
San Diego London Boston
New York Sydney Tokyo Toronto

Preface
The study of two-dimensional analytic geometry has gone in and out of fashion several times
overthepastcentury,howeverthisclassic(cid:12)eldofmathematicshasonceagainbecomepopular
duetothegrowingpowerofpersonalcomputersandtheavailabilityofpowerfulmathematical
softwaresystems,suchasMathematica,thatcanprovideaninteractiveenvironmentforstudy-
ing the (cid:12)eld. By combining the power of Mathematica with an analytic geometry software
system calledDescarta2D,the author has succeededin meshing an ancient(cid:12)eld ofstudy 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 ana-
lytic geometry can use this book and software for the study, researchor just plain enjoyment
of analytic geometry.
Mathematica provides an attractive environment for studying analytic geometry. Mathe-
matica supports both numeric and symbolic computations, meaning that geometry problems
canbe solvednumerically,producingapproximateorexactanswers,aswellasproducinggen-
eral formulas with variables. Mathematica also has good facilities for producing graphical
plots which are useful for visualizing the graphs of two-dimensional geometry.
Features
Exploring Analytic Geometry with Mathematica, Mathematica and Descarta2D provide the
following outstanding features:
(cid:15) The book can serve as classical analytic geometry textbook with in-line Mathematica
dialogs to illustrate key concepts.
(cid:15) A large number of examples with solutions and graphics is keyed to the textual devel-
opment of each topic.
(cid:15) Hints are provided for improving the reader’s use and understanding of Mathematica
and Descarta2D.
(cid:15) More advanced topics are covered in explorations provided with each chapter, and full
solutions are illustrated using Mathematica.
v
vi Preface
(cid:15) AdetailedreferencemanualprovidescompletedocumentationforDescarta2D,withcom-
plete syntax for over 100 new commands.
(cid:15) Complete source code for Descarta2D is provided in 30 well-documented Mathematica
notebooks.
(cid:15) ThecompletebookisintegratedintotheMathematicaHelpBrowserforeasyaccessand
reading.
(cid:15) A CD-ROM is included for convenient, permanent storage of the Descarta2D software.
(cid:15) A complete software system and mathematical reference is packaged as an a(cid:11)ordable
book.
Classical Analytic Geometry
Exploring Analytic Geometry with Mathematica begins with a traditional development of an-
alytic 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 (cid:15) Points (cid:15) Equations (cid:15) Graphs (cid:15) Lines (cid:15) Line Segments (cid:15) Cir-
cles (cid:15) Arcs (cid:15) Triangles (cid:15) Parabolas (cid:15) Ellipses (cid:15) Hyperbolas (cid:15) General Conics (cid:15)
Conic Arcs (cid:15) Medial Curves (cid:15) Transformations (cid:15) Arc Length (cid:15) Area (cid:15) Tan-
gent Lines (cid:15) Tangent Circles (cid:15) Tangent Conics (cid:15) Biarcs.
Eachchapterbeginswithde(cid:12)nitionsofunderlyingmathematicalterminologyanddevelops
the topic with more detailed derivations and proofs of important concepts.
Explorations
EachchapterinExploringAnalyticGeometrywithMathematica concludeswithmoreadvanced
topics in the form of exploration problems to more fully develop the topics presented in each
chapter. Therearemorethan100ofthesemorechallengingexplorations,andthefullsolutions
areprovidedontheCD-ROMasMathematicanotebooksaswellasprintedinPartVIIIofthe
book. Sampleexplorationsincludesomeofthemorefamoustheoremsfromanalyticgeometry:
Carlyle’s Circle (cid:15) Castillon’s Problem (cid:15) Euler’s Triangle Formula (cid:15) Eyeball The-
orem (cid:15) Gergonne’s Point (cid:15) Heron’s Formula (cid:15) Inversion (cid:15) Monge’s Theorem (cid:15)
Reciprocal Polars(cid:15) Reflection in a Point (cid:15) Stewart’s Theorem (cid:15) plus many more.
Preface vii
Descarta2D
Descarta2D provides a full-scale Mathematica implementation of the concepts developed in
Exploring Analytic Geometry with Mathematica. Areferencemanualsectionexplainsindetail
theusageofover100newcommandsthatareprovidedbyDescarta2Dforcreating,manipulat-
ing 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 (cid:12)les on the CD-ROM.
CD-ROM
The CD-ROM provides the complete text of the book in Abode Portable Document Format
(PDF)for interactivereading. Inaddition,the CD-ROMprovidesthe followingMathematica
notebooks:
(cid:15) Chapters with Mathematica dialogs, 24 interactive notebooks
(cid:15) Reference material for Descarta2D, three notebooks
(cid:15) Complete Descarta2D source code, 30 notebooks
(cid:15) Descarta2D packages, 30 loadable (cid:12)les
(cid:15) Explorationsolutions, 125 notebooks.
These notebooks have been thoroughly tested and are compatible with Mathematica Version
3.0.1 and Version 4.0. Maximum bene(cid:12)t 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
(cid:12)les can be accomplished without using Mathematica itself.
Organization of the Book
Exploring Analytic Geometry with Mathematica is a 900-pagevolume divided into nine parts:
(cid:15) Introduction (Getting Started and Descarta2D Tour)
(cid:15) Elementary Geometry (Points, Lines, Circles, Arcs, Triangles)
(cid:15) Conics (Parabolas,Ellipses, Hyperbolas, Conics, Medial Curves)
(cid:15) Geometric Functions (Transformations, Arc Length, Area)
(cid:15) Tangent Curves (Lines, Circles, Conics, Biarcs)
(cid:15) Descarta2D Reference (philosophy and command descriptions)
(cid:15) Descarta2D Packages (complete source code)
viii Preface
(cid:15) Explorations (solution notebooks)
(cid:15) Epilogue (Installation Instructions, Bibliography and a detailed index).
About the Author
Donald L. Vossler is a mechanical engineer and computer software designer with more than
20 years experience in computer aided design and geometric modeling. He has been involved
in solidmodeling since its inception inthe early 1980’sandhas contributedto the theoretical
foundationofthesubjectthroughseveralpublishedpapers. Hehasmanagedthedevelopment
of a number of commercial computer aided design systems and holds a US Patent involving
the underlying data representations of geometric models.
Contents
I Introduction 1
1 Getting Started 3
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 What’s on the CD-ROM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Mathematica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 Starting Descarta2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6 Outline of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Descarta2D Tour 9
2.1 Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Line Segments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5 Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.6 Arcs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.7 Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.8 Parabolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.9 Ellipses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.10 Hyperbolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.11 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.12 Area and Arc Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.13 Tangent Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.14 Symbolic Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.15 Next Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
II Elementary Geometry 25
3 Coordinates and Points 27
3.1 Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Rectangular Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
ix
x Contents
3.3 Line Segments and Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4 Midpoint between Two Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.5 Point of Division of Two Points . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.6 Collinear Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.7 Explorations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4 Equations and Graphs 39
4.1 Variables and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.3 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.4 Solving Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.5 Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.6 Parametric Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.7 Explorations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5 Lines and Line Segments 51
5.1 General Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.2 Parallel and Perpendicular Lines . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.3 Angle between Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.4 Two{Point Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.5 Point{Slope Form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.6 Slope{Intercept Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.7 Intercept Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.8 Normal Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.9 Intersection Point of Two Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.10 Point Projected Onto a Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.11 Line Perpendicular to Line Segment . . . . . . . . . . . . . . . . . . . . . . . . 72
5.12 Angle Bisector Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.13 Concurrent Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.14 Pencils of Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.15 Parametric Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.16 Explorations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6 Circles 85
6.1 De(cid:12)nitions and Standard Equation . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.2 General Equation of a Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.3 Circle from Diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.4 Circle Through Three Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.5 Intersection of a Line and a Circle . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.6 Intersection of Two Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.7 Distance from a Point to a Circle . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.8 Coaxial Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.9 Radical Axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.10 Parametric Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99