Exploring Analytic Geometry with Mathematica
Donald L. VosslerBME, Kettering University, 1978 MM, Aquinas
College, 1981
Anaheim, California USA, 1999
Copyright 1999-2007 Donald L. Vossler
PrefaceThe study of two-dimensional analytic geometry has gone
in and out of fashion several times over the past century, however
this classic eld 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 eld. By
combining the power of Mathematica with an analytic geometry
software system called Descarta2D, the author has succeeded in
meshing an ancient eld 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.
FeaturesExploring 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 readers 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.
v
vi
Preface 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 CD-ROM is included for convenient, permanent storage of
the Descarta2D software. A complete software system and
mathematical reference is packaged as an aordable book.
Classical Analytic GeometryExploring 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
denitions of underlying mathematical terminology and develops the
topic with more detailed derivations and proofs of important
concepts.
ExplorationsEach 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 on the CD-ROM 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: Carlyles Circle Castillons Problem Eulers
Triangle Formula Eyeball Theorem Gergonnes Point Herons Formula
Inversion Monges Theorem Reciprocal Polars Reection in a Point
Stewarts Theorem plus many more.
Preface
vii
Descarta2DDescarta2D 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 les on the CD-ROM.
CD-ROMThe CD-ROM provides the complete text of the book in Abode
Portable Document Format (PDF) for interactive reading. In
addition, the CD-ROM provides the following Mathematica notebooks:
Chapters with Mathematica dialogs, 24 interactive notebooks
Reference material for Descarta2D, three notebooks Complete
Descarta2D source code, 30 notebooks Descarta2D packages, 30
loadable les Exploration solutions, 125 notebooks. These notebooks
have been thoroughly tested and are compatible with Mathematica
Version 3.0.1 and Version 4.0. Maximum benet 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 les can be
accomplished without using Mathematica itself.
Organization of the BookExploring 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)
viii Explorations (solution notebooks) Epilogue (Installation
Instructions, Bibliography and a detailed index).
Preface
About the AuthorDonald 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 solid modeling since its inception in the early 1980s 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.
ContentsI Introduction. . . . . . . . . . . . . . . . . . . . .
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13 3 3 4 5 6 7 9 9 10 12 13 14 15 16 17 18 19 20 20 21 22 23
1 Getting Started 1.1 Introduction . . . . . . . 1.2 Historical
Background . 1.3 Whats on the CD-ROM 1.4 Mathematica . . . . . .
1.5 Starting Descarta2D . . 1.6 Outline of the Book . . 2
Descarta2D Tour 2.1 Points . . . . . . . . 2.2 Equations . . . . .
. 2.3 Lines . . . . . . . . . 2.4 Line Segments . . . 2.5 Circles .
. . . . . . . 2.6 Arcs . . . . . . . . . 2.7 Triangles . . . . . .
2.8 Parabolas . . . . . . 2.9 Ellipses . . . . . . . 2.10
Hyperbolas . . . . . 2.11 Transformations . . 2.12 Area and Arc
Length 2.13 Tangent Curves . . . 2.14 Symbolic Proofs . . 2.15 Next
Steps . . . . .
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II
Elementary Geometry
25
3 Coordinates and Points 27 3.1 Numbers . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2
Rectangular Coordinates . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 28
ix
x 3.3 3.4 3.5 3.6 3.7 Line Segments and Distance . . Midpoint
between Two Points . Point of Division of Two Points Collinear
Points . . . . . . . . . Explorations . . . . . . . . . . . . . . .
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Contents . . . . . . . . . . . . . . . 30 33 33 36 37
4 Equations and Graphs 4.1 Variables and Functions 4.2
Polynomials . . . . . . . 4.3 Equations . . . . . . . . 4.4 Solving
Equations . . . 4.5 Graphs . . . . . . . . . . 4.6 Parametric
Equations . 4.7 Explorations . . . . . .
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39 . 39 . 39 . 41 . 42 . 46 . 47 . 48 51 . 51 . 54 . 55 . 56 .
58 . 62 . 64 . 65 . 69 . 70 . 72 . 73 . 74 . 75 . 78 . 81 85 85 88
89 90 91 92 95 96 97 99
5 Lines and Line Segments 5.1 General Equation . . . . . . . . .
. . 5.2 Parallel and Perpendicular Lines . . 5.3 Angle between
Lines . . . . . . . . . 5.4 TwoPoint Form . . . . . . . . . . . 5.5
PointSlope Form . . . . . . . . . . . 5.6 SlopeIntercept Form . . .
. . . . . 5.7 Intercept Form . . . . . . . . . . . . 5.8 Normal
Form . . . . . . . . . . . . . 5.9 Intersection Point of Two Lines
. . . 5.10 Point Projected Onto a Line . . . . . 5.11 Line
Perpendicular to Line Segment 5.12 Angle Bisector Lines . . . . . .
. . . 5.13 Concurrent Lines . . . . . . . . . . . 5.14 Pencils of
Lines . . . . . . . . . . . . 5.15 Parametric Equations . . . . . .
. . 5.16 Explorations . . . . . . . . . . . . . 6 Circles 6.1
Denitions and Standard Equation 6.2 General Equation of a Circle .
. . 6.3 Circle from Diameter . . . . . . . . 6.4 Circle Through
Three Points . . . 6.5 Intersection of a Line and a Circle 6.6
Intersection of Two Circles . . . . 6.7 Distance from a Point to a
Circle . 6.8 Coaxial Circles . . . . . . . . . . . 6.9 Radical Axis
. . . . . . . . . . . . 6.10 Parametric Equations . . . . . . .
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Contents
xi
6.11 Explorations . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 101 7 Arcs 7.1 Denitions . . . . . . .
. . . . . . 7.2 Bulge Factor Arc . . . . . . . . . 7.3 ThreePoint
Arc . . . . . . . . . 7.4 Parametric Equations . . . . . . 7.5
Points and Angles at Parameters 7.6 Arcs from Ray Points . . . . .
. 7.7 Explorations . . . . . . . . . . . 8 Triangles 8.1 Denitions
. . . . . . . 8.2 Centroid of a Triangle 8.3 Circumscribed Circle .
8.4 Inscribed Circle . . . . 8.5 Solving Triangles . . . 8.6 Cevian
Lengths . . . . 8.7 Explorations . . . . . 105 . 105 . 107 . 110 .
111 . 112 . 113 . 114 117 117 120 122 123 124 128 128
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III
Conics. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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133. . . . . . . 135 135 135 136 139 140 141 142 145 145 147 147
150 151 153 153 155 156
9 Parabolas 9.1 Denitions . . . . . . . . . . . . . . 9.2
General Equation of a Parabola . . 9.3 Standard Forms of a Parabola
. . . 9.4 Reduction to Standard Form . . . 9.5 Parabola from Focus
and Directrix 9.6 Parametric Equations . . . . . . . 9.7
Explorations . . . . . . . . . . . .
10 Ellipses 10.1 Denitions . . . . . . . . . . . . . . . . 10.2
General Equation of an Ellipse . . . . 10.3 Standard Forms of an
Ellipse . . . . . 10.4 Reduction to Standard Form . . . . . 10.5
Ellipse from Vertices and Eccentricity 10.6 Ellipse from Foci and
Eccentricity . . 10.7 Ellipse from Focus and Directrix . . . 10.8
Parametric Equations . . . . . . . . . 10.9 Explorations . . . . .
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xii 11 Hyperbolas 11.1 Denitions . . . . . . . . . . . . . . . .
. . 11.2 General Equation of a Hyperbola . . . . . 11.3 Standard
Forms of a Hyperbola . . . . . . 11.4 Reduction to Standard Form .
. . . . . . 11.5 Hyperbola from Vertices and Eccentricity 11.6
Hyperbola from Foci and Eccentricity . . 11.7 Hyperbola from Focus
and Directrix . . . 11.8 Parametric Equations . . . . . . . . . . .
11.9 Explorations . . . . . . . . . . . . . . . . 12 General Conics
12.1 Conic from Quadratic Equation . . . . . 12.2 Classication of
Conics . . . . . . . . . . 12.3 Center Point of a Conic . . . . . .
. . . 12.4 Conic from Point, Line and Eccentricity 12.5 Common
Vertex Equation . . . . . . . . 12.6 Conic Intersections . . . . .
. . . . . . . 12.7 Explorations . . . . . . . . . . . . . . . 13
Conic Arcs 13.1 Denition of a Conic Arc 13.2 Equation of a Conic
Arc . 13.3 Projective Discriminant . 13.4 Conic Characteristics . .
. 13.5 Parametric Equations . . 13.6 Explorations . . . . . . . 14
Medial Curves 14.1 PointPoint . 14.2 PointLine . . 14.3 PointCircle
. 14.4 LineLine . . 14.5 LineCircle . 14.6 CircleCircle 14.7
Explorations
Contents 159 159 161 161 166 167 168 169 170 173 175 175 184 184
185 186 189 190 193 193 194 196 196 198 199 201 201 202 204 206 207
210 212
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IV
Geometric Functions
215
15 Transformations 217 15.1 Translations . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 217 15.2
Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 219 15.3 Scaling . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 222
Contents
xiii
15.4 Reections . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 224 15.5 Explorations . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 16 Arc
16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 16.9 Length Lines and Line
Segments . . . . . . Perimeter of a Triangle . . . . . . . Polygons
Approximating Curves . Circles and Arcs . . . . . . . . . .
Ellipses and Hyperbolas . . . . . . Parabolas . . . . . . . . . . .
. . . Chord Parameters . . . . . . . . . Summary of Arc Length
Functions Explorations . . . . . . . . . . . . 229 229 230 231 231
233 234 235 236 236
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17 Area 17.1 Areas of Geometric Figures 17.2 Curved Areas . . .
. . . . . 17.3 Circular Areas . . . . . . . 17.4 Elliptic Areas . .
. . . . . . 17.5 Hyperbolic Areas . . . . . . 17.6 Parabolic Areas
. . . . . . . 17.7 Conic Arc Area . . . . . . . 17.8 Summary of
Area Functions 17.9 Explorations . . . . . . . .
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237 . 237 . 240 . 240 . 242 . 245 . 246 . 248 . 249 . 249
V
Tangent Curvesto a Circle . . . . . to Conics . . . . . . to
Standard Conics . . . . . . . . . . . . . . . . . . . . . . . . . .
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253. . . . 255 255 266 273 280 283 283 285 286 287 288 289
18 Tangent Lines 18.1 Lines Tangent 18.2 Lines Tangent 18.3
Lines Tangent 18.4 Explorations
19 Tangent Circles 19.1 Tangent Object, Center Point . . . . . .
. . 19.2 Tangent Object, Center on Object, Radius . 19.3 Two
Tangent Objects, Center on Object . . 19.4 Two Tangent Objects,
Radius . . . . . . . . 19.5 Three Tangent Objects . . . . . . . . .
. . . 19.6 Explorations . . . . . . . . . . . . . . . . .
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xiv 20 Tangent Conics 20.1 Constraint Equations . . . . . . .
20.2 Systems of Quadratics . . . . . . 20.3 Validity Conditions . .
. . . . . . 20.4 Five Points . . . . . . . . . . . . 20.5 Four
Points, One Tangent Line . 20.6 Three Points, Two Tangent Lines
20.7 Conics by Reciprocal Polars . . . 20.8 Explorations . . . . .
. . . . . . 21 Biarcs 21.1 Biarc Carrier Circles . . . . . 21.2
Knot Point . . . . . . . . . . 21.3 Knot Circles . . . . . . . . .
. 21.4 Biarc Programming Examples 21.5 Explorations . . . . . . . .
.
Contents 293 293 294 296 296 298 301 306 310
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311 . 311 . 314 . 316 . 317 . 322
VI
Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . .
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323. . . . . . 325 325 326 326 337 338 339 341 367
22 Technical Notes 22.1 Computation Levels . . . . 22.2 Names .
. . . . . . . . . . . 22.3 Descarta2D Objects . . . . 22.4
Descarta2D Packages . . . . 22.5 Descarta2D Functions . . . 22.6
Descarta2D Documentation 23 Command Browser 24 Error Messages
VII
Packages. . . . . . . . . . . . . . . . . . . . . . . . . . . .
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385. . . . . . . . . . 387 395 399 405 411 415 421 427 429
437
D2DArc2D . . . . . D2DArcLength2D . D2DArea2D . . . .
D2DCircle2D . . . D2DConic2D . . . . D2DConicArc2D . . D2DEllipse2D
. . D2DEquations2D . D2DExpressions2D D2DGeometry2D . .
Contents D2DHyperbola2D . . . D2DIntersect2D . . . D2DLine2D . .
. . . . D2DLoci2D . . . . . . D2DMaster2D . . . . . D2DMedial2D . .
. . . D2DNumbers2D . . . . D2DParabola2D . . . . D2DPencil2D . . .
. . D2DPoint2D . . . . . . D2DQuadratic2D . . . D2DSegment2D . . .
. D2DSketch2D . . . . . D2DSolve2D . . . . . . D2DTangentCircles2D
D2DTangentConics2D D2DTangentLines2D . D2DTangentPoints2D
D2DTransform2D . . . D2DTriangle2D . . . . . . . . . . . . . . . .
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xv 445 453 457 465 469 473 477 479 485 489 497 505 511 515 519
523 531 537 539 545
VIII
Explorations. . . . . . . . . . . . . . . . . . . . . . . . . .
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555. . . . . . . . . . . . . . . . . . . . 557 559 561 563 565
567 569 571 573 575 577 579 581 583 585 589 591 593 595 597
apollon.nb, Circle of Apollonius . . . . . . . . . . . . . . .
arccent.nb, Centroid of Semicircular Arc . . . . . . . . . .
arcentry.nb, Arc from Bounding Points and Entry Direction
arcexit.nb, Arc from Bounding Points and Exit Direction .
archimed.nb, Archimedes Circles . . . . . . . . . . . . . . .
arcmidpt.nb, Midpoint of an Arc . . . . . . . . . . . . . . .
caarclen.nb, Arc Length of a Parabolic Conic Arc . . . . . .
caarea1.nb, Area of a Conic Arc (General) . . . . . . . . .
caarea2.nb, Area of a Conic Arc (Parabola) . . . . . . . . .
cacenter.nb, Center of a Conic Arc . . . . . . . . . . . . . .
cacircle.nb, Circular Conic Arc . . . . . . . . . . . . . . . .
camedian.nb, Shoulder Point on Median . . . . . . . . . . . .
caparam.nb, Parametric Equations of a Conic Arc . . . . .
carlyle.nb, Carlyle Circle . . . . . . . . . . . . . . . . . . .
castill.nb, Castillons Problem . . . . . . . . . . . . . . .
catnln.nb, Tangent Line at Shoulder Point . . . . . . . . .
center.nb, Center of a Quadratic . . . . . . . . . . . . . .
chdlen.nb, Chord Length of Intersecting Circles . . . . . .
cir3pts.nb, Circle Through Three Points . . . . . . . . . .
circarea.nb, One-Third of a Circles Area . . . . . . . . . .
xvi cirptmid.nb, CirclePoint Midpoint Theorem . . . . . . . .
cramer2.nb, Cramers Rule (Two Equations) . . . . . . . .
cramer3.nb, Cramers Rule (Three Equations) . . . . . . . deter.nb,
Determinants . . . . . . . . . . . . . . . . . . . elfocdir.nb,
Focus of Ellipse is Pole of Directrix . . . . . . elimlin.nb,
Eliminate Linear Terms . . . . . . . . . . . . . elimxy1.nb,
Eliminate Cross-Term by Rotation . . . . . . . elimxy2.nb,
Eliminate Cross-Term by Change in Variables elimxy3.nb, Eliminate
Cross-Term by Change in Variables elldist.nb, Ellipse Locus,
Distance from Two Lines . . . . ellfd.nb, Ellipse from Focus and
Directrix . . . . . . . . ellips2a.nb, Sum of Focal Distances of an
Ellipse . . . . . . elllen.nb, Length of Ellipse Focal Chord . . .
. . . . . . ellrad.nb, Apoapsis and Periapsis of an Ellipse . . . .
. . ellsim.nb, Similar Ellipses . . . . . . . . . . . . . . . . . .
ellslp.nb, Tangent to an Ellipse with Slope . . . . . . . .
eqarea.nb, Equal Areas Point . . . . . . . . . . . . . . . .
eyeball.nb, Eyeball Theorem . . . . . . . . . . . . . . . . .
gergonne.nb, Gergonne Point of a Triangle . . . . . . . . . .
heron.nb, Herons Formula . . . . . . . . . . . . . . . . .
hyp2a.nb, Focal Distances of a Hyperbola . . . . . . . . .
hyp4pts.nb, Equilateral Hyperbolas . . . . . . . . . . . . .
hyparea.nb, Areas Related to Hyperbolas . . . . . . . . . .
hypeccen.nb, Eccentricities of Conjugate Hyperbolas . . . .
hypfd.nb, Hyperbola from Focus and Directrix . . . . . . hypinv.nb,
Rectangular Hyperbola Distances . . . . . . . hyplen.nb, Length of
Hyperbola Focal Chord . . . . . . . hypslp.nb, Tangent to a
Hyperbola with Given Slope . . . hyptrig.nb, Trigonometric
Parametric Equations . . . . . . intrsct.nb, Intersection of Lines
in Intercept Form . . . . . inverse.nb, Inversion . . . . . . . . .
. . . . . . . . . . . . johnson.nb, Johnsons Congruent Circle
Theorem . . . . . knotin.nb, Incenter on Knot Circle . . . . . . .
. . . . . . lndet.nb, Line General Equation Determinant . . . . . .
lndist.nb, Vertical/Horizontal Distance to a Line . . . . .
lnlndist.nb, Line Segment Cut by Two Lines . . . . . . . .
lnquad.nb, Line Normal to a Quadratic . . . . . . . . . . .
lnsdst.nb, Distance Between Parallel Lines . . . . . . . .
lnsegint.nb, Intersection Parameters of Two Line Segments
lnsegpt.nb, Intersection Point of Two Line Segments . . .
lnsperp.nb, Equations of Perpendicular Lines . . . . . . . .
lntancir.nb, Line Tangent to a Circle . . . . . . . . . . . . .
lntancon.nb, Line Tangent to a Conic . . . . . . . . . . . . . . .
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Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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599 601 603 605 607 609 611 613 615 617 619 623 625 627 629 631 633
637 639 641 643 645 647 651 653 657 659 661 663 665 667 671 675 677
679 681 685 687 689 691 693 695 697
Contents mdcircir.nb, Medial Curve, CircleCircle . . . . . . . .
. mdlncir.nb, Medial Curve, LineCircle . . . . . . . . . .
mdlnln.nb, Medial Curve, LineLine . . . . . . . . . . mdptcir.nb,
Medial Curve, PointCircle . . . . . . . . . mdptln.nb, Medial
Curve, PointLine . . . . . . . . . . mdptpt.nb, Medial Curve,
PointPoint . . . . . . . . . mdtype.nb, Medial Curve Type . . . . .
. . . . . . . . monge.nb, Monges Theorem . . . . . . . . . . . . .
. narclen.nb, Approximate Arc Length of a Curve . . . . normal.nb,
Normals and Minimum Distance . . . . . . pb3pts.nb, Parabola
Through Three Points . . . . . . pb4pts.nb, Parabola Through Four
Points . . . . . . . pbang.nb, Parabola Intersection Angle . . . .
. . . . . pbarch.nb, Parabolic Arch . . . . . . . . . . . . . . . .
pbarclen.nb, Arc Length of a Parabola . . . . . . . . . . pbdet.nb,
Parabola Determinant . . . . . . . . . . . . pbfocchd.nb, Length of
Parabola Focal Chord . . . . . . pbslp.nb, Tangent to a Parabola
with a Given Slope . pbtancir.nb, Circle Tangent to a Parabola . .
. . . . . . pbtnlns.nb, Perpendicular Tangents to a Parabola . . .
polarcir.nb, Polar Equation of a Circle . . . . . . . . . .
polarcol.nb, Collinear Polar Coordinates . . . . . . . . .
polarcon.nb, Polar Equation of a Conic . . . . . . . . . .
polardis.nb, Distance Using Polar Coordinates . . . . .
polarell.nb, Polar Equation of an Ellipse . . . . . . . .
polareqn.nb, Polar Equations . . . . . . . . . . . . . . .
polarhyp.nb, Polar Equation of a Hyperbola . . . . . . .
polarpb.nb, Polar Equation of a Parabola . . . . . . . .
polarunq.nb, Non-uniqueness of Polar Coordinates . . . pquad.nb,
Parameterization of a Quadratic . . . . . . ptscol.nb, Collinear
Points . . . . . . . . . . . . . . . radaxis.nb, Radical Axis of
Two Circles . . . . . . . . . radcntr.nb, Radical Center . . . . .
. . . . . . . . . . . raratio.nb, Radical Axis Ratio . . . . . . .
. . . . . . . reccir.nb, Reciprocal of a Circle . . . . . . . . . .
. . recptln.nb, Reciprocals of Points and Lines . . . . . . .
recquad.nb, Reciprocal of a Quadratic . . . . . . . . . .
reflctpt.nb, Reection in a Point . . . . . . . . . . . . .
rtangcir.nb, Angle Inscribed in a Semicircle . . . . . . .
rttricir.nb, Circle Inscribed in a Right Triangle . . . .
shoulder.nb, Coordinates of Shoulder Point . . . . . . .
stewart.nb, Stewarts Theorem . . . . . . . . . . . . . .
tancir1.nb, Circle Tangent to Circle, Given Center . . . . . . . .
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xvii 699 703 705 707 711 713 715 717 719 721 723 725 727 729 731
733 735 737 739 743 745 747 749 751 753 755 757 759 761 763 765 767
769 771 773 775 777 779 781 783 785 787 789
xviii tancir2.nb, Circle Tangent to Circle, Center on Circle,
Radius tancir3.nb, Circle Tangent to Two Lines, Radius . . . . . .
. . tancir4.nb, Circle Through Two Points, Center on Circle . . .
tancir5.nb, Circle Tangent to Three Lines . . . . . . . . . . .
tancirpt.nb, Tangency Point on a Circle . . . . . . . . . . . . .
tetra.nb, Area of a Tetrahedrons Base . . . . . . . . . . . .
tncirtri.nb, Circles Tangent to an Isosceles Triangle . . . . . .
tnlncir.nb, Construction of Two Related Circles . . . . . . . .
triallen.nb, Triangle Altitude Length . . . . . . . . . . . . . .
trialt.nb, Altitude of a Triangle . . . . . . . . . . . . . . . .
triarea.nb, Area of Triangle Congurations . . . . . . . . . . .
triarlns.nb, Area of Triangle Bounded by Lines . . . . . . . . .
tricent.nb, Centroid of a Triangle . . . . . . . . . . . . . . . .
tricev.nb, Triangle Cevian Lengths . . . . . . . . . . . . . . .
triconn.nb, Concurrent Triangle Altitudes . . . . . . . . . . .
tridist.nb, Hypotenuse Midpoint Distance . . . . . . . . . . .
trieuler.nb, Eulers Triangle Formula . . . . . . . . . . . . . .
trirad.nb, Triangle Radii . . . . . . . . . . . . . . . . . . . .
trisides.nb, Triangle Side Lengths from Altitudes . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 791 793
795 797 799 801 803 807 809 811 813 815 817 819 823 827 829 833
835
IX
Epilogue
837839 843 845
Installation Instructions Bibliography Index
Exploring Analytic Geometry with Mathematicaby Donald L. Vossler
The study of two-dimensional analytic geometry has gone in and out
of fashion several times over the past century. This classic eld 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 eld. By combining the
power of Mathematica with an analytic geometry software system
called Descarta2D, the author has succeeded in meshing an ancient
eld 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. A classic
study in analytic geometry, complete with in-line Mathematica
dialogs illustrating every concept as it is introduced. Excellent
theoretical presentation Fully explained examples of all key
concepts Interactive Mathematica notebooks for the entire book.
provides a complete computer-based environment for study of
analytic geometry all chapters and reference material are provided
on the CD in addition to being printed in the book. Complete
software system: Descarta2D a software system, including source
code, for the underlying computer implementation, called Descarta2D
is provided Part VII of the book is a listing of the (30)
Mathematica les notebooks supporting Descarta2D; the source code is
also in on the CD Explorations More than 120 challenging problems
in analytic geometry are posed. Complete solutions are provided
both as interactive Mathematica notebooks on the CD and as printed
material in the book. Mathematica and Descarta2D Hints are provided
to expand the readers knowledge and understanding of Descarta2D and
Mathematica . Detailed reference manual Complete documentation for
Descarta2D Fully integrated into the Mathematica Help Browser
About the authorDonald 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 solid modeling since its inception in the early 1980s 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.
CD-ROM includedFull contents of book included on CD-ROM, which
will operate on Macintosh, Windows and UNIX machines with
Mathematica 3.0.1 or 4.0 installed.
