Table Of ContentCOORDINATE
GEOMETRY
BY
LUTHER PFAHLER EISENHART
PROFESSOR OF MATHEMATICS
PRINCETON UNIVERSITY
DOVER
PUBLICATIONS, INC.
NEW YORK NEW YORK
Copyright, 1939, by Ginn and Company
All Rights Reserved
This new Dover edition first published in 1960
is an unabridged and unaltered republication
of the First Edition.
This book is republished by permission of
Ginn and Company, the original publisher of
this text.
Manufactured inthe United States ofAmerica
Dover Publications, Inc.
180 Varick Street
New York New York
14,
Preface
The purposes and general plan of this book are set forth
in the Introduction. Its practicabilityas a text has been tested
and proved through use during two years in freshman courses
in Princeton University. Each year it has been revised as a
result of suggestions not only by the members of the staff but
also by the students, who have shown keen interest and help-
fulness in the development of the project.
An unusual feature of the book is the presentation of co-
ordinategeometryintheplaneinsuchmannerastoleadreadily
tothestudyoflinesandplanesinspaceasageneralizationofthe
geometry of the plane this is done in Chapter 2. It has been
;
our experience that the students have little, if any, difficulty in
handling the geometry of space thus early in the course, in the
way in which it is developed in this chapter.
We have also found that students who have studied deter-
minants in an advanced course in algebra find for the first time
in the definition and frequent use of determinants in this book
an appreciation of the value and significance of this subject.
In the preparation and revisions of the text the author has
received valuable assistance from the members of the staff at
Princeton in particular, Professors Knebelman and Tucker
and Messrs. Tompkins, Daly, Fox, Titt, Traber, Battin, and
Johnson the last two have been notably helpful in the prepa-
;
ration of the final form of the text and in the reading of the
proof. The Appendix to Chapter 1, which presents the relation
between the algebraic foundations of coordinate geometry and
axioms of Hilbert for Euclidean plane geometry, is due in the
main to Professors Bochner and Church and Dr. Tompkins,
particularly to the latter, at whose suggestion it was prepared
and incorporated. The figuresweredrawnbyMr. J. H. Lewis.
It remains for me to express my appreciation ofthe courtesy
and cooperation of Ginn and Company in the publication of
this book.
LUTHER PFAHLER EISENHART
iii
Contents
PAGE
INTRODUCTION
ix
CHAPTER
1
Points and Lines in tKe Plane
SECTION
1. The Equation of the First Degree in x andy 3
2. Cartesian Coordinates in the Plane 8
3. Distance between Two Points Direction Numbers and Di-
rection Cosines Angle between Directed Line Segments 11
4. Internal and External Division ofa Line Segment 17
5. An Equation ofa Line Parametric Equations ofa Line 20
6. Direction Numbers and Direction Cosines of a Line Angle
between Two Lines 27
7. The Slope ofa Line 33
8. Directed Distance from a Line to a Point 35
9. Two Equations ofthe First Degree in xandy - Determinants
of the Second Order 39
10. The Set of Lines through a Point 47
11. Oblique Axes 53
12. The Circle 54
13. Resume Line Coordinates 62
CHAPTER
2
Lines and Planes in Space Determinants
14. Rectangular Coordinates in Space 71
15. Distance between Two Points Direction Numbers and Di-
rection Cosinesofa Line Segment Angle between Two Line
Segments 76
Contents
SECTION PAGE
16. Equations of a Line Direction Numbers and Direction Co-
sines ofa Line Angle ofTwo Lines 82
17. An Equation ofa Plane 88
18. The Directed Distance from a Plane to a Point The Dis-
tance from a Line to a Point 95
19. Two Equations of the First Degree in Three Unknowns A
Line as the Intersection ofTwo Planes 98
20. Two Homogeneous Equations of the First Degree in Three
Unknowns 104
21. Determinants of the Third Order Three Equations of the
First Degree in Three Unknowns 106
22. Three Homogeneous Equations ofthe First Degree in Three
Unknowns 114
23. EquationsofPlanes Determined byCertainGeometricCon-
ditions Shortest Distance between Two Lines 119
24. The Configurations ofThree Planes 123
25. Miscellaneous Exercises The Sphere 127
26. Determinants ofAny Order 130
27. Solution of Equations of the First Degree in Any Number
of Unknowns Space of Four Dimensions 137
CHAPTER
5
Transformations of Coordinates
28. Transformations of Rectangular Coordinates 149
29. Polar Coordinates in the Plane 154
30. Transformations of Rectangular Coordinates in Space 160
31. Spherical and Cylindrical Coordinates 166
vi
Contents
CHAPTER
4
The Conies Locus Problems
SECTION PAGE
32. A Geometric DefinitionoftheConies 171
33. The Parabola 174
34. Tangents and Polars 177
35. Ellipses and Hyperbolas 182
36. Conjugate Diameters and Tangents ofCentral Conies 191
37. Similar Central Conies The Asymptotes of a Hyperbola
Conjugate Hyperbolas 196
38. The Conies as Plane Sections ofa Right Circular Cone 201
39. Equations of Conies Whose Axes Are ^Parallel to the Co-
ordinate Axes 203
40. The General Equation of the Second Degree Invariants 208
41. The Determination ofa Conic from Its Equation in General
Form 215
42. Center, Principal Axes, and Tangents of a Conic Defined
by a General Equation 221
43. Locus Problems 229
ti
CHAPTER
5
The Quadric Surfaces
44. SurfacesofRevolution TheQuadricSurfacesofRevolution 239
45. Canonical Equations ofthe Quadric Surfaces 243
46. The Ruled Quadrics 248
47. Quadrics Whose Principal Planes Are Parallel to the Co-
ordinate Planes 252
48. The General Equation of the Second Degree in x, y, and z
The Characteristic Equation Tangent Planes to a Quadric 256
vii
Contents
SECTION PAGE
49. Centers Vertices Points of Symmetry 264
50. The Invariants 7, /, D, and A 267
51. Classification of the Quadrics 271
APPENDIX TO CHAPTER
1 279
INDEX
293
Vlll
Introduction
Coordinate geometry is so called because it uses in the treat-
ment of geometric problems a system of coordinates, which
associates with each point of a geometric figure a set of num-
bers coordinates so that the conditions which each point
must satisfy are expressible by means of equations or inequali-
ties ordinarily involving algebraic quantities and at times
trigonometric functions. By this means a geometric problem is
reduced to an algebraic problem, which most people can handle
with greater ease and confidence. After the algebraic solution
has been obtained, however, there,remains its geometric inter-
pretation to be determined; for^ttigdPfpblem is geometric, and
algebra is a means to its solution, not the end.4' This method
was introduced by Rene Descartes in La Geometric, published
in 1636 accordingly coordinate geometry is sometimes called
;
Cartesian geometry. Before the time of Descartes geometric
reasoningonly wasused in the studyofgeometry. The advance
inthedevelopmentofgeometricideassincethetimeofDescartes
is largely due to the introduction of his method.
Geometry deals with spatial concepts. The problems of
physics, astronomy, engineering, etc. involvenotonly spacebut
usually time also. The method of attack upon these problems
is similar to that used in coordinate geometry. However,
geometricproblems, because ofthe absenceofthe timeelement,
are ordinarily simpler, and consequently it is advisable that the
student first become familiar with the methods of coordinate
geometry.
The aim of this book is to encourage the reader to think
mathematically. The subject matter is presented as a unified
whole, not as a composite of units which seem to have no
relation to one another. Each situation is completely analyzed,
and the various possibilities are all carried to their conclusion,
for frequently the exceptional case (which often is not pre-
sented to a student) is the one that clarifies the general case
the idea epitomized in the old adage about an exception and a
ix
