A Treatise on the Differential Geometry of Curves and Surfaces (Dover Phoenix Editions)

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I xSm*^i?c?^n

A TREATISE ON THE DIFFERENTIAL GEOMETRY OF PURVES AND SURFACES

BY

LUTHER PFAHLER EISENHART PROFESSOR OF MATHEMATICS IN PRINCETON UNIVERSITY

GINN AND COMPANY BOSTON

NEW YORK CHICAGO LONDON

CvA

COPYRIGHT,

1909,

BY

LUTHER PFAHLER EJSENHART ALL RIGHTS RESERVED 89-8

gftc

SUftengum

GINN AND COMPANY PRILTORS BOSTON

PKOU.S.A.

6 A-.ATH..

STAT.

LIBRARY

PEEFACE This book

is

a development from courses which I have given in

number of years. During this time I have come to more would be accomplished by my students if they had an otherwise adapted to the introductory treatise written in English and Princeton for a feel that

use of

men

beginning their graduate work.

the method Chapter I is devoted to the theory of twisted curves, in general being that which is usually followed in discussions of this I have introduced the idea of moving axes, subject. But in addition and have derived the formulas pertaining thereto from the previously

In this way the student is made familiar with a method which is similar to that used by Darboux in the first volume of his Lemons, and to that of Cesaro in his Geometria obtained Frenet-Serret formulas.

not only of great advantage in the treat ment of certain topics and in the solution of problems, but it is valu

This method

Intrinseca.

is

able in developing geometrical thinking.

book may be divided into three parts. The deals with the geometry of a sur first, consisting of Chapters II-YI, face in the neighborhood of a point and the developments therefrom,

The remainder

of the

such as curves and systems

of curves defined

by

differential

equa

which the

by large extent the method from the discussion of two quad properties of a surface are derived ratic differential forms. However, little or no space is given to the tions.

To a

is that of Gauss,

and their invariants. In algebraic treatment of differential forms in the first chapter, defined as addition, the method of moving axes, an to has been extended so as to be applicable investigation of the properties

of

surfaces

and groups

of

surfaces.

The extent

of the

no attempt has theory concerning ordinary points a discussion For the consider been made to exceptional problems. of such questions as the existence of integrals of differential equa tions and boundary conditions the reader must consult the treatises which deal particularly with these subjects. In Chapters VII and VIII the theory previously developed is as the quadrics, ruled applied to several groups of surfaces, such total curvature, and constant of surfaces minimal surfaces, surfaces, is

surfaces with plane

and spherical

so great that

lines of curvature. iii

PEEFACE

iv

The idea

of applicability of surfaces is introduced in Chapter III as a particular case of conformal representation, and throughout the

book attention

called to examples of applicable surfaces. However, the general problems concerned with the applicability of surfaces are discussed in Chapters IX and X, the latter of which deals entirely is

with the recent method

of

Weingarten and

its

developments.

The

remaining four chapters are devoted to a discussion of infinitesimal deformation of surfaces, congruences of straight lines and of

circles,

and

triply orthogonal systems of surfaces. It will be noticed that the book contains

student will find that applications of

the

w hereas r

certain of

others

many them

constitute

examples, and the are merely direct

extensions of the

formulas, theory which might properly be included as portions of a more ex tensive treatise. At first I felt constrained to give such references as

would enable the reader to consult the journals and treatises from which some of these problems were taken, but finally it seemed best to furnish no such key, only to remark that the Encyklopadie der mathematisclicn Wissenschaften may be of assistance. And the same

may

be said about references to the sources of the subject-matter of Many important citations have been made, but there has

the book.

not been an attempt to give every reference. However, I desire to acknowledge my indebtedness to the treatises of Darboux, Bianchi, and Scheffers. But the difficulty is that for many years I have con sulted these authors so freely that now it is impossible for except in certain cases, what specific debts I owe to each.

me

to say,

In its present form, the material of the first eight chapters has been given to beginning classes in each of the last two years; and the remainder of the book, with certain enlargements, has constituted an advanced course which has been followed several times. It is im suitable credit for the suggestions made and for me to

give possible the assistance rendered by my students during these years, but I am conscious of helpful suggestions made by my colleagues, Professors Veblen, Maclnnes, and Swift, and by my former colleague, Professor Bliss of Chicago. I wish also to thank Mr. A. K. Krause for making the drawings for the figures. It remains for me to express my appreciation of the courtesy shown by Ginn and Company, and of the assistance given by them during the printing of this book.

LUTHER PFAHLER E1SENHART

CONTENTS CHAPTEE

I

CURVES IN SPACE PAGE

SECTION

12.

PARAMETRIC EQUATIONS OF A CURVE OTHER FORMS OF THE EQUATIONS OF A CURVE LINEAR ELEMENT TANGENT TO A CURVE ORDER OF CONTACT. NORMAL PLANK CURVATURE. RADIUS OF FIRST CURVATURE OSCULATING PLANE PRINCIPAL NORMAL AND BINORMAL OSCULATING CIRCLE. CENTER OF FIRST CURVATURE TORSION. FRENET-SERRET FORMULAS FORM OF CURVE IN THE NEIGHBORHOOD OF A POINT. THE SIGN OF TORSION CYLINDRICAL HELICES

13.

INTRINSIC EQUATIONS.

14.

RICCATI EQUATIONS

15.

THE DETERMINATION OF THE COORDINATES OF A CURVE DEFINED

16.

MOVING TRIHEDRAL ILLUSTRATIVE EXAMPLES

1.

2. 3. 4. 5. 6.

7. 8.

9.

10. 11.

....

BY 17.

ITS INTRINSIC

FUNDAMENTAL THEOREM

1

3

4 6

8 9

10 12 14

16 18

20 22

25

27

EQUATIONS

30

.33

18.

OSCULATING SPHERE

37

19.

BERTRAND CURVES TANGENT SURFACE OF A CURVE INVOLUTES AND EVOLUTES OF A CURVE MINIMAL CURVES

39

20. 21. 22.

CHAPTER

41

43 .

47

II

CURVILINEAR COORDINATES ON A SURFACE. ENVELOPES 23.

24. 25. 26.

PARAMETRIC EQUATIONS OF A SURFACE PARAMETRIC CURVES TANGENT PLANE ONE-PARAMETER FAMILIES OF SURFACES. ENVELOPES v

52 54 56

....

59

CONTENTS

vi SECTION 27. 28. 29.

PAGE

DEVELOPABLE SURFACES. RECTIFYING DEVELOPABLE APPLICATIONS OF THE MOVING TRIHEDRAL ENVELOPE OF SPHERES. CANAL SURFACES

CHAPTER

....

61 04

66

III

LINEAR ELEMENT OF A SURFACE. DIFFERENTIAL PARAME TERS. CONFORMAL REPRESENTATION 30.

LINEAR ELEMENT

70

31.

ISOTROPIC DEVELOPABLE

72

32.

TRANSFORMATION OF COORDINATES ANGLES BETWEEN CURVES. THE ELEMENT OF AREA FAMILIES OF CURVES MINIMAL CURVES ON A SURFACE VARIATION OF A FUNCTION DIFFERENTIAL PARAMETERS OF THE FIRST ORDER DIFFERENTIAL PARAMETERS OF THE SECOND ORDER SYMMETRIC COORDINATES ISOTHERMIC AND ISOMETRIC PARAMETERS ISOTHERMIC ORTHOGONAL SYSTEMS CONFORMAL REPRESENTATION ISOMETRIC REPRESENTATION. APPLICABLE SURFACES CONFORMAL REPRESENTATION OF A SURFACE UPON ITSELF CONFORMAL REPRESENTATION OF THE PLANE SURFACES OF REVOLUTION CONFORMAL REPRESENTATIONS OF THE SPHERE

33. 34. 35.

36. 37. 38. 39.

40. 41.

42. 43. 44. 45. 46. 47.

72

....

74

78 81 82

84

....

88 91

93

95 98

.... .

.

100 101

104 107 109

.

CHAPTER IV GEOMETRY OF A SURFACE IN THE NEIGHBORHOOD OF A POINT

....

50.

FUNDAMENTAL COEFFICIENTS OF THE SECOND ORDER RADIUS OF NORMAL CURVATURE PRINCIPAL RADII OF NORMAL CURVATURE

51.

LINES OF CURVATURE.

52.

TOTAL AND MEAN CURVATURE EQUATION OF EULER. DUPIN INDICATRIX CONJUGATE DIRECTIONS AT A POINT. CONJUGATE SYSTEMS

48. 49.

53. 54.

57. 58.

FUNDAMENTAL FORMULAS

56.

117

118

EQUATIONS OF RODRIGUES

ASYMPTOTIC LINES. CHARACTERISTIC LINES CORRESPONDING SYSTEMS ON Two SURFACES GEODESIC CURVATURE. GEODESICS

55.

.

114

121

123

124 .

.......

126 128

130

...

131 133

CONTENTS

vii

PAGE

SECTION 59. GO. 61.

62.

GEODESIC TORSION SPHERICAL REPRESENTATION RELATIONS BETWEEN A SURFACE AND ITS SPHERICAL REPRE SENTATION HELICOIDS

137 141

143 146

CHAPTEE V FUNDAMENTAL EQUATIONS. THE MOVING TRIHEDRAL 63.

ClIRISTOFFEL SYMBOLS

152

64.

THE EQUATIONS OF GAUSS AND OF CODAZZI FUNDAMENTAL THEOREM FUNDAMENTAL EQUATIONS IN ANOTHER FORM

153

65. 66. 67. 68.

69.

70. 71.

72. 73. 74. 75. 76.

157

160

TANGENTIAL COORDINATES. MEAN EVOLUTE THE MOVING TRIHEDRAL FUNDAMENTAL EQUATIONS OF CONDITION LINEAR ELEMENT. LINES OF CURVATURE CONJUGATE DIRECTIONS AND ASYMPTOTIC DIRECTONS. SPHER ICAL REPRESENTATION FUNDAMENTAL RELATIONS AND FORMULAS PARALLEL SURFACES SURFACES OF CENTER FUNDAMENTAL QUANTITIES FOR SURFACES OF CENTER SURFACES COMPLEMENTARY TO A GIVEN SURFACE .

.

162 166

168 171 172

174 177 179 .

181

184

CHAPTER VI SYSTEMS OF CURVES. GEODESICS 77. 78. 79. 80.

81. 82.

83.

84. 85.

86. 87.

88.

89. 90.

ASYMPTOTIC LINES SPHERICAL REPRESENTATION OF ASYMPTOTIC LINES FORMULAS OF LELIEUVRE. TANGENTIAL EQUATIONS CONJUGATE SYSTEMS OF PARAMETRIC LINES. INVERSIONS SURFACES OF TRANSLATION ISOTHERMAL-CONJUGATE SYSTEMS SPHERICAL REPRESENTATION OF CONJUGATE SYSTEMS TANGENTIAL COORDINATES. PROJECTIVE TRANSFORMATIONS EQUATIONS OF GEODESIC LINES GEODESIC PARALLELS. GEODESIC PARAMETERS GEODESIC POLAR COORDINATES AREA OF A GEODESIC TRIANGLE LINES OF SHORTEST LENGTH. GEODESIC CURVATURE GEODESIC ELLIPSES AND HYPERBOLAS

189

.... ....

.

.

.

191

193

195 197

198 .

.

.

200 201 204 206

207 209

.... .

.

212

213

CONTENTS

viii

SECTION 91. 92. 93.

94.

PAGE

SURFACES OF LIOUVILLE INTEGRATION OF THE EQUATION OF GEODESIC LINES GEODESICS ON SURFACES OF LIOUVILLE LINES OF SHORTEST LENGTH. ENVELOPE OF GEODESICS

CHAPTEK

214 .

.

.

215 218

220

VII

QUADRICS. RULED SURFACES. MINIMAL SURFACES 95. 96.

97.

98. 99.

100.

101. 102. 103. 104. 105. 106. 107. 108.

109.

110. 111. 112. 113.

114.

CONFOCAL QUADRICS. ELLIPTIC COORDINATES FUNDAMENTAL QUANTITIES FOR CENTRAL QUADRICS FUNDAMENTAL QUANTITIES FOR THE PARABOLOIDS LINES OF CURVATURE AND ASYMPTOTIC LINES ox QUADRICS GEODESICS ON QUADRICS GEODESICS THROUGH THE UMBILICAL POINTS ELLIPSOID REFERRED TO A POLAR GEODESIC SYSTEM PROPERTIES OF QUADRICS EQUATIONS OF A RULED SURFACE LINE OF STRICTION. DEVELOPABLE SURFACES CENTRAL PLANE. PARAMETER OF DISTRIBUTION PARTICULAR FORM OF THE LINEAR ELEMENT ASYMPTOTIC LINES. ORTHOGONAL PARAMETRIC SYSTEMS MINIMAL SURFACES LINES OF CURVATURE AND ASYMPTOTIC LINES. ADJOINT MINI MAL SURFACES MINIMAL CURVES ON A MINIMAL SURFACE DOUBLE MINIMAL SURFACES ALGEBRAIC MINIMAL SURFACES ASSOCIATE SURFACES FORMULAS OF SCHWARZ .

.

226 .

....

.

.

.

.

CHAPTER

.

229

230 232 231 236

.

236 239 241

242

244 247 .

248 250

253 254

258 260

263 264

r

VIII

SURFACES OF CONSTANT TOTAL CURVATURE. TF-SURFACES. SURFACES WITH PLANE OR SPHERICAL LINES OF CUR

VATURE 115. 116. 117.

118. 119.

120. 121.

SPHERICAL SURFACES OF REVOLUTION PSEUDOSPHERICAL SURFACES OF REVOLUTION GEODESIC PARAMETRIC SYSTEMS. APPLICABILITY TRANSFORMATION OF HAZZIDAKIS TRANSFORMATION OF BIANCHI TRANSFORMATION OF BACKLUND THEOREM OF PERMUTABILITY

270 272 275

278 280 284 286

CONTENTS

ix

PAGE

SECTION

289

122.

TRANSFORMATION OF LIE

123.

JF-SURFACES.

124.

EVOLUTE OF A TF-SuitFACE SURFACES OF CONSTANT MEAN CURVATURE RULED IF-SuRFACES SPHERICAL REPRESENTATION OF SURFACES WITH PLANE LINES OF CURVATURE IN BOTH SYSTEMS SURFACES WITH PLANE LINES OF CURVATURE IN BOTH SYSTEMS SURFACES WITH PLANE LINES OF CURVATURE IN ONE SYSTEM.

125.

126.

127.

128. 129.

.

.

FUNDAMENTAL QUANTITIES

291 .

.......

131. 132.

133. 134.

296 299

300 302

305

SURFACES OF MONGE 130.

294

MOLDING SURFACES SURFACES OF JOACHIMSTHAL SURFACES WITH CIRCULAR LINES OF CURVATURE CYCLIDES OF DUPIN SURFACES WITH SPHERICAL LINES OF CURVATURE

307 308

310

312 IN

ONE 314

SYSTEM

CHAPTER IX DEFORMATION OF SURFACES 321

142.

PROBLEM OF MINDING. SURFACES OF CONSTANT CURVATURE SOLUTION OF THE PROBLEM OF MINDING DEFORMATION OF MINIMAL SURFACES SECOND GENERAL PROBLEM OF DEFORMATION DEFORMATIONS WHICH CHANGE A CURVE ON THE SURFACE INTO A GIVEN CURVE IN SPACE LINES OF CURVATURE IN CORRESPONDENCE CONJUGATE SYSTEMS IN CORRESPONDENCE ASYMPTOTIC LINES IN CORRESPONDENCE. DEFORMATION OF A

342

143.

RULED SURFACE METHOD OF MINDING

144.

PARTICULAR DEFORMATIONS OF RULED SURFACES

345

135.

136. 137.

138. 139.

140.

141.

.

323

327 331

333 336 338

344

CHAPTER X DEFORMATION OF SURFACES. THE METHOD OF WEINGARTEN 351

148.

REDUCED FORM OF THE LINEAR ELEMENT GENERAL FORMULAS THE THEOREM OF WEINGARTEN OTHER FORMS OF THE THEOREM OF WEINGARTEN

149.

SURFACES APPLICABLE TO A SURFACE OF REVOLUTION

145. 146.

147.

.... .

.

353 355 357 362

CONTENTS

x SECTION 150. 151.

PAGE

MINIMAL LINES ON THE SPHERE PARAMETRIC SURFACES OF GOURSAT. SURFACES APPLICABLE TO CERTAIN PARABOLOIDS

364

366

CHAPTER XI INFINITESIMAL DEFORMATION OF SURFACES 152.

GENERAL PROBLEM

153.

CHARACTERISTIC FUNCTION ASYMPTOTIC LINES PARAMETRIC ASSOCIATE SURFACES

154.

155. 156. 157. 158. 159.

....

373 374

376

378

PARTICULAR PARAMETRIC CURVES RELATIONS BETWEEN THREE SURFACES S, S v S SURFACES RESULTING FROM AN INFINITESIMAL DEFORMATION ISOTHERMIC SURFACES

CHAPTER

379

382

385 387

XII

RECTILINEAR CONGRUENCES 160.

DEFINITION OF A CONGRUENCE.

SPHERICAL REPRESENTATION

392

161.

NORMAL CONGRUENCES. RULED SURFACES OF A CONGRUENCE

393

LIMIT POINTS.

395

PRINCIPAL SURFACES 163. DEVELOPABLE SURFACES OF A CONGRUENCE. 164. ASSOCIATE NORMAL CONGRUENCES 162.

165. 166. 167.

168. 169.

170. 171. 172. 173.

FOCAL SURFACES

DERIVED CONGRUENCES FUNDAMENTAL EQUATIONS OF CONDITION SPHERICAL REPRESENTATION OF PRINCIPAL SURFACES AND OF DEVELOPABLES FUNDAMENTAL QUANTITIES FOR THE FOCAL SURFACES ISOTROPIC CONGRUENCES CONGRUENCES OF GUICIIARD PSEUDOSPHERICAL CONGRUENCES IT-CONGRUENCES CONGRUENCES OF RIBAUCOUR

........ ...... .

CHAPTER

398

401

.

.

403 406

407 409 412

414 415 417 420

XIII

CYCLIC SYSTEMS 174. 175. 176.

GENERAL EQUATIONS OF CYCLIC SYSTEMS CYCLIC CONGRUENCES SPHERICAL REPRESENTATION OF CYCLIC CONGRUENCES

426 431 .

432

CONTENTS

XI

PAGE

SECTION 177. 178.

179.

180.

SURFACES ORTHOGONAL TO A CYCLIC SYSTEM NORMAL CYCLIC CONGRUENCES CYCLIC SYSTEMS FOR WHICH THE ENVELOPE OF THE PLANES OF THE CIRCLES is A CURVE CYCLIC SYSTEMS FOR WHICH THE PLANES OF THE CIRCLES PASS THROUGH A POINT

436 437 439

440

CHAPTEE XIV TRIPLY ORTHOGONAL SYSTEMS OF SURFACES 181.

TRIPLE SYSTEM OF SURFACES ASSOCIATED WITH A

CYCLIC

SYSTEM

446

182.

GENERAL EQUATIONS. THEOREM OF DUPIN

183.

EQUATIONS OF LAME TRIPLE SYSTEMS CONTAINING ONE FAMILY OF SURFACES OF REVOLUTION TRIPLE SYSTEMS OF BIANCHI AND OF WEINGARTEN THEOREM OF RIBAUCOUR

184.

185.

186. 187. 188.

....

THEOREMS OF DARBOUX TRANSFORMATION OF COMBESCURE

INDEX

.

447 449 451

452

457 458 461

467

DIFFERENTIAL GEOMETRY CHAPTER

I

CURVES IN SPACE 1.

Parametric equations of a curve.

Consider space referred to

fixed rectangular axes, and let (x, y, z) denote as usual the coordi nates of a point with respect to these axes. In the plane 2 = draw a circle of radius r and center (a, b). The coordinates of a point P on the circle can be expressed in the form

x

(1)

a

-{-

r cos u,

y

= b H- r sin u,

2

=

0,

where u denotes the angle which the radius to P makes with the to 360, the point P describes As u varies from positive o>axis.

the circle.

The

quantities

a, 5,

r determine the position

and

size

of the circle,

whereas u determines the position of a point upon

In this sense

a variable or parameter for the equations (1) are called parametric

And

circle.

it is

equations of the circle. straight line in space

A

point on a,

/3,

Let

P

it,

The

7.

P (a, Q

6,

c),

and

is

its

determined by a direction-cosines

latter fix also the sense of the line.

be another point on the line, and let the P be denoted by u, which is positive Q

distance

P

or negative. The rectangular coordinates of are then expressible in the form (2)

it.

x

= a + ua,

y

= b + u(B,

z

c

P

+ wy. FIG. 1

To each value

u there corresponds a point on the line, and the coordinates of any point on the line are expressible as in (2). These equations are consequently parametric of

equations of the straight line. When, as in fig. 1, a line segment

pendicular to a line

OZ

PD,

of constant length

at D, revolves uniformly about

OZ

,

per

as axis,

CURVES IN SPACE

2

and at the same time

P is

locus of

D

moves along

called a circular helix.

2-axis, the initial position of

between the u,

latter

it

If

with uniform velocity, the the line OZ be taken for the

PD for the positive

and a subsequent position

a>axis,

of

PD

and the angle

be denoted by

the equations of the helix can be written in the parametric form

x

(3)

=a

z

= bu,

b is

of translation of D.

and

D

radian,

a sin u,

y

determined by the velocity of rotation of Thus, as the line PD describes a moves the distance b along OZ.

where the constant

PD

cos u,

the above equations u is the variable or parameter. Hence, with reference to the locus under consideration, the coordi indicate this by writing these nates are functions of u alone.

In

all of

We

equations

The functions / /2 / have x,

,

eral case

forms when the locus is a But we proceed to the gen when /r /2 / are any func

definite

circle, straight line or circular helix.

and consider equations

(4),

,

tions whatever, analytic for all values of u, or at least for a certain domain.* The locus of the point whose coordinates are given by (4),

u takes

as

all

values in the domain considered,

is

a curve.

Equa

tions (4) are said to be the equations of the curve in the parametric all the points of the curve do not lie in the same plane form.

When

called a space curve or a twisted curve ; otherwise, a plane curve. It is evident that a necessary and sufficient condition that a

it is

curve, defined linear relation

by equations (4), be plane, between the functions, such ofi + 5f2 +

(5) a, b, c,

dition

is satisfied

u

in (4) be replaced

0,

by any function of

v,

say

*fc*^(*0,

(6)

equations

*

+d=

that there exist a

as

d denote constants not all equal to zero. This con by equations (1) and (2), but not by (3).

where If

c/3

is

(4)

E.g. in case

when

it is

assume a new form,

u

complex,

is

two fixed values; supposed to be real, it lies on a segment between within a closed region in the plane of the complex variable.

it lies

EQUATIONS OF A CURVE It is evident that the values of x, y,

3

given by

z,

value

(7) for a

(4) for the corresponding value

are equal to those given by i u obtained from (6). Consequently equations (4) and (7) define the same curve, u and v being the respective parameters. Since

of

,

of

there

is

lytic, it

in

no restriction upon the function

except that

,

it

be ana

follows that a curve can be given parametric representation

an infinity of ways. 2. Other forms of the equations of a curve.

tions (4) be solved for w, giving parameter, equations (7) are

x

(8)

= x,

In this form the curve

is

y

u

equa

$(#), then, in terms of x as

= F (x),

z

2

= F (x).

really defined

8

by the

be a plane curve in the o?y-plane,

or, if it

If the first of

last

its

two equations,

equation

is

in the

customary form

y =/(*)

(9)

The

points in space whose coordinates satisfy the equation lie on the cylinder whose elements are parallel to the

y = F (x) z

and whose cross section by the xy-pl&ne is the curve y = F2 (x). In like manner, the equation z = F3 (x) defines a cylinder whose

2-axis

Hence the curve with the common to two cylinders equations (8) with perpendicular axes. Conversely, if lines are drawn through the points of a space curve normal to two planes perpendicular to one another, we obtain two such cylinders whose intersection is the given curve. Hence equations (8) furnish a perfectly gen elements are parallel to the #-axis. is

the locus of points

eral definition of a space curve.

in

In general, the parameter u can be eliminated from equations (4) such a way that there result two equations, each of which in

volves

all

three rectangular coordinates.

Qfa

(10)

Moreover,

if

y, z)

=

two equations

0,

<S>

a (a;,

Thus, y, z)

=

0.

of this kind be solved for

y and

we

z as

get equations of the form (8), and, in turn, of of u. Hence (4), by replacing x by an arbitrary function It will curve. a of also the are equations (10) general equations

functions of

x,

the form

be seen later that each of these equations defines a surface.

CURVES IN SPACE

4 It

should be remarked, however, that when a curve

two cylinders

as the intersection of

may happen

two surfaces

or of

(8),

is

defined (10), it

that these curves of intersection consist of several

parts, so that the

new

equations define more than the original ones.

For example, the curve defined by the parametric equations x

(i)

is

=

=

y

w,

w2

z

,

=

w3

,

a twisted cubic, for every plane meets the curve in three points. Thus, the plane

+

ax

by

-f

cz

+

=

d

meets the curve in the three points whose parametric values are the roots of the e(l uation

CM*

This cubic

lies

+

&n"

+

an

+

=

d

0.

upon the three cylinders y

=

x2

z

,

=

x3

=

y3

,

z2

.

and second cylinders is a curve of the sixth degree, of the sixth degree, whereas the last two intersect in a curve of the ninth degree. Hence in every case the given cubic is only a part of

The

intersection of the first

of the first

and third

it is

that part which lies on all three cylinders. u from equations (i), thus

the curve of intersection

Again,

we may

eliminate

xy

(ii)

=

y*

z,

=

xz,

and the second a hyperboliclies on both of these surfaces, parabolic cone. The straight line y = 0, z = but not on the cylinder y = x 2 Hence the intersection of the surfaces (ii) consists of

which the

first

defines a hyperbolic paraboloid

.

of this line

and the cubic. The generators of the paraboloid are defined by x

=

a,

for all values of the constants a

generator of the 3. is

first

=

and

ay 6.

;

y

From

z

6,

(i)

we

=

bx

;

see that the cubic meets each

family in one point and of the second family in two points.

Linear element.

the limit,

z

when

By

definition the length of an arc of a curve toward which the perimeter of an

it exists,

inscribed polygon tends as the

number

and their which such a limit

of sides increases

lengths uniformly approach zero. Curves for does not exist will be excluded from the subsequent discussion.

Consider the arc of a curve whose end points m mined by the parametric values U Q and # and let intermediate points with parametric values u^ w 2 ,

,

l

k

of the chord

mkmk+l

ma are mv m

,

,

deter

2,

.

,

be

The length

is

=V2,r/;.^,

L1

)-/v(oi

2 .

= i,

2,

3

LINEAR ELEMENT By

mean value theorem

the

5

of the differential calculus this

is

equal to

where

f

.

t

= wt +

0-

ay

M

)e

l

if

c

+

to this line is equal to \(bx"- ay")e*

bz )e

Hence,

=,

are the direction-cosines.

distance from

(23)

b

write the equa

MM

l

2

+

-]

2

+

]

+ [(az - cx )e +

2

1

.

-]

be considered an infinitesimal of the

}*.

first order,

this distance also is of the first order unless

in

which case

it is

of the second order at least.

But when

these

equations are satisfied, equations (22) define the tangent at M. Therefore, of all the lines through a point of a curve the tangent is

nearest to the curve.

* Whenever the functions x y z appear in a formula it is understood that the arc s is the parameter otherwise we use /{, /2 /3 indicating by accents derivatives with respect to the argument u. ,

;

,

,

,

CURVES IN SPACE

8 5.

When

Normal plane.

Order of contact.

the curve

such

is

that there are points for which

^=4

(24)

x

z

y

M

the distance from

to the tangent is of the third order at least. l In this case the tangent is said to have contact of the second order, whereas, ordinarily, the contact is of the first order. And, in gen eral, the tangent to a curve has contact of the wth order at a point,

if

the following conditions are satisfied for n

xw


V"y

^=^=^rr

25 )

When

the parameter of the curve is any whatever, equations (24), (25) are reducible to the respective equations ~jTf

fl

//

J%

Jl

-f(.n-\)

J%

J\

f (rt-1)

J

f(n-l)

Ja

2,

The plane normal contact

to the tangent to a curve at the point of normal called the plane at the point. Its equation is

is

(X

(26)

where

a,

/3,

x)

a

+ (Y

7 have the values

y}

ft

+ (Z

z)

7

=

0,

(20).

EXAMPLES Put the equations

1.

of the circular helix (3) in the

form

(8).

Express the equations of the circular helix in terms of the arc measured from a point of the curve, and show that the tangents to the curve meet the elements of the circular cylinder under constant angle. 2.

Show

3.

the curve

is

that

if

at every point of a curve the tangency

a straight

is

of the second order,

line.

Prove that a necessary and

sufficient condition that at the point (x 2/o) of = / "(BO) f(x) the tangent has contact of the nth order is/"(x ) the tangent crosses the = also, that according as n is even or odd r=/()(z ) = curve at the point or does not. 4.

,

the plane curve y .

.

.

;

Prove the following properties of the twisted cubic the cubic one and only one meets the (a) Of all the planes through a point of 3 cubic in three coincident points its equation is 3 u*x - 3 uy + z - w = 0. on a plane has a the orthogonal projection (6) There are no double points, but 5.

:

;

double point. a variable chord of the cubic and by each of (c) Four planes determined by four fixed points of the curve are in constant cross-ratio.

CUKVATUKE

FIKST

9

M

Let Jf, be two l points of a curve, As the length of the arc between these points, and A0 the angle between the tangents. The limiting value of Curvature.

6.

A0/As

as

M

l

Radius of

approaches

curvature.

first

namely dd/ds, measures the

Jf,

rate of

M

as the point of con change of the direction of the tangent at tact moves along the curve. This limiting value is called the

curvature of the curve at M, and

its reciprocal the radius of the latter will be denoted by p. first In order to find an expression for p in terms of the quantities defining the curve, we introduce the idea of spherical representa first

curvature

;

We

take the sphere * of unit radius with center at the origin and draw radii parallel to the positive directions ofthe tangents to the curve, or such a portion of it that no two tion as follows.

tangents are parallel. The locus of the extremities is a curve upon the sphere, which is in one-to-one correspondence with the

In this sense

given curve.

we have

a spherical representation, or

spherical indicatrix, of the curve.

The angle A# between the tangents to the curve at the points M, M^ is measured by the arc of the great circle between their

m

on the sphere. If ACT denotes the representative points m, l of of the the arc and m^ spherical indicatrix between length then by the result at the close of 3,

m

dO

v = lim

da

A,

0.

and the resulting equation be divided by

77

form

(X- x)a + (Y- y)l + (Z-z)c = Q, the current coordinates. When the

through the tangent at

where

of the

Jf,

y"b

+

77

z"c

first

approaches zero,

=

0.

and higher and in the

OSCULATING PLANE Eliminating

a,

c >,

from equations

11

we

(32), (33), (34)

obtain, as

the equation of the osculating plane,

X-x Y-y Z-z x

(35)

1

y

=

0.

x"

y"

From

this

we

plane

when

find that

in terms of a general

the curve

parameter

x_ x Y

is

w,

y

is defined by equations (4) the equation of the osculating

z_ z

(36)

//

fi

/,"

The plane defined by either of these equations is unique except when the tangent at the point has contact of an order higher than In the latter case equations (33), (34) are not independent, as follows from (24); and if the contact of the tangent is of the wth order, the equations =

the

first.

^+^+^

G

0?

up to and including n are not independent of this equation and (33) are inde But for r = n + pendent, and we have as the equation of the osculating plane at this

for all values of r

one another.

~\.,

X-x Y-

singular point,

x

When

Z-z =

y

0.

plane, and its plane is taken for the rry-plane, to Z = 0. Hence the osculating plane reduces the equation (35) of a plane curve is the plane of the latter, and consequently is the same for all points of the curve. Conversely, when the osculating

a curve

is

plane of a curve is the same for all its points, the curve is plane, for all the points of the curve lie in the fixed osculating plane. The equation

of the osculating plane

of

the twisted cubic

(2)

is

readily

reducible to

where JT, F, Z are current coordinates. From the definition of the osculating plane and the fact that the curve is a cubic, it follows that the osculating plane meets the curve only at the point of osculation. As equation (i) is a cubic in w, it follows that through a point (o, 2/o, ZQ) not on tne cnrve there pass three planes which osculate the cubic. Let MI, w 2 u 3 denote the parameter values of these points. ,

Then from

(i)

we have

=

3

XG,

3

?/o,

2t

i

\

n

CURVES IN SPACE

12 By means

of these relations the equation of the plane through the corresponding is reducible to

three points on the cubic

(X -

XQ) 3

7/0

- Y(

This plane passes through the point

y

(x

)

,

+

3x

2/0,

ZQ)

(Z 5

-

z

)

= 0.

hence we have the theorems

:

The points of contact of the three osculating planes of a twisted cubic through a point not on the curve lie in a plane through the point. The osculating planes at three points of a twisted cubic meet in a point which lies in the plane of the three points.

By means of these theorems we can establish a dual relation in space by mak ing a point correspond to the plane through the points of osculation of the three osculating planes through the point, and a plane to the point of intersection of the three planes which osculate the cubic at the points where it is met by the plane. In particular, to a point on the cubic corresponds the osculating plane at the point,

and

vice versa.

8.

Principal normal and binormal.

finity of

Evidently there are an in

normals to a curve at a point.

Two

of these are of par

the normal, which lies in the osculating plane at the point, called the principal normal; and the normal, which is perpendicular to this plane, called the binormal.

ticular interest

:

If the direction-cosines of the

we have from X

:

/

binormal be denoted by X,

/>t,

z>,

(35) :

v

= (y

z"-

z

: y")

(z

z

x"~

:

z")

(*

/ -y

x").

In consequence of the identity

common

the value of the

ratio

is

reducible by means of (19) and

We

take the positive direction of the binormal to p.* be such that this ratio shall" be -f- p then (28) to

;

\ = p(y z"-z f

(37)

When (38)

the parameter u

x=

v z"),

= p(x y

general, these formulas are

:

~ P dycPz

dzcfy

~~df~

*For

is

x

x"~

282

or in other form /oof

y"),

^ = P (z

SV = /

0,

as

is

^~ P

dzd*xdxd?z

~ P dxd y

~

ds

3

seen by differentiating

2x"*=

1

dyd^x ds*

with respect to

s.

PRINCIPAL NORMAL AND BINORMAL By

definition the principal

normal

13

perpendicular to both the the convention that its positive is

tangent and binormal. We make direction is such that the positive directions of the tangent, prin cipal normal and binormal at a point have the same mutual ori entation as the positive directions of the x-, y-, z-axes respectively.

These directions are represented in fig.

2 by the lines

Hence,

if

Z,

m,

MT, MC, MB.

denote the direc

n,

tion-cosines of the principal normal,

we have*

m

(39)

=4-1,

/JL

from which

it

= mv = \ = ftn a I

I

FIG. 2

follows that

fjij

nu,

ft

n\

Iv,

7

vft^

m=

va

\7,

n

=

yl

#n,

777*,

/Ji

i^

= Z/i = Xp = am

m\, yuo:,

ftl.

,

Substituting the values of #, /3, 7; X, /u-, i^ from (19) and (37) in the m, w, the resulting equations are reducible to expressions for ,

Hence, when the parameter u (42)

is

general,

we have

l=-(W-

or in other form, 2

_ In consequence of (29) equations (42) (43)

or

by means of

dzd 2 s

be written:

da

dft

dj

ds

ds

ds

(27),

da

Hence the tangent to the principal

may

2

m

dft

dy

do-

da-

to the spherical indicatrix of a curve is parallel

normal to the curve and has the same sense. *C. Smith, Solid Geometry, llth

ed., p. 31.

CURVES IN SPACE

14 9.

Center of

circle.

Osculating

first

We

curvature.

have defined

M

the osculating plane to a curve at a point to be the limiting of the determined the and by a plane position by tangent at of the as the latter the curve. curve, approaches point l along

M

M

We

M

consider

now

the circle in this plane which has the same tan

M as the curve, and passes through M The limiting posi M approaches called the osculating

gent at

{

tion of this circle, as to the

curve at M.

7I/,

l

circle

It is evident that its center

normal at M. Hence, with reference F the coordinates of (7 denoted by Q cipal

,

X =x +

Y

rl,

where the absolute value of

y

M

when

l

X

,

-f

rm,

,

to

Z

Q,

C

is

on the prin

any fixed axes in space, are of the form

Z^=z +

rn,

r is the radius of the osculating circle.

In order to find the value of of the circle,

.

is

we

r,

does not have

return to the consideration limiting position, and

its

we

let X, F, Z\ Zj, m^ n^ r^ denote respectively coordinates of the cen ter of the circle, the direction-cosines of the diameter through and the radius. If x v y v z l be the coordinates of M^ they have the

M

values (17), and since rl If

l

is

on the

notice that

we have i

2 e*x".

-)

after reducing the

.

above equation

2

,

1-r^Z/ where

circle,

= 2(A - xtf = 2(7-^- ex -

^x ^ = 0, and through by e we have

we

divide

M

limi-t r l

becomes

r,

^x\

=

(),

and higher orders in e. In the becomes 2z 7, that is - and this equation

involves terms of the

77

+*?

first

,

reduces to

so that r

is

equal to the radius of curvature.

On

this

account the

of curvature and its center the osculating circle is called the center of first curvature for the point. Since r is positive the center circle

of curvature

is

consequently

its

(44)

on the positive half of the principal normal, and coordinates are

X =x +

pl,

Y =y + pm,

Z = z + pn. Q

CENTER OF CURVATURE The

15

normal to the osculating plane at the center of curva

line

called the polar line or polar of the curve for the corre sponding point. Its equations are

ture

is

X-x-pl = Y-y-pm = Zz

/45\

\

In

line for

C

2

fig.

pn

^

v

JJL

represents the center of curvature and

CP

the polar

M.

A curve may be looked upon as the path of a point moving under the action of a system of forces. From this point of view it is convenient to take for parameter the time which has elapsed since the point passed a given position. Let t denote this parameter. As t is a function of 8, we have dx

ds

_dx ~

_

ds

_ dt~

Hence the

may

ds

dz

~dt

~dt~

dy

C

ds~dt~ *dt

_

ds

y ~dt

rate of change of the position of the point with the time, or its velocity,

be represented by the length

manner, by means of

laid off

on the tangent

In like

we have

(41),

d^ _ ~ From

to the curve.

dt

this it is seen that the rate of

d?s

n

/ds\ 2

7

change of the velocity at a point, or the

be represented by a vector in the osculating plane at the point, through the latter and whose components on the tangent and principal normal

may

acceleration,

d*s -

1

, and -

df*

1.

/dY I

)

P \dtj

EXAMPLES

Prove that the curvature of a plane curve defined by the equation

Show

(x,

y)dx

ex

cy p 2.

M

(J/

2

+

N

that the normal planes to the curve,

x

a sin 2

it,

y

=

a sin u cos w,

z

=

a cos M,

pass through the origin, and find the spherical indicatrix of the curve. 3.

The

4.

Derive the following properties of the twisted cubic

straight line

is

In any plane there planes can be drawn. (a)

the only real curve of zero curvature at every point.

is

one

line,

:

and only one, through which two osculating

fixed osculating planes are cut by the line of intersection of any two osculating planes in four points whose cross-ratio is constant. and four fixed points of the curve (c) Four planes through a variable tangent (6)

Four

are in constant cross-ratio. (d)

What

is

the dual of

(c)

by the

results of

7 ?

CURVES IN SPACE

16 5.

Determine the form of the function

curve x 6.

=

w,

y

=

sin w, z



so that the principal normals to the

(u) are parallel to the yz-plane.

Find the osculating plane and radius of x a cos u -f 6 sin w, y = a sin u

first

+

10. Torsion. Frenet-Serret formulas. less a

curvature of

6 cos w,

z

It has

=

c sin 2 u.

been seen

that,

un

curve be plane, the osculating plane varies as the point

moves along the curve. The change in the direction depends evidently upon the form of the curve. The ratio of the angle A^ between the binormals at two points of the curve and their curvi linear distance As expresses our idea of the mean change in the direction of the osculating plane.

And

so

we take

the limit of

one point approaches the other, as the measure of of this rate the change at the latter point. This limit is called this ratio, as

the second curvature, or torsion, of the curve, and

its

radius of second curvature, or the radius of torsion. will be denoted by r.

inverse the

The

latter

In order to establish the existence of this limit and to find an expression for

we draw

in terms of the functions defining the curve,

it

radii of the unit sphere parallel to the positive binormals

of the curve

and take the locus of the end points

a second spherical representation of the curve.

of these radii as

The

coordinates of

points of this representative curve on the sphere are X, /*, v. Pro ceeding in a manner similar to that in 6, we obtain the equation

i_

(46)

where

r dcr l is

2

ds*

the linear element of the spherical indicatrix of the

binormals. In order that a real curve have zero torsion at every point, the cosines

X,

/*,

v

must be constant. By a change of the fixed axes, which evidently has no effect upon the form of the curve, the cosines can be given the values X = 1, /* = v = 0. const. Hence a necessary It follows from (40) that a = 0, and consequently x and sufficient condition that the torsion of a real curve be zero at every point is that the curve be plane.

In the subsequent discussion

we

shall

respect to s of the direction-cosines a,

deduce them now.

From

(4T)

a

(41)

=i,

/3

need the derivatives with

&

7;

we have

=, y-.

I,

m, w;

X, p, v.

We

FRENET-SERRET FORMULAS In order to find the values of X respect to

s

/*

,

i/,

,

we

17

differentiate with

the identities,

X 2 +At2+z 2 = 1?

+

a\

,

/A

+

= 0,

7*

and, in consequence of (47), obtain

From

these,

+

= 0,

The

by

the proportion (40), follows

1

fjLfj,

vv

1

f :

fjL

:v

r

is

,: >

is

I

=

(46).

=

V

t

:

=

l*>

T

T

T

If the identity

from

the latter equation.

V A =

(48)

the result

1/r, as is seen

is

not determined by thus sign by writing the above proportion

algebraic sign of r its

0.

=l:m:n,

the factor of proportionality

We fix

r

+

X and

+ /V + yv =

\

XX

^7

vfi

be differentiated with respect to

s

reducible by (40), (47), and (48) to

I

(49)

m

and n Similar expressions can be found for Gathering to fundamental in gether these results, we have the following formulas .

the theory of twisted curves, and called the Frenet-Serret formulas

Y

\^ v v ^-/l+2),

(50)

As an example, we If the equation

:

derive another expression for the torsion.

\

= p(y z

be differentiated with respect to

"

s,

z

y")

the result

may be

written

and similar ones for WI/T, rz/r be multiplied by ?, wz, n respectively and added, we have, in consequence of (50) and (41), If this equation

y

x

z z"

(51)

y"

x

"

"

y

z

"

CUKVES IN SPACE

18

The last three of equations (50) give the rate of change of the direction-cosines of the osculating plane of a curve as the point of osculation moves along the curve. From these equations it follows that a necessary and sufficient condition that this rate of change at a point be zero is that the values of s for the point make the determinant in equation (51) vanish. At such a point the osculat

ing plane

said to be stationary.

is

Form

of curve in the neighborhood of a point. The sign of have made the convention that the positive directions of the tangent, principal normal, and binormal shall have the same 11.

We

torsion.

relative orientation as the fixed

M

2-axes respectively.

x-, y-,

When we

take these lines at a point for axes, the equations of the curve Q can be put in a very convenient form. If the coordinates be ex

pressed in terms of the arc measured from = (41) that for s

M^ we

have from (19)

and

P

When

the values of

I

and X from

the fourth of equations (50),

x

(5 2)

"

this

for

z"

-z

-

for

"

y

/




must be remarked that the new axes of coordinates are not necessarily real, so that when it is important to know whether the It

curves are real (73).

We

An

will be advisable to consider the general formulas

example of

this will be given later.

apply the preceding results to several problems. is plane the torsion is zero, and conversely. For this case equa

shall

When

it

the curve

tion (65) reduces to

=

-

ds

which the general

of

= where a

is

integral

is

p

-if J 1

ae

P

an arbitrary constant, and by

spherical indicatrix of the tangent.

=

ae~

(27)