Differential geometry a first course

A First Course
D Somasundaram

Alpha
Science


Differential Geometry
A First Course

www.pdfgrip.com


www.pdfgrip.com


Differential Geometry
A First Course

D Somasundaram

©
Alpha Science International Ltd.
Harrow, U.K.

www.pdfgrip.com


D. Somasundaram
Department of Mathematics
Erstwhile Madras University

P.G. Extension Centre
Salem, Tamil Nadu, India
Copyright © 2005
Alpha Science International Ltd
Hygeia Building, 66 College Road,
Harrow, Middlesex HA1 1BE, U.K.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval
system or transmitted in any form or by any means, electronic, mechanical, photocopying,
recording or otherwise, without prior written permission of the publisher.
ISBN 1 -84265- 182-X
Printed in India

www.pdfgrip.com


Preface

This book is a detailed introduction to the classical theory of curves and surfaces
which is offered as a core subject in mathematics at the post-graduate level in most
of our universities. Based on Serret-Frenet formulae, the theory of space curves is
developed. The theory of surfaces includes the first fundamental form with local
intrinsic properties, geodesies on surfaces, the second fundamental form with local
non-intrinsic properties and the fundamental equations of the surface theory with
several applications. A variety of graded examples and exercises are included to
illustrate all aspects of the theory.
Relevant motivation of different concepts and the complete discussion of
theory and problems without omission of steps and details make the book selfcontained and readable so that it will stimulate self-study and promote learning
among the students in the post-graduate course in mathematics.
I take this opportunity to thank Mr. N.K. Mehra, the Managing Director of
Narosa Publishing House, for his personal care and excellent co-operation in the

publication of this book. Suggestions for the further improvement of the book will
be most welcome.
Finally I wish to dedicate this book to the fond memory of my beloved parents
Tmt. D. Vishalakshi and Thiru A. Durairaj and to my most revered Professor
V. Ganapathy Iyer.
D. Somasundaram

Madurai

www.pdfgrip.com


www.pdfgrip.com


Contents

Preface

v

1. Theory of Space Curves

1

1.1
1.2
1.3
1.4
1.5

1.6
1.7
1.8
1.9
1.10
1.11
1.12
1.13
1.14
1.15
1.16
1.17
1.18
1.19
1.20

Introduction
Representation of space curves
Unique parametric representation of a space curve
Arc-length
Tangent and osculating plane
Principal normal and binormal
Curvature and torsion
Behaviour of a curve near one of its points
The curvature and torsion of a curve as the intersection
of two surfaces
Contact between curves and surfaces
Osculating circle and osculating sphere
Locus of centres of spherical curvature
Tangent surfaces, involutes and evolutes

Betrand curves
Spherical indicatrix
Intrinsic equations of space curves
Fundamental existence theorem for space curves
Helices
Examples I
Exercises I

2. The First Fundamental Form and Local Intrinsic
Properties of A Surface
2.1
2.2
2.3
2.4

Introduction
Definition of a surface
Nature of points on a surface
Representation of a surface

www.pdfgrip.com

1
2
3
7
10
15
18
26

31
35
37
43
48
57
61
67
69
76
80
99

101
101
101
103
106


viii
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13

2.14
2.15
2.16
2.17

Contents

Curves on surfaces
Tangent plane and surface normal
The general surfaces of revolution
Helicoids
Metric on a surface—The first fundamental form
Direction coefficients on a surface
Families of curves
Orthogonal trajectories
Double family of curves
Isometric correspondence
Intrinsic properties
Examples II
Exercises II

3. Geodesies on a Surface
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8

3.9
3.10
3.11
3.12
3.13
3.14
3.15
3.16
3.17

Introduction
Geodesies and their differential equations
Canonical geodesic equations
Geodesies on surfaces of revolution
Normal property of geodesies
Differential equations of geodesies using normal property
Existence theorems
Geodesic parallels
Geodesic polar coordinates
Geodesic curvature
Gauss-Bonnet theorem
Gaussain curvature
Surfaces of constant curvature
Conformal mapping
Geodesic mapping
Examples III
Exercises III

4. The Second Fundamental Form and Local
Non-intrinsic Properties of a Surface

4.1
4.2
4.3
4.4
4.5

Introduction
The second fundamental form
Classification of points on a surface
Principal curvatures
Lines of curvature
www.pdfgrip.com

108
109
114
117
118
123
131
133
136
138
142
143
158

160
160
160

173
174
178
182
192
199
201
202
220
227
232
237
242
249
276

279
279
279
284
290
296


ix

Contents

4.6
4.7

4.8
4.9
4.10
4.11
4.12
4.13
4.14

The Dupin indicatrix
Developable surfaces
Developables associated with space curves
Developables associated with curves on surfaces
Minimal surfaces
Ruled surfaces
Three fundamental forms
Examples IV
Exercises IV

5. The Fundamental Equations of Surface Theory
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9

Introduction

Tensor notations
Gauss equations
Weingarten equations
Mainardi-Codazzi equations
Parallel surfaces
Fundamental existence theorem for surfaces
Examples V
Exercises V

Hints and Answers to Exercises
1.20
2.17
3.17
4.14
5.9

302
313
321
329
331
335
353
358
379

382
382
382
383

388
390
399
404
416
442

444
444
446
448
450
454

Exercises I
Exercises II
Exercises III
Exercises IV
Exercises V

References

455

Index

456

www.pdfgrip.com

1
Theory of Space Curves

1.1

INTRODUCTION

Differential Geometry is the study of properties of space curves and surfaces.with
the help of vector calculus. This geometry examines in more details the curves in
space and surfaces, whereas the differential geometry of the plane curves deals
with the tangents, normals, curvature, asymptotes, involutes, evolutes etc. which
have analogues for space curves. Though we have these analogues for a space
curve, the curvature at a point of a space curve and the tangent plane at a point to a
surface play a dominant role in the differential geometry. Thus the differential
geometry studies the pointwise properties of space curves and surfaces as distinct
from the algebraic geometry whose sole aim is to describe the properties of the
configuration as a whole.
In this chapter on space curves, first we shall specify a space curve as the
intersection of two surfaces. Then we shall explain how we shall arrive at a unique
parametric representation of a point on the space curve and also give a precise
definition of a space curve in E3 as set of points associated with an equivalence
class of regular parametric representations. With the help of this parametric
representation, we shall define tangent, normal and binormal at a point leading to
the moving triad (t, n, b) and their associated tangent, normal and rectifying
planes. Since the triad (t, n, b) at a point is moving continuously as P varies over
the curve, we are interested to know the arc-rate of rotation of t, n, and b. This
leads to the well-known formulae of Serret-Frenet.
Then we shall establish the conditions for the contact of curves and surfaces
leading to the definitions of osculating circle and osculating sphere at a point on

the space curve and also the evolutes and involutes. Before concluding this
chapter, we shall explain what is meant by intrinsic equations of space curves and
establish the fundamental theorem of space curves which states that if curvature
and torsion are the given continuous functions of a real variable sy then they
determine the space curve uniquely. We shall illustrate all the notions developed
with a particular type of space curves known as helices. In all our discussion, the
basic formula of Serret-Frenet occupies the central position.

www.pdfgrip.com


2

Differential Geometry—A First Course

1.2

REPRESENTATION OF SPACE CURVES

Whenever we use the word space, it means the Euclidean space of dimension three
denoted by Ey In this space a single equation generally represents a surface and so
we need two equations to specify a curve. Thus we first introduce a space curve as
the intersection of two surfaces
Fl(x,y,z) = 0,

F2(*,;y,z) = 0

...(I)

Though we are able to fix the curve in space with the help of the equations (1),

we are unable to obtain the representation of different points on the whole curve.
Since a space curve is a set of points with a sense of description, it is desirable to
know the coordinates of a point on the curve as functions of single parameter. So
the question naturally arises whether one can obtain the parametric representation
of a curve with the help of equation (1). First let us examine this question and then
its converse.
Let the functions F{ and F2 of (1) satisfy the following conditions of the
implicit function theorem.
(i) Let F{ and F2 have continuous partial derivatives of first order.
(ii) At least one of the following three Jacobian determinants
d(F^F2)
d(y,z)

d(Fl9F2)
'

d(z,x)

d(FlyF2)
'

d(x,y)

is different from zero at a point (x0, y0, z0) on the curve.
Under these conditions one can obtain the solution of two variables in terms of
the third variable. Hence without loss of generality, let us assume that the first
Jacobian is different from zero. Then we can solve for y and z in terms of x and
obtain y =/,(*) and z -f2M as their solutions. Having obtained these values ofy
and z in terms of x, we take the parametric representation as
x=u,

y=fx{u\

z=f2(u)

...(3)

The above equations treating* as a parameter are true for the restricted values
of JC under the conditions (2). So they cannot give the parametric representation of
the whole curve. Thus having started with the definition of a curve as the
intersection of two surfaces, we are unable to arrive at a satisfactory representation
of the whole curve.
Next let us examine the converse question in a little more detailed manner. Let
us assume that the curve is given in the following parametric form
* = *(«), y = y(u),

z = z(u)9

ux
...(4)

and find the curve as the intersection of two surfaces. Now solving for u in terms
of x from the first equation as u = f{x) and substituting for u in the other two
equations, we obtain
y = y[fMl

z = z[f(x)]

...(5)

Hence the equations of two surfaces specifying the curve are given by (5).

www.pdfgrip.com


3

Theory of Space Curves

The difficulty with this method of elimination is that the intersection of the
above two surfaces not only contain the given curve, but also it may contain some
extra curves as shown by the following example.
Example 1. Consider the cubic curve with parametric representation
x = uyy = u > z = u

...(1)

A most straight forward method of elimination gives
y-x2

and xz = y2

...(2)

We know that y2 = xz represents a parabolic cylinder and x2 = y is a cone with
the vertex at the origin. Hence the two surfaces specify the curve and their
intersection not only contain the cubic curve (1) but also thez-axis as the equations
(2) satisfy x = 0, y = 0.
Another method of elimination of the parameter u in (1) givesxy = z andxz -y2

which are the equations of the two required surfaces specifying the cubic curve.
Though y2 = xz represents the same parabolic cylinder as in (2), xy = z represents
a paraboloid. Their intersection contains not only the given cubic curve but also the
x-axis as the two equations satisfy the conditions y = 0, z = 0.
Thus we see that different types of eliminations of the parameters in the
parametric equations not only give the curve as the intesection of the two surfaces
but also include extraneous curves like z-axis or jc-axis. As a result, we are unable
to arrive at a one to one correspondence between the points on the curves and the
curves given by the intersections of two surfaces whose equations we obtain from
the parametric representations.
So, of the two methods of defining a space curve, we find that the definition of
a curve as the intersection of the two surfaces gives very little information about
the curves. Though it fixes the curve in space, it fails to give the coordinate
representation of different points and the sense of description of the curve. So it is
not amenable to the treatment of vector calculus.
Summarising the above discussion, we conclude that when the curve is defined
as the intersection of two surfaces, the parametric representation obtained from the
two surface equations does not give the whole curve and when the curve is
represented parametrically, the equations we obtain from the elimination of
parameters lead two surface equations giving not only the curve in question but
also some extraneous curves. Hence the two definitions are not equivalent.
So between the two definitions of a space curve, we choose the parametric
representation which is most advantageous. Since the parametric representation
gives the position of different points on the curve and also the sense of variation
along the curve, we can represent the points on the curve vectorially and use vector
analysis for studying the properties of space curves.

1.3 UNIQUE PARAMETRIC REPRESENTATION OF A
SPACE CURVE
The greatest advantage of parametric representation is that it gives the sense of

description of the curve as parameter varies in a given interval. However it should
www.pdfgrip.com


4

Differential Geometry—A First Course

be noted that we have different methods of parametrising the same curve. The
problem is how we are going to deal with the different parameters representing the
same point on the curve. We collect together different parameters representing the
same point into a class and choose a representative parameter from the class which
yields the same property as the other parameters of the class. Hence a curve will
therefore be specified by all possible parametric representations which are
equivalent in that all describe the same curve in the same sense. We shall make
these notions precise in the following definitions.
Definition 1. Let / be a real interval and ra be a positive integer. A real valued
function/defined on / is said to be of class ra, if/has continuous rath derivative at
every point of/. We call such functions Cm functions.
Note 1. When a function is infinitely differentiate, then / i s said to be of
class infinity or the function itself is called C°° functions.
Note 2. If/is a real valued function of several variables, then it is of class ra
if it admits all continuous partial derivatives of order ra.
Definition 2. A function/is said to be analytic over /, if/is single valued and
/possesses continuous derivatives of all orders at every point of /. This class of
functions is denoted by (0. The function itself is called a C® function. These
functions have power series representations in the neighbourhood of every point
of/.
Definition 3. A vector valued function R = R(w) defined on / is said to be of
class ra, if it has continuous rath order derivative at every point of/.

If we represent R vectorially as R = (JC, y> z), then the above definition implies
that each of the components JC, y, z is of class ra and x, y, z are functions of u.
Definition 4.

If R = — never vanishes on /, then the vector valued
du
function R = R(w) is said to be regular. This implies that x, y, z will never vanish
on / simultaneously.
Definition 5. A regular vector valued function R = R(w) of class ra is called
a path of class ra.
Note 1. As the parameter u varies, R(w) gives the position vector of different
points on the curve. Thus a path can be considered as the locus of a moving point
giving the manner in which the curve is described.
Note 2. Since the definition of a path depends not only on a vector valued
function R(w), but also on the interval /, there are as many paths as there are regular
vector valued functions of class ra defined on /. Likewise a single path may be
defined by different Cm functions defined on different intervals I{, I2 etc. For
example we can have a path of the same class defined on Ix and I2 corresponding to
two different vector valued functions. Hence to arrive at a unique single path of the
given class corresponding to the parameter u defined on /, it is desirable to
partition the paths into mutually disjoint classes .of the same type and choose a
representative path of the same class with a unique parameter. We shall achieve
this by an equivalence relation among the paths of the same class as follows.

www.pdfgrip.com


5

Theory of Space Curves

Definition 6. Two paths Rj and R2 of class Cn defined on I{ and/ 2 are said to
be equivalent if there exists a strictly increasing function 0 of class m which maps
/j onto 12 such that Rj = R 2 ° 0
If we takeRj = (xltyu Z\) and R2 = (x2, y2, z2), then the above conditions are the
same as
xx{u) = x2((p(u)X yx(u) = y2((p(u))i z,(w) = z2(0(w))
First let us verify that the notion of equivalence of paths of the same class as
defined in the previous paragraph is an equivalence relation.
(i) To prove the relation is reflexive, let R, be a path of class m defined on /
and let us take 0 = /. The identity function is an increasing function on /
and R, = R,o/ so that R{ is equivalent to itself. Thus the relation of
equivalence of paths is reflexive.
(ii) Let Rj and R2 be the paths of the same class m defined on /, and I2
respectively. Let R, be equivalent to R2. We shall show that R2 is
equivalent to R,. Since R{ is equivalent to R2, there exists a strictly
increasing function 0 from /, onto I2 such that R, = R2o 0. Since 0 is a
strictly increasing function on /, onto I2 with 0 # 0 , the inverse function
0 _1 exists as a strictly increasing function on I2 onto I{. Hence we have
R2 = R, o 0" ! which shows that R2 is equivalent to R,. Hence this relation
is symmetric,
(iii) To prove the relation is transitive, let the path Rj be equivalent to R2 and
R2 be equivalent to R3. Then there exists a strictly increasing function 0
defined on lx onto I2 such that
R, = R 2 o0.

...(1)

In a similar manner, there exists a strictly increasing function i/^on I2 onto
73 such that

R2 = R 3 o ^ .

...(2)

Since 0:1{ —> I2 and y/: l2 —> 73 are strictly increasing functions yo 0is a welldefined strictly increasing function on Ix onto 73. From (1) and (2) we have
Rj = R2 o0= R 3 o ^ o 0
Since y/o(p\s strictly increasing function on I{ onto 73, Rj is equivalent to R3,
proving that the relation is transitive.
Thus the notion of equivalence of paths of the same class Cm is an equivalence
relation. This relation introduces a partition on the paths of the same class splitting
the paths of the same class into mutually disjoint classes such that the paths within
the same class are equivalent to one another. Using these mutually disjoint classes,
we shall define a space curve and the parametric representation as follows.
These different equivalent classes of paths of class m determine the curves of
class m. Thus any path R determines a unique curve and R is called the parametric
representation of the curve. The variable u is called the parameter. Further
R = (*» y* z) wherex =x(u),y =y(u), z = z{u) is called the parametric representation
of the curve. The function 0 of two equivalent paths is called the change of
parameter. Though 0 preserves the sense of description of the curve, it gives the
www.pdfgrip.com


6

Differential Geometry—A First Course

changes in the manner of description of the curve. Summarising the above, we
define a curve as follows.
Definition 7. Any curve of class m in E3 is defined to be any set of points in
E3 associated with an equivalence class of regular parametric representations of

class m having one parameter.
Since the properties of a curve depend on a particular parameter chosen, every
property of a path is not a property of the curve. We are concerned with those
properties of a curve which are common to all parametric representations. This
means those properties which are invariant under a parametric transformation.
Before proceeding further, we shall illustrate the equivalent representations by
the following two examples.
Example 1. The following are the two equivalent representations of a circular
helix.
(i) Rj(w) = (a cos w, a sin w, bu), ue Ix = [0, n)
1 - v2 2av „ ,
_i *
, V G I2 = [0, oo)
r,
T , 2b tan
v
2
2
1+v 1+v
j
v
To show thatRj is equivalent toR 2 , we find a change of parameter to I2 such that Rj(w) = R2[0(w)].
(ii) R2(v) =

Let us takev = 0(w) = tan — u which is an increasing function from/j onto/ 2 so
that we can take this function as a change of parameter.
Hence

R2(v) = R2[0(w)]

2 1

~

u

1 - tan — u

2a tan2 — ~,
-i
— ,2fctan l
1 + tan - w 1 + tan 2
2

u
vt a n -L

= (a cos w, b sin w,few)= R^w)
Thus there exists a change of parameter 0(w) such thatR^w) = R2[0(w)] so that
R2 is equivalent to R{.
Example 2.

Let I{ = [ 0, - j and / 2 = (0, 1).

Let Rt(w) = (2 cos2 w, sin 2w, 2 sin u) be defined on /j
If 0: / 2 -» /i is defined as w = 0(v) = sin _1 (v), find the parametric
representation R2(v) equivalent to R{(u).
Now 0: 72 -> / t is u = 0(v) = sin_1(v). Then we have v = sin u.
Hence R^w) = [2(1 - sin2 w), 2 sin u cos w, 2 sin u]
Now

R2(v) =Ri[0(v)]

www.pdfgrip.com


7

Theory of Space Curves

R2(v) = [2(1 - v2), 2v>/l - v2 , 2v] which is the required parametric
transformation equivalent to 1^ (w).

1.4 ARC-LENGTH
We define the arc length of a curve and derive a formula for the arc length. Using
this formula of arc length, we show that the arc length is invariant under parametric
representation so that the arc length of a curve can be used as a parameter in our
study of properties of a space curve. Hence the arc length of a space curve plays the
important role as a natural parameter.
Definition 1. Let R = R(w) be a path with parameter we /. As u varies over
[a, b] c /, the path is an arc of the original path joining a and b. Let A be the
subdivision of [a, b] as follows
A = {a = w0 < ux Corresponding to this subdivision A of [a, b], let
n

L(A) = £ IROO-RK.,)!
/=!

Then L(A) gives the sum of the lengths of the sides of the polygon inscribed

within the curve by joining the successive points on the path. Any addition of new
points like u- in the side w/_1 u{ increases L(A),
since |R(M/) - R(M|._ {)\ < |R(w ã) - R(ô/-,)| + |R(ô/) - R(M-)|
Since a * by as A varies over all possible subdivisions of [a, b], we obtain a nonempty subset (L(A)} of real numbers. The least upper bound of this set {L(A)} of
real numbers is defined to be the length of the arc between a and b.
The following theorem asserts the existence of finite upperbound for the set
(L(A)} and gives a formula for it.
Theorem 1. If R = R(w) is the parametric representation of a curve where
u G [a, b], the length of the curve
s = S(u)= \"\R(u)\du
Proof.

...(1)

For a subdivision A = {a = u0 < u{ < u2 ... < un - b\
n

we have

L(A) = £

|R(M(.)-!*(«,._,)!

...(2)

Since R is at least of class C1, we have
|R(M,.)-R(M|._,)|= f"'R(«) du\
| J « f -i

Using (3) in (2), we obtain,

L(A)=V

R(u)du\

www.pdfgrip.com

.-(3)


8

Differential Geometry—A First Course

By Schwarz inequality, we get
Y

P R G O du\ < Y

["'\R(u)\du =

f\R(u)\du

..(4)

i=\

i=\

so that we have L(A)<

f | R | du
Ja

Since the right hand side of (4) is finite and independent of A, the set {L(A)} for
all possible subdivisons A of [a, b] is a bounded set of real numbers and it is
bounded above. So the least upper bound of (L(A)} exists as a finite quantity.
Next we shall show that this upperbound is actually (1) given in the theorem.
If S = S(u) denotes the arc length from a to w, then S(u) - S(u0) gives the arc
length between u0 and u. Since we have defined the arc length as the least upper
bound of (L(A)}, we have from (4)
S(u)-S(u0)<

iU\R(u)\du

...(5)

Since the length of the chord joining R(w) and R(w0) is less than the arcual
length, we have
\R(u)-R(u0)\
...(6)

From (5) and (6), we have
R( W )-R(« 0 )
U-


Wn

UQ

U-

UQ

...(7)

*>UO

Taking the limit as u —> w0, we get from (7)
|R(«o)|^S(iio)<|R(«o)|
Hence S(u0) exists and has the value S(u0) = |R(w0)|
...(8)
Since (8) is equally true for any parameter uQ in /, we conclude from (8)
(i) S is a function of the same class as the curve
(ii) As S(a) = 0, s = S(u) =

\"\R(u)\du

.(9)

Ja

where s denotes the length of the curve from a to u.
Corollary. In terms of cartesian parametric representation,
s = \ y]x2 + y2 + z2 du.
Ja

Proof. In cartesian parametric representation,

let R(u) = x(u)i + y(u)j + z(u)k.
Then |R(w)| =

yjx2+y2

+ z2 . Using this expression in (9), we get

'= fV* 2 +> ,2 +^ 2 du
Ja

www.pdfgrip.com


9

Tlieory of Space Curves

Further, since s = |R|, s2 = x2 + y2 + z2 which gives in terms of differential
ds = dx2 + dy2 + dz2.
Note, Since s * 0, we can take 5 as a new parameter. The change of
parameter from s to u is given by S(u) in (9). From (9), we can obtain u = Q(s) so
that the curve parametrised with respect to s is R = R[(p(s)].
Theorem 2. The arc length of a curve is invariant under parametric
transformation.
Proof. Let R{(u) be the parametric representation of the given curve with
parameter u. Let us consider the parametric transformation t = 0(w). Let the
parametric representation corresponding to t be R2(0- Since R, and R2 are
equivalent representations R,(w) = R 2 (0 = R2[0(«)L As u varies from a to b,
2

t = 0(w) varies from (j)(a) to
...(1)
du

Now using (1) in the arc-length formula, we get

J |R,|rf«= J|R 2 (t)|

du

...(2)

Since t = (j)(u) is a strictly increasing function and 0(w) i=- 0, we have
1^1
\du\

dt
du

..(3)

b

Using (3) in (2), we obtain

|R,|dw=

|R
that the ;arc

~ 2l|x dt which proves
.
Ha)

length is invariant for a change of parameter from u to t.
Note. When we change the parameter from u to f, the formula for arc length
retains its form with / instead of u. This is a very important property of the arc
length.
Example 1. Find the arc length of one complete turn of the circular helix
r(w) = (a cos w, a sin «, bu), -oo< u <<*>

...(1)

where a > 0 and obtain the equation of the helix with s as parameter.
From (1) r(w) = (- a sin w, a cos w, £)
s = S(u) = f |r(w)| du = f yja2 + b2 du = cw

So

Jo

...(2)

Jo

where c = yja2 + b 2 .
If a helix starts from w0, it makes one complete turn when u = u0 + 2n. Hence
the arc length corresponding to one complete turn is
u0 + 2n

s=

[

yja2 + b2 du = yja2 + b2 • In = Inc.

"0

www.pdfgrip.com


Differential Geometry—A First Course

10

To obtain the equation of the helix with s as parameter,
we have from (2), s = cu so that u = —
c
Using (3) in (1), we get

...(3)

r(.y) = \a cos - , a sin - , — which is the required equation.

1.5

TANGENT AND OSCULATING PLANE

If 7 is the given curve, then its parametric representation r = r(w) is the equation of
the curve in the sense that it gives the position vector of different points on 7. We

use R to represent position vectors of points in space not necessarily on 7 in order
to distinguish it from r = r(w) on 7 We also assume 7 is of class > 1 which means
that r(w) has continuous derivatives of all orders so that r = r(w) has power series
expansion at a point u0 in the neighbourhood u.
r(w0 + h) = r(w0) + — r(w0) + — f (w0) + ... + — r('°(w0) + 0(hn)
I!
2!
n!
where lim

= 0 where u-Un = h.
hn
°
In our study, we usually include first two or three terms in the above expansion.
Definition 1. Unit tangent vector to 7 at P. Let P and Q be two
neighbouring points on 7 with parameters u0 and u respectively. The parameters of
P and Q are very close together in the sense that when Q -» P, u - u0 —> 0. The unit
vector along PQ tends to the unit vector atP as Q —> P. This unit vector denoted by
t is defined to be the unit tangent vector at P. The sense of t is that of increasing s.
Definition 2. The line through P parallel to t is called the tangent line to 7 at
P. If R is the position vector of any point on this tangent line to 7 at P, then the
vector
is called the tangent vector to 7 and P.
From the above definition of the tangent vector at P, we have the following
properties.
A->O

dx
(i) The unit tangent vector t = — where we have chosen s as the parameter
ds

measured from P.
Proof. The unit vector along the chord PQ is
r(M)-r(M 0 )

|r(w)-r(w 0 )|
Since t is the unit vector along the chord PQ as Q —> P,
weget

t = lim r ( " ) ~ ^ o ) =
u->u0 |r(w) - r(w0)|

r(K)-ry
u->u0 \u-u0\

Hm

www.pdfgrip.com

\u-u0\
\r(u) - r(w0)|


11

Theory of Space Curves

Since the curve is of class m > 1, we get

t=

m
\Hu)\

Since s = r(w), we get t = —— = — • — = — .
s(u)
du as as
dx
Note, r = — is also a tangent vector, but not necessarily a unit vector.
du
(ii) Equation of the tangent line at P. Let R be the position vector of any point
Q on the tangent line at P. Then if the length of PQ is c, then the vector PQ = ct.
Hence the equation of the tangent line at P is
R = r + ct
which can also be written as R = r + cr where r is parallel to t.
Definition 3. Osculating plane. Let 7 be a curve of class m > 2 and P and Q
be two neighbouring points on 7 The limiting position of the plane that contains
the tangential line at P and passes through the point Q as Q —> P is defined as the
osculating plane at P.
Note. When y is a straight line the osculating plane is indeterminate at each
point. So we avoid this particular case in our discussion.
Definition 4. The point P on the curve for which r " = 0 is called a point of
inflexion and the tangent line at P is called inflexional.
Theorem 1. Let 7 be a curve of class m > 2 with arc length s as parameter. If
the point P on 7 has parameter zero, the equation of the osculating plane is
[R - r(0), r'CO), r"(0)] = 0 where r" * 0
If r"(0) = 0, let us assume that the curve 7 is analytic. Then the equation of the
plane at an inflexional point is
[fl-rCOXr'CO), r (k) (0)] = 0.
Proof. Using the arc length s as parameter, let 0 and s be the parameters of P
and Q. Let R be the position vector of the point on the plane containing the tangent

line at P and passing through Q. Then if r is the position vector of P, then the
vectors R - r(0), t = r'CO), and r(^) - r(0) are coplanar vectors. Hence the
condition of coplanarity gives the equation as
[R-r(0),r/(0),rW-r(0)] = 0

...(1)

Since the curve is of class m > 2, we have by Taylor's Theorem,
r(j) = r(0) + sr'(O) + -s2 r"(0) + 0(.s2) as s -> 0.
Using (1) in (2), we get
1 .
.1
R - r(0), r'(0), s r'(0) + -s2 r"(0) + 0(s2)
www.pdfgrip.com

=0

...(2)


Differential Geometry—A First Course

12

Neglecting the terms of higher order, the above equation becomes
2

[R-r(0),r'(0),*r'(0)] + R - r ( 0 ) , r m ^ r " ( 0 )

=0

...(3)

Since r'(0) x r'(0) and s is a scalar, the first term of (3) vanishes and so we get
[R-r(0),r'(0), r"(0)] = 0

...(4)

as the equation of the osculating plane provided the vectors r'(0) and r"(0) are
linearly independent. So to complete the proof, we have to prove r'(0) and r"(0)
are linearly independent. Since t = r' is a unit vector r'~ = 1. Differentiating this
relation, we have r' r" = 0 which shows that neither r' nor r" can be a constant
multiple of the other so that they are linearly independent unless r"(0) = 0.
If r"(0) = 0, then the point/5 is an inflexional point so we derive the equation of
the osculating plane at an inflexional point with an assumption that the curve y is
analytic.
Differentiating r' 2 = 1, we have r' r" = 0
Differentiating this relation once again, we have
r •r +r •r

=U

Since r" = 0, we get r' r'" = 0 at P.
Since r' cannot be zero, r' and r"' are linearly independent, unless r'" = 0.
Repeating this process of differentiation, let us assume that r (k) is first nonvanishing derivative of r such that r' r(k) = 0. So if r (k) * 0, we have from Taylor
Theorem
r(s) - r(0) = - ^ ^ + — r(k)(0) + 0 ( / ) as s -* 0.

...(5)

Using (5) in (1), we get
R - r ( 0 ) , r / ( 0 ) f - ^ l j + — r (k) (0) = 0
1!
k\
As in the previous case the above equation reduces to
[R - r(0), r^O), r(k)(0)] = 0 as the equation of the osculating plane at an inflexional
point.
If r(k) = 0 for all k > 2, then since the curve is analytic, we infer that t is constant
and therefore the curve is a straight line.
Corollary. If P is not point of inflexion, any vector lying in the osculating
plane is a r' + b r" for some constants a and b.
Proof. Since P is not a point of inflexion r" * 0. From (4) of the theorem r7
and r" lies in the osculating plane and pass through P. Hence any vector in the
osculating plane is a linear comlination of r' and r" so that we can take it as
ar' + bx" for some constants a and b. It is of importance to note that r" lies in the
osculating plane.

www.pdfgrip.com


13

Theory of Space Curves

Theorem 2. If w is the parameter of the curve y, then the equation of the
osculating plane at any point P with position vector r = r(w) is
[ R - r , r, r] =0.
dt
ds

dr du
du ds

r
s

Proof.

r =

Further

„ _ d (x\ du _ ( i r - vs) 1
du\s)ds
s2
s

Using these values of r' and r" in (4) of the previous theorem we have
r

r

YS

=0

S\

S S

Since r x r = 0 and s is a scalar, simplifying the above equation, we obtain
[R - r, r, r] = 0 as the equation of the osculating plane.
Corollary. If R = (X, Y, Z) and r = (JC, yy z) then the equation of the osculating
plane given by the scalar triple product in the theorem takes the form
X-x

Y-y

Z-z\

x

y

z

x

y

z

=0

Note 1. A tangent at a point P on a space curve is the line passing through the
two consecutive points on the curve. Likewise the osculating plane can be defined
" as the limiting position of the plane PQR of three consecutive points P, Q, R as Q
and R approach P. Using this definition, we shall derive the equation of the
osculating plane as follows.
Let r = r(w0)> r, = r(w,) and r 2 = r(u2) be three consecutive points on the curve.

Let Ra = p be the equation of the plane passing through the above three points.
Then if/(w) = R a - p , we have
/(Mo) = 0,/(M,) = Oand/(M2) = 0

Now we have the intervals [u0> w,] and [w,, u2]. Hence by Rolle's Theorem,
there exist points v, G [W0, U{] and v2 G [W,, U2] such thatf\v{) = 0 and/ / (v 2 ) = 0.
Since/' satisfies the conditions of the Rolle's Theorem in [v,, v2], there exists
av 3 G [uj, v2] such that/ 7/ (v 3 ) = 0. Hence when Q and R approach P,ult u2, v,, v2,
v3 approach u0. Writing u for u0 in the limiting case, we have
/(«) = r a -p = 0,/'(M) = v a = 0 , / »

= v a =0

...(1)

Using R • a = p in the first equation, we get
/( M ) = ( R - r ) . f l = 0
Thus from (1), we find the vectors is perpendicular to ( R - r ) , r and r so that
they are coplanar. So we write / ? - r = A r + ^ r where A and ji are constants.
Eliminating A, ji or using the condition of coplanarity, the equation of the
osculating plane is
www.pdfgrip.com


14

Differential Geometry—A First Course

[R-r, r,r] = 0
Note 2. In the case of the plane curves, the plane through the three

consecutive points on the curve is the plane itself. Hence the osculating plane
coincides with the plane of the curve itself.
The following example shows that at a point of inflexion, even a curve of class
oo need not possess an osculating plane.
Example 1. Let 7 be a curve defined by
r(w) =(w,0, e _l/ " 2 )

whenw>0

r(w) =(w, e-{/u\0)
whenw<0
and
r(w) =(0,0,0)
whenw = 0
First we shall show that the given curve 7 is of class infinity with u = 0 as an
inflexion point. Taking/(w) = e~l/"2, we prove/^O) = 0 for all k > 2
-\/u2

Now

r(0) = l i m * B ) - * 0 ) - l i m l - = 0
«->o
u
«->o u
l/« 2

AG) = l i m '

/(MW/(0)

«-»0

= lim

u

«-»0

^

=0

u

2

/"'(0) = limlfA--Vk l/>/«
" 2 ==,;mlimV - 6 y

3

== 0

. * •

In a similar manner
Now

r"(w) = (0,/"(w), 0) if w < 0

and

r"(w) = (0, 0,/"(w)) if u > 0
Hence when u —> 0, r"(w) = (0, 0, 0) which proves that u = 0 is an inflexional
point.
Since/ (k) (0) = 0 for all jfc, r(k)(0) = 0 for all k > 2. Hence 7 is a curve of class
infinity with u = 0 as an inflexion point.
Now let us find the equation of the osculating plane when u > 0
r(W) = («,0,^ l / M 2 ),r , (w) = | l , 0 , - ^ e -\/u

and

r » =

0,0,1 4 - 4 W
V

Hence the equation of the osculating plane is
-\/u2

X-u

K

Z-e

1

0

4e_1/":

2

0
www.pdfgrip.com

=0

2


Theory of Space Curves

15

Expanding the above determinant along the last column,

U

U J

Since - y — - ^ \e~{'u * 0 , the equation of the osculating plane whenw>0 is
Y= 0. In a similar manner the equation of the osculating plane when u < 0 is Z= 0.
So the osculating plane at u = 0 is indeterminate. This proves that at a point of
inflexion, even a curve of class °° need not possess an osculating plane.
Example 2. Find the equation of the osculating plane at a point u of the circular
helix
r = (a cos w, a sin w, bu)

...(1)

From the given equation, we have
r = (- a sin w, a cos ut b)

...(2)

r = (- a cos u,-a sin w, 0)
Using (1), (2), and (3), the equation of the osculating plane is
X - a cos u
- a sin u
-acosu

...(3)

Y- asm u Z - bu
a cos u
-a.s'mu

b
0

Expanding the above determinant along the last column and simplifying, the
equation of the osculating plane is b(X sin u-Y cos u - au) + aZ = 0.

1.6

PRINCIPAL NORMAL AND BINORMAL

Besides the tangent at P, we shall define the normal and binormal at P of the curve

leading to the moving triad of coordinate system at P.
Definition 1. Let P be a point on the curve y. The plane through P orthogonal
to the tangent at P is called the normal plane at P.
Since the osculating plane at P passes through the tangent at P, the normal
plane is perpendicular to the osculating plane at any point of the curve.
Definition 2. The line of intersection of the normal plane and the osculating
plane is called the principal normal at P. The unit vector along the principal-normal
is denoted by n. The sense of n may be chosen arbitrarily, provided it varies
continuously along the curve.
Using the above definitions, we have
(i) The equation of the normal plane. Let r = r(w) be a point on the curve and
R be the position vector of any point on the plane. Then (R - r) lies in the
normal plane. Since (R - r) is perpendicular to t, we get (R - r) • t = 0 as
the equation of the normal plane,
(ii) The equation of the principal normal at P. If the position vector of any
point P on the curve is r and if R is the position vector of any point Q on
www.pdfgrip.com