Differential geometry a first course

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 E 3 as set of points

A First Course

D Somasundaram

Alpha Science

Differential Geometry

A First Course

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

Relevant motivation of different concepts and the complete discussion of theory and problems without omission of steps and details make the book self-contained 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

Contents

Preface v

1 Theory of Space Curves 1

1.1 Introduction 1

1.2 Representation of space curves 2

1.3 Unique parametric representation of a space curve 3

1.4 Arc-length 7

1.5 Tangent and osculating plane 10

1.6 Principal normal and binormal 15

1.7 Curvature and torsion 18

1.8 Behaviour of a curve near one of its points 26

1.9 The curvature and torsion of a curve as the intersection

of two surfaces 31

1.10 Contact between curves and surfaces 35

1.11 Osculating circle and osculating sphere 37

1.12 Locus of centres of spherical curvature 43

1.13 Tangent surfaces, involutes and evolutes 48

1.14 Betrand curves 57

1.15 Spherical indicatrix 61

1.16 Intrinsic equations of space curves 67

1.17 Fundamental existence theorem for space curves 69

viii Contents

2.5 Curves on surfaces 108

2.6 Tangent plane and surface normal 109

2.7 The general surfaces of revolution 114

2.8 Helicoids 117 2.9 Metric on a surface—The first fundamental form 118

2.10 Direction coefficients on a surface 123

3 Geodesies on a Surface 160

3.1 Introduction 160 3.2 Geodesies and their differential equations 160

3.3 Canonical geodesic equations 173

3.4 Geodesies on surfaces of revolution 174

3.5 Normal property of geodesies 178

3.6 Differential equations of geodesies using normal property 182

4 The Second Fundamental Form and Local

Non-intrinsic Properties of a Surface 279

4.1 Introduction 279 4.2 The second fundamental form 279

4.3 Classification of points on a surface 284

4.4 Principal curvatures 290

4.5 Lines of curvature 296

4.6 The Dupin indicatrix

4.7 Developable surfaces

4.8 Developables associated with space curves

4.9 Developables associated with curves on surfaces

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 E 3 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 s y 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

1.2 REPRESENTATION OF SPACE CURVES

Whenever we use the word space, it means the Euclidean space of dimension three

denoted by E y 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

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 F 2 of (1) satisfy the following conditions of the

implicit function theorem

(i) Let F { and F 2 have continuous partial derivatives of first order

(ii) At least one of the following three Jacobian determinants

d(F^F 2 ) d(F l9 F 2 ) d(F ly F 2 ) d(y,z) ' d(z,x) ' d(x,y)

is different from zero at a point (x 0 , y 0 , z 0 ) 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 -f 2 M as their solutions Having obtained these values ofy

and z in terms of x, we take the parametric representation as

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

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)

Theory of Space Curves 3

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

A most straight forward method of elimination gives

We know that y 2 = xz represents a parabolic cylinder and x 2 = 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 -y 2

which are the equations of the two required surfaces specifying the cubic curve

Though y 2 = 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

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 C m 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 C m functions defined on different intervals I { , I 2 etc For

example we can have a path of the same class defined on I x and I 2 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

Theory of Space Curves 5

Definition 6 Two paths Rj and R2 of class C n 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 1 2 such that Rj = R2° 0

If we takeRj = (x lt y u Z\) and R2 = (x 2 , y 2 , z2), then the above conditions are the

same as

x x {u) = x 2 ((p(u)X y x (u) = y 2 ((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 I 2

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 I 2 such that R, = R2o 0 Since 0 is a

strictly increasing function on /, onto I 2 with 0 # 0 , the inverse function

0_ 1 exists as a strictly increasing function on I 2 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 l x onto I 2 such that

In a similar manner, there exists a strictly increasing function i/^on I 2 onto

73 such that

Since 0:1 { —> I 2 and y/: l 2 —> 73 are strictly increasing functions yo 0is a

well-defined strictly increasing function on I x onto 73 From (1) and (2) we have

Rj = R2 o 0 = R3o ^ 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 C m 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

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 E 3 is defined to be any set of points in

E 3 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 I x = [0, n)

(ii) R2(v) = 1 - vr ,2 2av „ , _i * T , 2b tan v

= (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)]

Theory of Space Curves 7

R2(v) = [2(1 - v2), 2v>/l - v 2 , 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

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 •) - R(«/-,)| + |R(«/) - R(M-)|

Since a * b y as A varies over all possible subdivisions of [a, b], we obtain a

non-empty 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

By Schwarz inequality, we get

Y P R G O du\ < Y ["'\R(u)\du = f\R(u)\du (4)

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(u 0 ) gives the arc

length between u 0 and u Since we have defined the arc length as the least upper

bound of (L(A)}, we have from (4)

Hence S(u 0 ) exists and has the value S(u 0 ) = |R(w0)|

Since (8) is equally true for any parameter u Q 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

Ja

(8)

.(9)

where s denotes the length of the curve from a to u

Corollary In terms of cartesian parametric representation,

s = \ y]x 2 + y 2 + z 2 du

Ja

Proof In cartesian parametric representation,

let R(u) = x(u)i + y(u)j + z(u)k

Then |R(w)| = yjx 2 +y 2 + z 2 Using this expression in (9), we get

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

Tlieory of Space Curves 9

Further, since s = |R|, s 2 = x 2 + y 2 + z 2 which gives in terms of differential

ds 2 = dx 2 + dy 2 + dz 2

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) = R2(0 = R2[0(«)L As u varies from a to b,

t = 0(w) varies from (j)(a) to <p(b) and R,(w) = R2(0 (1)

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, £)

So s = S(u) = f |r(w)| du = f yja 2 + b 2 du = cw (2)

Jo Jo

where c = yja 2 + b2

If a helix starts from w0, it makes one complete turn when u = u 0 + 2n Hence

the arc length corresponding to one complete turn is

u 0 + 2n

s= [ yja 2 + b 2 du = yja 2 + b 2 • In = Inc

"0

To obtain the equation of the helix with s as parameter,

we have from (2), s = cu so that u = — (3)

c

Using (3) in (1), we get

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 u 0 in the neighbourhood u

r(w0 + h) = r(w0) + — r(w0) + — f (w0) + + — r('°(w0) + 0(h n )

I! 2 ! n!

where lim = 0 where u-Un = h

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 u 0 and u respectively The parameters of

P and Q are very close together in the sense that when Q -» P, u - u 0 —> 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

Theory of Space Curves 11

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

(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

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

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

non-vanishing derivative of r such that r' r(k) = 0 So if r(k) * 0, we have from Taylor

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

Theory of Space Curves 13

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

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, y y 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 r2 = r(u 2 ) 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/(M 2 ) = 0

Now we have the intervals [u 0> w,] and [w,, u 2 ] Hence by Rolle's Theorem,

there exist points v, G [W0, U { ] and v 2 G [W,, U 2 ] such thatf\v { ) = 0 and//(v2) = 0

S i n c e / ' satisfies the conditions of the Rolle's Theorem in [v,, v 2 ], there exists

av3G [uj, v2] such that/7/(v3) = 0 Hence when Q and R approach P,u lt u 2 , v,, v2,

v3 approach u 0 Writing u for u 0 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

[ 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

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 u t b) (2)

r = (- a cos u,-a sin w, 0) (3)

Using (1), (2), and (3), the equation of the osculating plane is

X - a cos u Y- asm u Z - bu

- a sin u a cos u b -acosu -a.s'mu 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

the normal, then the vector PQ = An where A is a scalar Then R = r + An

is the equation of the normal at P

Definition 3 The normal at P orthogonal to the osculating plane is called the

binormal at P The unit vector along the binormal is denoted by b The sense of the

unit vector b along the binormal is chosen such that t, n, b form a right handed system of axes

The behaviour oft, n, b at a point P on the curve is the same as the unit vectors

ij\ k, along the coordinate axes Hence we have

b = t x n , t = n x b , n = b x t

and tn = nb = bt = 0, tt = bb = nn=l

Definition 4 The plane containing the tangent and binormal is called the

rectifying plane

From the above definitions, we have

(i) The equation of the rectifying plane If r = r(w) is a point on the curve and

R is the position vector of any point on the rectifying plane, then (R - r) is

in the rectifying plane and orthogonal to n Hence (R - r) • n = 0 is the

equation of the rectifying plane

Note Since the binormal is orthogonal to the osculating plane, the equation

of the osculating plane can be put in the form (R - r) • b = 0

(ii) The equation of the binormal Let r = r(w) be the position vector of any

point P on the curve and let R be the position vector of any point Q on the binormal Then PQ = /Jb where ji is a scalar Then R = r + jib is the equation of the binormal at P

Note Since r, r lie in the osculating plane, r x r is perpendicular to the

osculating plane Since the binormal is perpendicular to the osculating plane, the binormal is parallel t o r x r

Since r gives the direction of the tangent and r x f gives the direction of the binormal r x (r x r) gives the direction of the principal normal Thus we get

r x (r x r) = (r- r ) r - (r- r ) r is the direction of the principal normal

Note When we take the arc length as parameter, then we have Z 1 = 1 and consequently r' • r" = 0 Using this in the above formula, the principal normal is parallel to r"

Summarising the above we conclude that if we choose the arc length as parameter, then r', r" and r' x r " give the directions of the tangent, principal normal and binormal at a point on the curve

Example Find the directions and equations of the tangent, normal and binormal

and also obtain the normal, rectifying and osculating planes at a point on the circular helix

r= h os 0' asin @'7)

Theory of Space Curves 17

Now r = sin - , - cos - , -, (-a (s\ a fs} b)

(1), (2) and (3) give the directions of the tangent, normal and binormal

Next let us find the equations of the tangent, normal and binormal

The equation of the tangent is R = r + At which gives

Let us find the equations of three planes

The equation of the normal plane is (R - r) • t = 0 which gives

a

(2)

(3)

ac cos \- \Y-acs\n\- \ X+(Zc-bs)b = 0 The equation of the rectifying plane

is (R - r) • b = 0, which gives cos - \ X + sin - Y- a = 0

The equation of the osculating plane is (R - r) • b = 0 Since b has the direction

r' x r", we get the above equation as (R - r) • (r' x r") = 0 (4) Writing (4) in the determinant form, we obtain

which is the equation of the osculating plane

1.7 CURVATURE AND TORSION

At each point of the curve, we have defined an orthogonal triad t, n, b forming a right handed system and also we have noted that at each point this moving triad determines three fundamental planes which are mutually perpendicular

It is important to note that t, n, b vary from point to point on the curve as the point moves on the curve as shown in the following figure

Rectifying plane

Definitoin 1 The arc rate at whch the tangent changes direction as the point

P moves along the curve is called the curvature vector of the curve and it is denoted

by/C

Theory of Space Curves 19

As we have already noted, the osculating plane of a plane curve is the same at all points but the osculating plane changes from point to point on a space curve So

we shall now define a quantity measuring the arc-rate of change of the osculating plane

Definition 2 The torsion at a point P of a curve is defined as the arc-rate at

which the osculating plane turns about the tangent atP as P moves along the curve

It is denoted by r 11/T | denoted a is called the radius of torsion

Having defined the torsion at a point on the space curve, the next question is to devise a method of measuring it The natural method of measuring the turning of a plane is to measure the turning of its normal Since the binormal is orthogonal to the osculating plane, we shall use the binormal to measure torsion Thus the arc-

Theorem 1 (Serret-Frenet Formulae) If (t, n, b) is the moving orthogonal

triad of unit vectors at a point P on a space curve y, then

dt dn db (i) — = Kit (ii) — = r b - Kt (iii) — = - rn

ds ds ds

Proof We first prove (i) and (iii) and then derive (ii) from them

To prove (i), differentiating t • t = 1 with respcet to s at a point P of the curve,

we have t • t ' = 0 so that t' is perpendicular to t

dv

Since t = — , t' = r" As r " lies in the osculating plane, t' also lies in the

ds

osculating plane Therefore t' is a vector perpendicular to t and lies in the

osculating plane Hence t' is parallel to the principal normal By definition |t'| = K, being the curvature at P on the curve Since we know the magnitude K and the

direction n oft', we can write t' = ± /en By convention we take t' = KTI

(ii) As in the above case, we find the vector b' Differentiating b- b = 1, we have

b b ' = 0 so b' is perpendicular to the binormal at P and hence b ' lies in

the osculating plane Since b • t = 0, differentiating this and using (i) we get

b'-1 + b- (K*n) = 0 As b• n = 0, we get b'-1 = 0 showing that b ' is perpendicular to

t Therefore b ' is a vector perpendicular to t and lies in the osculating plane Hence

b ' is parallel to the principal normal at P By definition |b'| = T, being the torsion at

P Since we know the magnitude r and the direction n of b', we can write b ' = - r n

where the negative sign is introduced because as a convention torsion is regarded

as positive when the rotation of the osculating plane as s increases is in the

direction of a right handed screw moving in the direction oft

To prove (iii), let us consider n = b x t

Differentiating both sides of the above vector with respect to s,

Theorem 2 A necessary and sufficient condition for a curve to be a

straight-line is that K = 0 at all points of the curve

Proof The condition is necessary Let us take the curve to be a straight line

and let its vector equation be r = as + b where a and b are constant vectors

Differentiating this equation, we get r' = t = a

As a is a constant vector, we have r" = t' = 0 Since the curvature vector r "

vanishes at all points of the curve, its magnitude K= 0 at all points of the curve

To prove the sufficiency of the condition, let us assume that K= 0 at all points

of the curve This implies that the curvature vector t' = r " = 0 Integrating the

above equation twice, we obtain r = as + b where a and b are constant vectors This

proves that the curve is a straight line

Theorem 3 A necessary and sufficient condition that a given curve be a

plane curve is that r = 0 at all points of the curve

Proof The condition is necessary Let us take the curve to be a plane curve

and show that r = 0 Since the curve lies in a plane, the osculating plane at every point of the curve is the plane containing the curve itself so that the binormal b is a constant Since b is constant — - 0 which implies

all points of the curve

Conversely let r = 0 at all points of the curve On this assumption, we shall

prove that the curve is a plane curve S.ince T= 0, — = 0 at all points of the curve

ds

so that b is a constant vector Now for any vector r,

d , ix dx db ,

— (r b) = b + r = t b + r b'

Theory of Space Curves 21

Since t b = 0 and 1/ = 0, we have — ( r b) = 0 for any point r on the curve

ds

Hence r • b = constant = c (say) If r = (x(s), y(s), z(s)) and b = (fej, b 2 , &3), then

r • b = c gives xb x + yb 2 +zb 3 = c which shows that r(s) = (x(s) 9 y(s), z(s)) lies on the

plane b x X + b 2 Y+ b 3 Z = c This proves that the curve is a plane curve and the

condition is sufficient

Definition 3 If ris non-zero, then the curve is called a twisted curve In the

following, we shall give the formulae for curvature and torsion first in terms of the

arc length and then in terms of a general parameter u We shall use dashes to denote

differentiation with respect to s and dots to denote differentiation with respect tow

Theorem 4 If r = r(s) is the position vector of a point P with arc-length as

parameter on a curve c, then

(i) K ^ r " r "

\r' r" r'"l

(ii) T =L r' „ ' , , or K^tr'.r".!-'"]

r -r

Proof, (i) We know that r' = t and r " = KU (1)

Hence r " r " = (fen)• (/en) = /c2, since n n = 1

Since t is a unit vector, (1) gives | r| = s (2)

Differentiating (1) again with respect to w,

r = — 2 " = — ( t j ) = — ( t j ) — = t f j 2 + t j

du du ds du

Using (1) and.(3), we get r x f = t i x (KUS 2 + ti")

Since t x n = b and t x t = 0, we get

[r, r , r ] = - i V r From the property of the scalar triple product,

Theory of Space Curves 23

From Theorem 4, [r', r", r'"] = K*TSO that [r, r, r] = K^TCW')" 6

Further we obtain [r', r", r"'l = (u'f [ r , r , r ]

Corollary A necessary and sufficient condition that a curve is a plane curve

i s [ r , r, r] = 0

Proof To prove the necessity of the condition, let us assume that the curve is

a plane curve; Then T = 0 by Theorem 3 Since [r, f, r] = K 2 T, T = 0 implies

[r, f , f ] = 0

To prove the converse, let [r, f, f ] = 0 Then K 2 T = 0 Hence either K = 0 or

T= 0 We shall prove that T = 0 at all points of the curve Let r * 0 at some point of

the curve Then x * 0 in some neighbourhood of that point Since K = 0 in this

neighbourhood, the arc of the curve in this neighbourhood is a straight line This implies that T = 0 on this line in the neighbourhood of this point contradicting the hypothesis that T* 0 This contradiction proves thatr= 0 at all points of the curve

so that the curve is a plane curve Thus the condition is sufficient also

Note In terms of the dash derivatives, we have [i/, r", i*"'] = u' 6 [r, r, r]

Since u = — * 0, [r, f, r ] = 0 if and only if [r', r", rw ] = 0 which is the necessary

ds

and sufficient condition for a space curve to be a plane curve

Example 1 Find the curvature and torsion of the circular helix

r = {a cos w, a sin «, bu)

From the equation of the helix, we have

Note Sometimes, it is easier to deal with dash formula for curvature and

torsion than the dot formula To illustrate this, we find curvature and torsion of the circular helix by dash formula

Using the arc-length as parameter, position vector of any point on the circular helix is

Theory of Space Curves 25

Differentiating (5) with respect to u again

b — ( i3 K ) - i4 Km = 2(6«, - 3 , 0) (6)

du

Taking dot product of (4) and (6), we get

(jft+i 2 icn)-[b—(i 3 £)-i 4 KTn] = -12

Since b t = b n = n t = 0and n n = 1, we get

- S 6 K 2 T= - 12 so that /C 2 T= -r (7)

i6

Taking dot product of (5) with itself on both sides and using (3)

• c ^4'9" ' :9" ' :1' <8) (l + 4w2+9w4)3

Using (8) in (7), we get T = — - — — - 2—

-(9w4 + 9u l + 1)

Example 3 Find the curvature and torsion of

r = {a cos 6, a sin 0, aO cot a) (1) Instead of using the formulae for K and r, we shall find K and T from the

derivatives of (t, n, b) with the help of Serret-Frenet formulae

Differentiating (1) with respect to s, we get

2 + a 2 cot2 a so that— = a cosec a

Taking dot product on both sides of (2) with itself

= a 2 + a 2 cot2 a so thai

d9

From (2) we have t = sin a (- sin 0, cos ft cot a) (3)

Differentiating (3) with respect to s,

dt ds

= sin a(- cos 6, - sin ft 0)

ds dd

Using Serret-Frenet formula, we get

Kii = (- cos ft - sin ft 0) (4)

a

Taking dot product on both sides of (4) with itself,

o sin4 a sin2 a

IT = — 2 ~ ~ ~ o r K =

a a

Using this value of fcin (4), n = (- cos 0, - sin 0, 0)

Since we know t and n, we have

b = t x n = (cos a sin ft - cos a cos ft sin a)

Hence — = - rn = (cos a cos ft sin 6 cos a, 0) — (4)

ds ds

Taking dot product on both sides of (4) with itself,

9 , 2 / 1 2 • 2n 2 Nsin2a sin a cos a

T = (cos 6 cos a + sin 0 cos a) —~— glving Tas T=

a a

1.8 BEHAVIOUR OF A CURVE NEAR ONE OF ITS

POINTS

Using Serret-Frenet formula, we shall study the behaviour of a curve in the

neighbourhood of a point on the curve In the following theorem, we obtain the

coordinates of a point near the given point on the curve with reference to the

coordinate axes along t, n, b and then deduce a number of properties from them

Theorem 1 Let the curve be of class m > 4 At a point P on the curve, let the

coordinate axes, ox, oy, oz be taken along t, n, b If X, Y 9 Z are the coordinates of the

neighbouring point Q on the curve, then

, , / m r^O) r"(0) 2 r'"(0) 3 r(/v)(0) 4 , 4N A *"(1)

r(^) = r(0) + - ^ - ^ J + — — r + — ^ - ^ r + * — r + o(r) as 5 -> 0

1! 2! 3! 4!

where s is the small arc PQ and r (0) = 0

To study the equation (1), let us find r', r", r'" and r(,v) at the origin 0

(i) 1/(0) =? t

(ii) r,/(0) = t,= x-n

Theory of Space Curves 27

If X, y, Z are the coordinates of the neighbouring point Q with position vector

r(s) with reference to the coordinate system 0(*, y, z) in the direction oft, n, b, then

express the coordinates X, Y> Z in terms of the arcual length PQ = s •

(X, K, Z) is called Serret-Frenet approximation of the curve

Using the above equation, we have the following deductions

Neglecting the powers of/, we have

(X 2 +K 2 + Z 2) =S 2 K 2 S 4 =S 2 (l K 2 S 2 )

12 I 12 J

Thus (x2+r2+z2)1/2 =/i ^x-Vj

Using the Binomial expansion on the right hand side,

(x 2 + Y 2 +Z 2 } ~ s 1 K-V which shows that when *•* 0, the arc length

PQ differs from the chord PQ by a term of the third order in s

(iii) We obtain the approximate equations of the projections of the curve on

three planes In each case we obtain the lowest power of s and eliminate s

between the two equations Y = — X

(b) The projection of the curve on the rectifying plane is

z=— x\y=o

6

Theory of Space Curves 29

KT 3

From the equations, we obtain as in (i) X ~ s and Z s

6 Eliminating s between them, Z = — X3

r - 1 7T 7T

As in Examples 1 of 1.4 and 1.7, s = V2w, K- T= — and ^ = -y=r at w = — so

2 V2 2 the curve can be represented as

f \ ( S S S

r(s) = [ « » - £ , a n - j - ^

Using the Theorem 1, we have

4 6^ 7 2 , Since we know fcand T, we can find the projections of the curve on the planes using (a), (b) and (c) of (iii)

Theorem 2 The length of the common perpendicular d between the tangents

at two neighbouring points with the arcual distance s between them is

3

KTS

approximately d = 12

Proof Let 0 and P be two adjacent points on the curve with OP having the

arcual length s

Let the position vector of P be r = r(s) Let us choose the cartesian coordinate

system at O in the direction of the triad (t, n, b) and so r(0) = 0

The unit tangent vectors at O and P are r'(0) and r'(s) The common perpendicular of these two tangents at O and P is nothing but the shortest distance

between them Since the shortest distance between them is perpendicular to both

r'(0) and r'(s), it is parallel to r'(.s) x r'(0) Thus if d is the shortest distance between them, it is the projection of PQ = r(^) - r(0) on r'(.s) x r'(O)

Since t, n, b give the fixed direction of the coordinate axes at 0, differentiating

the above equation with respect to s,

Neglecting the terms of higher order > 4, we get

|I"(J) x r'CO)!2 = J V + Kits 3 = K V [ 1 + —s)

K' V/2 ( 1 K"

so that \r\s) x r'(0)| = KS 11 + — s \ = KS 1 + s\ approximately

Thus r W t r ' W x r'(O)] = (KS2 s + —K 3 V n2 <

SK + K