Weakened Bertrand Curves

7/29/2019 Weakened Bertrand Curves

1/15

T o h o k u M a t h . J o u r n .Vol. 19, No. 2, 1967

WEAKENED BERTRAND CURVESH O N - F E I LAI

(Received November 30, 1966)

1. Introduction. In the discussion of Bert rand curves (in fact nearly allcurves) in elementary classical differential geometry, it is always assumed(explicitly or implicitly) that the curvature is nowhere zero. In th is paperwe drop this requirement on the Bertrand curves and investigate into theproperties of two types of similar curves (the Frenet - Bertrand curves and theweakened Bertrand curves) under weakened conditions. The properties oft h e Frenet - Bertrand curves tu rn out to be strikingly similar to those of theBertrand curves.

We first put down our convention of defining regular curves. We alsouse the convention that if f is a real function on a set A, by f=0 we meant h a t f is everywhere zero on A, and by f 0 we mean that f is nowherezero on A.

D E F I N I T I O N 1.1. A parametrized curve (simply called a curve) in E3is a point set in E3 together with an equivalence class of continuous andlocally injective surjections

( , , ) : L - > defined by

where two such mappings ( , , % ), ( , , ) which are defined on two intervalsL , L respectively are said to be equivalent if th ere exists a continuous, strictlym o n o t o n i c increasing function

: L- >L

with L as image set ( must then be a bijection from L onto L) such that

E a c h of the mappings ( , , %) is called a parametrization of .


2/15

142 H N - F Ei LAID E F I N I T I O N 1.2. A Cr curve (r 1, 2, , or oo or ) is a curve which

admits a parametrization ( , , % ) in which , , % are of class C r.D E F I N I T I O N 1.3. A Cr regular curve is a curve which admits a para-

metrization ( , , ) in which , , are of class Cr and % are of class Cr. (It follows that '2- \- '2- \ - '2 = l, where the dashdenotes differentiation with respect to s).

REMARK 1. AC 1 regular curve which is also a Cr curve need not bea Cr regular curve if r > 1. For example, consider th e analytic curve :

x = t\ y = t\ z = 0,which is a C 1 regular curve because it admits the representation

a n d the function x is of class C 1. However, cannot be a Cr regular curvefor any r > 1.

REMARK 2. A Cr regular curve can have a parametrization ( , , %)which is of class Cr and such t h a t 2 + 2 + %2 is zero at some points. Forexample, consider the straight line

x = t\ y = 0, 2 = 0 .I n the following, we shall only consider C regular curves, although the

results obtained also hold for any Cr regular curve with sufficiently large r.


3/15

WEAKENED BERTRAND CURVES 143DEFINITION 1.4. A Bertrand curve : x(s\ se L is a C regular curve

with non- zero curvature for which th ere exists an oth er (different) C regularcurve : x(s\ where x(s) is of class C an d J C ' ( s ) ^ 0 (5 being th e arc lengthof only), also with non- zero curvature, in bijection with it in such amanner that th e principal normals to , at each pair of correspond ing point scoincide with th e line joining th e corresponding point s. Th e curve is calleda Bertrand conjugate of .

I t easily follows that the relation between a Bertrand curve and its con-jugate is symmetric.

I n th is paper we weaken the conditions. We shall adopt th e definitionof a C Frenet curve ([2]) as a C regular curve : x(s\ s L, for whichthere exists a C family of Frenet frames, that is, right-handed orthonormalframes t(s), n(s), 6(5), where t{s) = x\s\ satisfying the Frenet equations

for some C scalar functions k (s\ k2(s) which are called th e curvature andpseudo- torsion of .DEFINITION 1.5. A Frenet- Bertrand curve : x{s) (briefly called a FB

curve) is a C F renet curve for which th ere exists an othe r C F renet curve : x(s), where x(s) is of class C an d x'(s) 0, in bijection with it so that,by suitable choice of the Frenet frames (see [4]), the principal normal vectors71, n at correspond ing points on , both lie on th e line joining th e cor-responding poin ts. T h e curve is called a FB conjugate of .

Again th e relation between a FB curve and its conjugate is symm etric.DEFINITION 1.6. A weakened Bertrand curve : x{s\ se L (briefly

called a WB curve) is a C regular curve for which th er e exists another Cregular curve : x(s\ s L, where s is the arc length of , and a homeomor-phism : L > L such that( i ) th ere exist two (disjoint) closed subsets Z, N of L with void interiors suchthat C00 on L\N, (ds/ds) = 0 on Z, ~ e C on o'(L\Z\ and (ds/ d7) = 0 on (iV),(ii) th e line joining correspond ing po ints 5, s of and is orthogonal to


4/15

144 HON-FEI LAIand at thepoints 5,s respectively, and is along theprincipal normal to or at thepoints s, s whenever it is well defined. The curve is calleda WB conjugate of .

Thus for a WB curve we not only drop the requirement of being aFrenet curve, but also allow (ds/ds) to be zero on a subset with void interior((ds/ds) = 0 on an interval would destroy the injectivity of the mapping ).Since (ds/ ds) = 0 implies that (ds/ds) does not exist, theapparently artificialrequirements in (i) are in fact quite natural. The relation between a WBcurve and its conjugate is again symmetric.It is clear that a Bertrand curve is necessarily a FB curve, and a FBcurve is necessarily a WB curve. It will be proved in Theorem 3.1 thatunder certain conditions a WB curve is also a FB curve, while it will beclear from Lemma 2.2 that a FB curve need not be a Bertrand curve.

2. Frenet- Bertrand curves. In this section we study the structureandcharacterization of FB curves. We begin with a lemma, themethod used inwhich is classical (see [3]).

LEMMA 2.1. Let : x(s), szL be a FB curve and : x(s) a FB con-jugate of . Let all quantities belonging to bemarked with a tilde. Let(2.1) c(s)= x(s)+ (s) n(s).Then the distance | | between corresponding points of , is constant, andthere is a constant angle o such that t t =cos a and( i ) (1Xk^)sm.a = Xk2cos a,(ii) (l+S k^sina = Xk2cos a, with S=l,(iii) (l- &OC l + ) = cos2 a,(iv) k2\ = sin2 a.

A

PROOF. From (2.1) it follows that X(s) =(x(s)- x(s))- n(s) is of class C.Differentiation of (2.1)with respect to s gives

(2. 2) 1 = (l-Xk^t + '/ - Xk2b .Since by hypothesis we have n=8n with = 1 , scalar multiplication of (2.2)


5/15

WEAKENED BERTRAND CURVES 145by n gives

' = 0, = con s t an t .The re f ore(2.3 ) 1 = (l-Xk^t + Xk2b .Bu t by definition of FB curve we have s 0, so t h a t t is C function of s.H e n c e

4- ( t ) = k t n T + tik sn)as= 0 .

Consequent ly f is constan t , and t h e r e exists a con s t an t an g le a s u c h t h a t(2 .4 ) = co s tf + sin & .T a k i n g th e vector p roduct of (2. 3) an d (2. 4), we obta in

( 1 Xk^)s \nct = \ k2 cos a,which is (i) .N o w w r i t e

x = x xn .T h e r e f o r e(2.3 ' ) t = '[(l+X*kSt - l 2 6] .O n t h e ot he r han d , eq ua t ion (2 . 4) gives

b = x n= sin at + Sco s


6/15

146 HON-FEI LAIOn th e o ther hand, comparison of (2. 3) and (2. 4) gives

(2.5) 1 - Xki = Tcostf ,(2. 6) Xk2 = s sin a.Similarly (2. 3'), (2.4') give

(2. 5') 7(1+ 8 I ) = cos a,(2. 6')The properties (iii) and (iv) then easily follow from (2. 5) and (2. 5'), (2. 6) an d(2.6').

LEMMA 2.2. A necessaryand sufficient condition for a C regular curve to be a FB curve with a FB conjugate which is a line- segment is that should be either a line- segment or a non- planar circular helix.

PROOF. Necessity: Let : x(s) have a FB conjugate V: x(s) which isa line- segment. Then k1=0. U sing Lemma 2.1 (iii) an d (i), (ii), we have(2.7) 1 - *! = cos2 a,and then(2. 8) cos2 a sin a = Xk2 cos a,(2. 9) sin a = k2 cos a.From (2. 9) it follows that cos a 0. H ence (2. 8) is equivalent to(2. 8') &2 = sin a cos a.

Case 1. sin a = 0. Then cos = 1 , so that (2. 7) implies that & = 0,and is a line- segment. We no te also that (2. 8') implies that k2 = 0.

Case 2. sin a ^ 0. Th en cos ^ 1 , and (2. 7), (2. 8') imply that k l9 k2are non- zero constants, and is a non- planar circular h elix.Sufficiency: If is a non- planar circular helix

x = (a cos , a sin , 6), where t=s/*Ja2- \ - b2 \


7/15

WEAKENED BERTRAND CURVES 147we may take

n = ( cos t, sin t, 0) . Now put = a ,T h e n the curve with

X = JC + / lwill be a line- segment along the 2-axis, and can be made into a FB conjugateof if n is defined to be equal to n.

T h e case where is a line- segment is trivial.DEFIN ITIO N 2.1. By a plane arc we mean a plane curve which containsn o line- segments.L E M M A 2.3. If a FB curve has a FB conjugate which is a plane arc,then is a plane arc on the same plane.PROOF . Let a FB curve : x(s\ s L, have a FB conjugate : x(s)

which is a plane arc. Since does not con tain any line- segments, th e (closed)subset M of L on which k = 0 has a void interior, so that L\M is dense inL . But since is a plane curve, we must have

s' \ k2 = Ix, x"9 x'''\ = 0(see [4]). Therefore k2 = 0 on L\M and hence also on L, by continuity.Therefore Lemma 2.1 (iv), (i) give first

sin a = 0 ,a n d then

k2 = 0.T h u s is a plane curve. Moreover, cannot contain any line- segments,otherwise by Lemma 2.2 must also contain a line- segment . Th us is ap l a n e arc, which obviously lies on th e same plane as .

D E F I N I T I O N 2.2. A Bertrand arc is a FB curve which contains no line-segments or plane arcs and whose FB conjugate also has this property.N o t a t i o n s : We use the symbols S, % ), 33 to denote line- segment, plane

arc, non- planar circular helix and Bertrand arc respectively.


8/15

148 HON-FEI LAIDEFIN ITIO N 2.3. A FB curve is said to be of 2 type if it is a non-planar circular helix and its FB conjugate is a line- segment. Similarly we

define FB curves of 2>, 22, $5 and 2333 types.L e m m a 2.2 can t hen be restated: a FB curve which is a line- segment is

of either 22 or 2> type. On the other hand, a FB curve which is a non-planar circular helix need not be of )2 type, for it is well-known t h a t an o n - p l a n a r circular helix has c Bertrand conjugates (c being the cardinalityof the continuum) which are also non- planar circular helices.T h e o r e m s 2.1 and 2.2 below show t h a t , apart from th e cases treat ed inL e m m a 2.2, a FB curve either is a plane curve consisting of arcs of 22 or $ type or is a non- planar curve consisting of arcs of 2, 2 or 3323 type.

We shall have to use the following ([4])L E M M A A. Let fi, ,fn be continuous (real- valued) functions ofwhich f is defined on a proper interval L of the real line, and f (2gz5/z)

is defined on the set Gi^.^{s L: f(s) ^ 0, ,/i_1(5)^=0). Then there existn-\-l open sets B , B2, , Bn, Gn of L with the following properties :/ 1 = 0 on Bl9fx 0 and f2 = 0 on B2,

/ i^O, / 2 ^ 0 , . . . , fn^ Q and fn=0 on Bn,/ i ^O , / 2 ^ 0 , . . . , and fn 0 on Gn;

andI = Sj u 2 u u ~ B n u Gn,

where the closure operation is taken in R. Thus, the component intervalsof JB1? B2, , Bn, Gn, taken together, form a countable family of disjointproper intervals each of which is open in L, and the union of these com-ponent intervals is a dense subset of L.

THEOREM 2.1. / / a FB curve contains a plane arc, then (i) it is aplane curve with zero pseudo- torsion and has a dense subset which is theunion of a countable number of FB curves of 22 or $5 type, and the FBconjugate lies on the sarne plane as (ii) the curvatures of and of are bounded below or bounded above.

PROOF . (i). Let : x(s), s z L be a FB curve which contains a plane


9/15

WEAKEN ED BERTRAND CURVES 149arc cor responding to an interval I C L . L e t : x(s) b e t h e FB conjugate of un de r d iscussion .As proved in Lem m a 2.3, we have s in t f= 0 (see n ota t ions in Lem m a 2.1).C on sequent ly by L em m a 2.1 (i) an d (ii) , k 2 = k 2 = 0 o n L , an d , a r e p la n ecurves on the same plane .By Lemma A, L has a dense subse t which is th e un ion of two coun tablefam ilies 33i, 332 of disjoint intervals open in L such tha t k 1= 0 o n e a c h m e m b e rof 351 and k x 0 on each member of S5 2. By Lem m a 2.2, th e par t of cor -respon ding to any com pon ent interval of 35 is a l ine - segm ent , an d by Lem m a2.3, th e par t of cor respon ding to any com pon ent interval of 332 is a planearc .

( ii) S in c e si n t f = 0, c o s = l . H e n c e b y L e m m a 2.1 (iii) w e h a ve1 Xk 1 O. I t follows t h a t k is e i ther bounded below or bounded above by. S imi la r a rgum ents apply to .*THEOREM 2.2. If a FB curve contains no plane arc and is not a

line segment, then (i) it is non- planar and has a dense subset which is theunion of a countable number of FB curves of 2 ) , j%> or 3323 type, (ii) th epseudo- torsion k 2 is nowhere zero.

PROOF . L et : x{s\ s L be a FB curve which contains no plane arcsand is not a line-segment. Then there exists s0 L such that &i(50) 0. Ina neighbourhood of s0 on which k x 0 , we have a t leas t on e poin t wher e k 2is non - zero, o th erwise t his par t of would be a plan e arc . I t follows fromLemma 2 .1 ( i ) tha t sina O, an d th en from Lem m a 2.1 ( iv) an d ( iii ) th a tk 2 is n owh ere zero on L a n d t h a t k l9 k x cann ot be bo th zero a t any po in t .T h e l a t t e r fact shows th a t conta ins no a rc of type . I t follows fromLem m a 2.2 th a t any int erval of L on which k 0 corresponds to an arc of2 ) type , an d any interval on which k 1= 0 corresponds to an arc of ) 2 type .

Now consider an interval I cL on which k , k Q. T h e c o r r e s p o n d i n gpar ts 1? x of , t h en conta in n o l ine - segmen ts. S ince th ey a lso cann otcontain any p lane a rcs by hypothes is an d Lem m a 2 .3 , j i s a Be r t r an d a rcof 3333 type. M oreover, since is n ot a l in e- segmen t, i t m ust con tain at leaston e arc of ) S > or 3333 type, a n d is th erefore n on - plana r .T h e proof is th en com pleted by app lying L em m a A with n = 2, f x = k l9A = K

R EM AR K. I t should be no ted tha t a l though a non - p lana r FB curve maycontain arcs of & ) an d >Z types, no two of these arcs can be directly joinedtoge the r a t a po in t . T his follows immedia te ly from considera t ion of the


10/15

150 HON-FEI LAIcontinuity of k (or k). It also follows from Lemma 2.1 (i) that for a Bertrandcurve containing an arc of 2^ type, cos o 0.

Following the methods of [4], it is not difficult to construct a FB curvewith a countably infinite number of arcs of )2, 2> and 2325 types. We needonly to notice t hat on an arc of 2 ) (respectively |) 8) type, we have (ku k2,k u k2) = (0, - tan a, - sin2 a, sin a cos a) (respectively = (- sin 2 a,\ A A X / \ A- sin a cos a, 0, tan en), and that for fixed a with 0 < a < - ^- , thefunctions k2, ku k2 are uniquely determined by the function h using equations(i) to (iv) of Lemma 2.1.

We now study the converse problem: to find sufficient conditions for aF r e n e t curve to be a FB curve. In view of Theorems 2.1, 2.2, we need onlyconsider plane Frenet curves with zero pseudo- torsion whose curvatures arebounded below or bounded above, and non- planar Frenet curves whose pseudo-torsions are nowhere zero. (The case of a line- segment has already beentreated in Lemma 2.2.) *

T H E O R E M 2.3. Let : x(s\ seL be a plane C Frenet curve with zeropseudo- torsion and whose curvature is either bounded below or boundedabove {which is always the case if L is compact). Then is a FB curve,and has c FB conjugates which are plane curves, (c being the cardinalityof the continuum).

P R O O F . Let be a curve satisfying the conditions of the hypothesis.T h e n there are c non- zero numbers such that 1( s)< l / on L or ^i(5)>l/ on L. For any such , consider the plane curve (since n always lies ona plane containing ) with position vector

x = x + n.ThenJC' = (l-Xk^t.

Since 1\kx 0, is a C regular curve, and t t. It is then a straight-forward matter to verify that is a FB conjugate of .

T H E O R E M 2.4. Let : JC(S), s


11/15

WEAKENED BERTRAND CURVES 151for some constants , a with ^ O . Then is a non- planar FB curve.

PROOF. Define th e curve with position vectorx = x + \n.

Then, den oting differentiation with respect to 5 by a dash, we h aveJC' = ( l- i&O* + Xk2b.

Since k2 0, it follows th at is a C regular curve. Let quant ities belongingto be marked with a tilde. Th en

Hence

and, using (*),8t = cos at + sintfft,

where = + 1 or 1 according as \k2 an d since have th e same or oppositesigns. (N otice that from (*) we ha ve sin a 0). Therefores(dt/ ds) = (&! cos a k2 sin ) n .

N ow define n = n, k = { / s){kx zosc k2 sin ). Th ese are C functions of 5(and hence of 5), and

Further define b = txn an d k2 = (db/ds) n. The se ar e also C funct ionson L. It is then easy to verify th at with t h e frame {, n, b] and the func-tions kl9 k2, the curve I 1 becomes a C F renet curve. But n an d n lie on theline joining corresponding points of and . Th us is a FB curve and a FB conjugate of .

3. Weakened Bertrand curves.DEFINITION 3.1. Let D be a subset of a topological space X. A function

on X into a set Y is said to be D- piecewise constant if it is constant oneach component of Z


12/15

152 HON-FEI LAIWe shall use th e following lemm a ([1]).L EMMA B. L et X be a proper interval on the real line and D an open

subset of X. T hen a necessary and sufficient condition for every continuous,D- piecewise constant real function on X to be constant is that X\D shouldhave empty dense- in- itself kernel.

We n otice, however, th at if D is dense in X, any C 1 and D - p iecewisecons tant real fun ction on X must be cons tant , even i f D h a s n o n - e m p t ydense- in- i tself kern el .THEOREM 3.1. A W B curve for which N and Z have empty dense- in-

itself kernels {notations as in Definition 1.6) is a FB curve.P R O O F . L et : x(s\ s L be a WB curve and : JC(7), l a W B con-

juga te of . I t follows from th e defin i t ion tha t an d each has a C familyof tangent vectors t(s\ t(s). L et(3 .1) x(s) = x( ( s) ) = x(s) + (s) n(s) ,w h e r e n(s) is som e un i t vector funct ion a nd ( s ) ^ O is som e scalar funct ion.L e t D = L \ N , 7) = L \ ( Z) . Then 7(5) C on D, and 5(7) < C on D.

S t e p 1. T o p r o ve = c o n st a n t .S in c e = | J C J C | , i t is continuous on L a n d is of class C on everyinterval of D on which i t i s n owhere zero . Let P= [s z L : X(s) 0} and Xany componen t o f P. T h e n P, an d hen ce also X, is open in L . L e t / b ea n y c o m p o n en t in t e rva l of X Z T h e n on I , \ (s ) a n d n(s) axe of class C,a n d from (3.1) we have

JC'(5) = x(s) + '( 5) n(s) + ( $ ) n(s) .N ow by definition of a W B curve we have x'(s) n(s) = 0 = x'(s) n(s). H e n c e ,us ing the ident i ty n\s) n(s) = 0, we have

0 = '(5) (5) fl(5).Therefo re \ = cons tant on I .H e n c e i s cons tant on each com pon ent ( in terval) of th e set XnD. B u tby hypothesis X\ D ha s em pty dense- in- i tself kern el . I t follows f r o m L e m m a Bt h a t i s cons t an t (and non - zero ) on X. Since i s con t inuous on L , X


13/15

WEAKENED BERTRAND CURVES 153must be closed in L. But X is also open in L. Therefore by connectednesswe must have X=L, that is, is constant on L.

Step 2. To prove the existence of two frames {t(s\ n(s), b(s)}, {((s), n(s),b(s)} which are Fren et frames for , on D, D respectively.Since is a non- zero constant, it follows from (3.1) that n(s) is continuouson L and C on D, and is always orthogonal to t(s) (by definition of WBcurve). Now define 6(5) = t(s) X n(s). Then [t(s\ n(s), b(s)} forms a right-handed orthonormal frame for which is continuous on L and C on ZNow from the definition of WB curve we see that there exists a scalarfunction k^s) such that

t'(s) = kit/) n(s) on L .H e n c e k^s) t'(s) n(s) is continuous on L and C on D. Thus the firstF r e n e t formula holds on D. It is then straightforward to show th at th ereexists a C function k2(s) on D such that the Frenet formulas hold. Th us[t(s)9 n(s)9 b(s)} is a Frenet frame for on D.Similarly there exists a right- handed orthonormal frame {t(s\ n(s\ b(s)}for which is continuous on L and is a Frene t frame for on D. Moreover,we can choose

n( {s)) = n(s).Step 3. To prove that N= 0, Z =We first notice that on D we have

( )f = ^5 Xi + *i n T = 0,so that t t is constant on each component of D and hence on L by Lemma B.Consequently there exists a constant angle a such that

t(s) = cos a t(s) + sin a b(s) on L .F u r t h e r ,

($) = n(s)and so

6(5) = sin a t(s) + cos a b(s).Thus (s), #1(5), 6(5) are also of class C on Z On the other hand, t, n, b

are of class C with respect to s on D. Writing (3.1) in the form


14/15

154 H O N - F E I LAIJC = X l

a n d differentiating with respect to s on Df) ~\D), we havet = 7 '[ (1 + Xk.jt - Xk2 6 ] .

But = cos at sin rt 6 .

H e n c e(3.2) 7(l+ i) = costf.Since k1(s) = (dt/ds) n is defined and continuous on L and


15/15

WEAKENED BERTRAND CURVES 155The au thor wishes to thank Prof. Y. C. Wo n g for suggest in g this problem.

REFERENCES[1] H.F. LAI, O n piecewise- constant functions (to be published;.[2] K. NOMIZU, O n Frenet equations for curves of class C, Tohohu Math. Journ., (2)11(1959), 106-112.[3] C . E . WEATHERBURN, Differential geometry of three dimensions, vol. 1, Cambridge,1927.[ 4] Y. C. WONG AND H. F. LAI, A critical examination of the theory of curves in three-dimensional differential geometry, T hoku. Math. Journ., 19(1967), 1-31

D E P A R T M E N T O F MATHEMATICS,U N I V E R S I T Y O F H O N G K O N G ,H O N G K O N G .

Documents

Weakened Bertrand Curves