On two notions oî curvature

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X V II (1974) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE X V II (1974)

Z. W^

(Krakow)

On two notions oî curvature

Introduction.

There are two possible definitions of the curvature of the arc in terms of circles determined by three points of the arc, con­

cerning either the limit of radii or the limit of centres. In this note we construct an arc such that the former limit exists but the latter one does not exist.

To formulate the result more precisely we give now both notions called in this note g-curvature and C-curvature. Consider two sequences P n, Qn tending to P 0 as n oo, P n ф Р 0, Qn ф P 0, Qn Ф P n, n = 1, 2, ...

Denote by Cn the centre of the circumference passing through P n, Qn, P 0, and by gn its radius. If l/gn has the limit for any such sequences we say, that the ^-curvature exists and is equal to the limit. If Cn has the limit С, С ф Р 0, we say that the C-curvature exists and is equal to jC —P 01-1.

In this note we give an example of a space curve, which has ^-curvature at an interior point P 0 but has no C-curvature at this point. The idea of this example was suggested by A. Plié.

1. As in Golqb and Plis(1) we consider a curve of the type

(1) a = а2еЩа),

where a = cq, a = a 2-\-ias and oq, a2, a3 are real. Kow we put ( в (a) = ln( — ln|a|) for 0 < |a| < 1 ,

<2⁾

1/9(0) = 0.

Consider the points P corresponding to a = p and Q corresponding to a = yp, where 0 < p , у < 1. Write I = |P|, rj = \P — Q\, Ç = \Q\.

We have

I = (p2+p*)*,

(3) n = j^(l - y f p 2 + ( l - y 2) V + sin2 ’ C = (y2p 2 + /i>4)}.

(x) S. Golq.b and A. P lis, A rem ark on the curvature of non-plane curves, Colloq.

Math. 9 (1962), p. 127-130.

534 Z. W ç gl o w s ki

The radius

of the circumference passing through P , Q and the origin is equal to

(4) q = fyCUi + rj + C)~i (£ + ,n — £ ) '* ( £ + £ — r])~i (v + C — £)“ *.

We can easily obtain the following inequalities:

(

6

(6)

sin21 In In yp ln^>

1 41n2^>

3--У У

2

I,

sm l — y

In p 2y|lnp|

Using the Maclaurin developement we obtain

^ = p [ l + l / + 0 (p 4)],

(7) 7] = (1 - y )p |l + \p2^(1 + y )2 + — ■ sin2| l n + 0(P4)j>

C = y p [ i - H r 2P2 + 0 ( y 4p 4)],

where 0 (pn) denotes a function such that 0 (pn)jpn is bounded when p tends to zero.

4y2 lnyu

The te r m --- sinH ln--- ( 1 - y ) 2 2 In p because of (5).

By (3) and (7), we give

is tending to zero when p -> 0,

t - n ^ ( f — t W + v r 1

= jj)2 [1 - (1 - y)2] +i>* [ l - (1 - y2f - 4 / sin2 J1 n ^ - 1 } X

1(2 - y )p -h O ip 3)]-1

= y p l l + 0(p*)],

f - C = ( l * - £ 2 ) ( f + t r 1 =

[

2( 1 -

? +

Ч 1 - / ) Ж 1 +

)

+ 0 (

*)Г'

= ( l - y ) p [ l + 0 ( p * ) ] ,

v + Ç-Ç

= [fa + ÉT-ÉMfo + C + fT 1-

Let us consider the first brackets

(î? + t)2- l 2 = 7?2 + t 2- l 2 + 2î7C.

Two notions of curvature 535

Using (3) we have

*?2 + £2- ! 2

= (1 — y f p 2 + (1 — y2)2p4 + ly 2p4sin2 |1п-^П^ + y2p 2 + y4p4 —p 2 —p4 In p

= p

[ ( l

+ + l - l] + p

|^ (l-y

+ / - l + 4y

s in H ln ^ ^ -j

= - 2y ( l - y ) p 2 + 2y ( l ~ y ) p 4\ - y (1 + y ) в т Ц 1п ~ ^ \

L 1

— y

= - 2у ( 1 - у ) р 2- 2у{1 - у ) р * [

+

) + e(p)], where e(p) denotes a fnnctïon tending to zero when p ->

.

2y In yp

The term --- sin

f i n --- tends to zero when p -> 0, because

1

— y In p

of (

). Applying (7) we obtain

2

ijf = 2y ( l - y ) p 2{l + ip * Г(1 + у

+ у

+

=

y

- y )p ‘ {

+ èj)«

+ y

+ y* + e(p)]}.

4y2 lnyp

The term --- sin

A In--- tends to zero when p -> 0 , because (

- y

Inp

of (5), and instead of it we can write the function e(p).

Now we obtain

v2 + ^{C 2 — i2} + = 2y(i — y)p4[i+y + y2—y — y2+£(p)]

= y ( l - y ) p 4[ l + £(p)l- From (7) we have

£ + V + £ = 2 p [l + 0 (p 2)].

Then

V + C—i = y ( l — y )p 4

+ e (p)] •

p

+

(p

= i y { l - y ) P 3l l + s (P)'}- JSTow лее can write

= y ( l —y)p3[ l + 0 (p 2)],

£+??+£ = 2p [1 + 0 (p 2)],

i+ r j - C = 2 ( l - y ) p [ l + 0 (p 2)],

£+£-i? = 2 y p [ l+ 0 ( p 2)],

( 8 )

636 Z. Wç g l o ws k i

We substitute (

) into (4); we have

e ^{= №} + 0 ( p 2) l - [ i + e ( p ) r 1 which tends to ^ as p -*

.

2.

Now let consider the other case. As in 1 point P corresponds to a — p and Q corresponds to a = yp, 0 < p < 1, but now — 1 < у < 0.

We can obtain as in 1,

I

= p [ l + }p2 + 0(p*)],

[ Г 4v2 In I vu11 1

(9) 7] = (1 y)P |l + i f f 2[^(1 + 7)2 + sin2iIn -T ° (P T ji C

- y p [ l + $ y 2P2 + 0(p*)l

and similar inequalities :

(10)

sin2|ln Щ ур 1

Inp

1 + 7

1 41n2p J

(11)

sin

\ In Щур\

lnp < 1 + 7

—

|ln^| *

Applying these inequalities and using the same method as in

we get

№ = - y ( i - y ) P s [ i + 0 (p 2)],

£+i?+C = 2 ( l - y ) p [ l + 0 (p*)],

l + C-î? = - b y ( l - y ) p s[ l + e(p )], V + C -.S = ~ 2y p [ l + 0 (p*)]

(we must remember that y is now negative).

Then using formula (4) we obtain easily

e^H i+o^m i + efr)]-1

which tends to | as p -> 0. The two other cases i. e. when — 1 < p < 0 and

<

< l o r —

<

<

may be treated as those in

and

, respec­

tively. When

— —1 we have the simplest case because the points P and Q lay on a parabola.

3.

From

and

is clear that curve (

), (

) has at the point (

,

,

) a ^-curvature — it is equal to

. But using the property /5(a)-* oo as a

it is not difficult to show that the curve has no O-curvature at this point.