P Pr P PX P P in an w-dimensional Euclidean space On the osculating circle o! a curve

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Seria I : PRACE MATEMATYCZNE X I I I (1969) ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I : COMMENTATIONES MATHEMATICAE X I I I (1969)

Stan isław Fu d a li (Wrocław)

On the osculating circle o! a curve in an w-dimensional Euclidean space

A notion of the osculating circle of a curve L at a fixed point Р 0(х]у xo) of L is usually defined in terms of the second derivatives of functions describing Jarnik [2] has shown that a weaker hypothesis is sufficient for the existence of an osculating circle for a curve in the plane E 2. In the present paper his result will be generalized to a case of a curve in an dimensional Euclidean space IP.

1. Preliminaries. By a continuous locally convex curve in IP we mean a curve L with the property that for each p in L there is an open neighbourhood U c L of p such that no line segment xy, where x, у eU, meets TJ in any of its interior points.

We consider a fixed cartesian coordinate system in EP and a con­

tinuous locally convex curve L represented parametrically by the equa­

tions x% = xl(x1) (i = 1 , 2 , ..., n) with the orientation corresponding to increasing parameter x 1. By a regular triplet {Рх\Р3\Рг) of L we mean three distinct points P x{x\, . x^), P %{x\, . . x3), P z(x\, x3) (where x\ — %г(®к)) of a convex subarc of L such that P 3 lies between P x and P 2.

No regular triplet is collinear. The unspecified summation will mean the summation over the set {1 , ..., n] ; the double sum JT1 will mean .

ij %<i

Let {P1IP3IP2} be a countable set of regular triplets of L and let (PxIP3IP2) be an arbitrary regular triplet of L. The, plane H2(x\,xl,xl) containing points P X, P 2, P 3 is uniquely oriented by the two vectors P3PX= £{x\ — xl)ki and P3P2 = £ ( x l — x\)ki. On the other hand, each of these vectors gives an orientation to a certain (n—1)-dimensional hyperplane. Let Hłl-1(p \x\, x\) be the hyperplane containing the centre of the line segment P 3P r and oriented by the vector P3Pr (r — 1, 2 ).

Since P x, P 2, P 3 belong to the regular triplet, then the plane H2(x}, r 2, r 3) and hyperplanes Hn~1(p \xl, x\) have only one common point P c{xlc, ..., x”).

Obviously, P c is a centre of the circle K{x\,x\,x\) containing points -P\ ? -Pi > P 3 •

In the sequel we shall use the abbrevations

sion we shall omit the subscripts, e.g., we shall write Vli instead of Vl\

etc. and Vp instead of Vlps (the latter for s = 3 only). Let us note that

By (1.1), the equations of hyperplanes Hn 1(p\x\,x\) (r — 1, 2 ) may be written as

The plane H2 may be denoted by n —2 equations of (n—l)-dimen- sional hyperplanes

(1.4) ( J - x i3)V n- ln + (xn- 1- x ^ - 1)V ni+ (x n- -x ^ V in- 1 = 0 (j = 1, .. ., n - 2 ) as their common part. The system of n equations (1.3) and (1.4) defines the point P c. The determinant of this system is equal to W = (y n- ln)rl- 3 JT'(L^)2 ^ o. The solution of system (1.3) and (1.4) is an n-triplet (x\, . .. , Xc), each member of which may be written in the form

(Jc = 1 , 2 , ..., n).

In this case the radius -й(ж{, x\, Ж3) of the circle K {x \, x\, x\) can be expressed by

I t is known that if points P 1 ,P l,P l of the triplet (Р?|Рз|Р£) con­

verge along the curve to a point P 0 (indepedently one from another) and the limit K r of circles K a{x\, x\, x\) exists, then K r is the osculating circle of i at P 0. However, one may expect that the osculating circle K r depends on the position of P 0 with the respect to triplets (P1IP3IP2) and just for this reason four definitions describing 4 types of convergence

К * = Vtp, Vij = and 1Vii = - гГ \

(1.3) (r = 1, 2).

(1.6)

O sculating circle o f a curve 5

of (P1JP3IP2) to P 0 are introduced. We shall prove the existence of an osculating circle in each of the cases denoted by these definitions, without referring to second-order continuous derivatives of the functions describing L.

De f in it io n 1. If (P“ |Рз]Р2) converges along the curve to a point P 0e_L, then by the osculating circle K r of L at P 0 we mean the limit of the circles K a{x\, x\, x\), if it exists. Thus

K r = lim K a{x\,x\,x\) (r = 1 , 2 , 3 , 4 ) (p“ip“ip2a)—>p0

and we have:

a) K x if P 0 does not belong to any arc P “P 2;

b) K 2 if P 0 belongs to each arc P “P 2;

c) K 3 if P 3 = P 0 in the regular triplet (P“[P£|P£);

d) Ж4 if P “ = P 0 in the regular triplet.

Thus K r (r — 1, 2, 3, 4) exists if and only if the corresponding finite limits of (1.5) and finite and positive limit of (1.6) exist, the convergence being understood in the respective sense. These limits define the centre and the radius of K x, K 2, K 3, K x, respectively.

Now, by a regular p air (PX|P2) of L we mean two distinct points P x and P 2 lying on a convex arc of L and such that P x precedes P 2 on L.

Let {P1IP2} be a countable set of regular pairs of L. The next defini­

tions will be obtained by considering the circle K a(t2, x\) containing an arbitrary regular pair (P“|P2) of L and tangent to I at P 2. Clearly, since P x precedes P 2, the point P x does not lie on the straight line t2, tangent to L at P 2.

Let H2(t2, a;}) be the plane containing the line t2 and the point P x.

It is uniquely oriented by the tangent vector 12 = JT x\ hi and the vector P2PX= ^ ( x l — xl)ki and may be described by n —2 equations of (n—1)- dimensional hyperplanes

(1.7) (xf - P 1)1Vn- ln + (xn- 1- x ^ - 1)1Vn)’-\-(xn- x ^ ) 1V}'n--1 = 0 (j = 1 , 2) as their common part. Let Hn~1(p\t2) and Hn~1(p \x\, x\) be the hyper­

planes normal to the vectors t 2 and P2PX, respectively, and such that the first contains the point P 2 and the second — the centre of line seg­

ment P 2P X. Their equations have the form

0- 8) J ? {x%—x\)x\ — 0 and J T [хг— V\2 = 0 , respectiлгely.

The system of equations (1.7) and (1.8) defines the point P c{xxc i Xo), which is the centre of circle K (t2, x\). The determinant of this system

is equal to В = (W”" 1 п)п~г^ ( ‘ 7 " )2. Hence D Ф 0, because (РХ\Р2) is a regular pair. The solution of system (1.7) and (1.8) is an %-tuplet i,i {x\, ..., Xc) each member of which may be written in the form

(1.9). xkc {t2,x\) = xk2

1Vkj x\ — x\

* 2i j

i у a (k = 1, 2, ..., n).

In this case the radius B (t2, ж}) of the circle K (t2, x\) can be expres­

sed by

(1.1 0)

B (t2, x\) = —

1

2

De fin it io n 2. If (P°|P£) converges along the curve to a point P 0eL, then by the osculating circle K r of L at P 0 we mean the limit of the circles K a(t2, x\), if it exists. Thus

K r = lim K a(t2, x\) (r = 5, 6, 7, 8) (p«ip2a)^p0

and we have:

a) K 5 if P 0 does not belong to any arc Р 4Р£;

b) K 6 if P 0 belongs to each arc P^Pl)

c) K 7 if P “ = P 0 in the regular pair (P“|P£);

d) K & if P 2 = P 0 in the regular pair.

It follows that K r (r = 5, 6, 7, 8) exists if and only if the finite limits of (1.9) and finite and positive limit of (1.10) exist, the convergence being understood in the respective sense. These limits define the centre and the radius of K 5, K 6, K 7, K B, respectively.

It is clear that the existence of K x or K 2 implies the existence of both K 3 and and the existence of K 5 or K 6 implies the existence of both K 7 and K B. We should note that these definitions depend on the fixed coordinate system; it may happen that K p (p — 1, 2, ..., 8) exists in one coordinate system and does not exist in another one.

(2.1)

(2.1')

2. Auxiliary lemmas. We shall often use the formulas

vij' v\vi~viv{

x\- x\

ly i x\

x\ — x\

4 v j 2- x i v i , fp^ _. /уД

u/] ¹I²

П - У 1 Г , v 1 V\, — x\

x\- ■x\ Vjn-

V 3 — V ]V1 Y2 y j x\ — x\ 1 ’

, rpJd/2 x\ — x\ ^V]

obtained from the identity a b —cd = (a ~ d ) c —(c—b)a.

Osculating circle o f a curve 7

Lem m a 1. Let the derivatives xk {xl) (k = 1, ...,n ) be finite and con­

tinuous at x\. Then

(A) the finite and positive limit of (1.6) [resp. (1.10)] exists i f and only i f the limits

(2^.2⁾

{resp.

(2^.2^')

lim

v \- v k

__ /уД

tX^i U/2 — B k {h = 1, ..., n)

lim ^{* 12'} 11 i x\- h’X2~*X0

*}**2

^2 _ 1 jgh (к = 1 , . . . , w)]

exist, are finite, and do not vanish simultaneously,

(B) the finite and simultaneously not vanishing limits of (1.5) [resp.

(1.9)] exist i f and only i f there exists the finite and positive limit o f (1.6) [resp. (1.10)].

We first prove (A) for the limits of (1.6) and (2.2).

I. Let limits (2.2) exist. By continuity of derivatives хк(ж1) at x\, using the mean-value theorem, we get

(2.3) lim V!ps Xk (k = 1, ...,n \ p , s = 1, 2, 3).

p ^ S

Hence

(2.4) _{i i} lim _i V V\

2 *

Ą rlĄ-'*h (^1—Жз) (^2—жз) < 0 By (2.1), (2.2) and (2.3) we also have

rH \ 2

(2.5) lim V — r) = у V {Blx>0—B Jxl0f .

x> l xl - xl 4 ^ •

(х¹- я:з)(а^ - а!з)<0

Taking into account that lim {x\ — x\f = 0 and using (2.5) vl xl-*xl

xl * xl and (2.4) we obtain from (1.6)

(2.6) lim B{x\, xl, x\) = [JT ( x l ) f 2 [$ JT 1 (BlxJ0 — B 1x o)2] 1/2 = G,

ля^ лД лД . 1 Ó

(x\-x\){Ą-x\)<b

where G is finite and positive.

II. Now, let the finite and positive limit of (1.6) exists. Hence limit (2.6) exists. It means that

lim M(x\,x\,x\) =

a h a j-oj! *

X 1* 2

x lim (x\ — x\)2 + lim

QT1

j

" 1,1

y v =

where 0 < G < oo. By continuity of derivatives xk(x l ) at x\ and by the mean-value theorem we get (2.3) and (2.4), all limits being finite. Taking into account that \0\ < oo and that lim(#2 — x\f = 0, we infer that there exists the finite limit

lim

*\ĄĄ- г,?

12

4

2в i,i

{х[ - хъ)(х2- хъ)<*

This yields the part (A) (for the limits of (1.6) and (2.2)) of the Lemma.

For the limits of (1.10) and (2.2') the part (A) of Lemma can be proved in much the same manner as for the limits of (1.6) and (2.2), but then we apply (2.1') instead of (2.1).

P r oo f of (В). I. Assuming that the finite and positive limit of (1.6) [resp. (1.10)] exists we infer that (2.6) also exists. Consequently, by (A) limits (2.2) [resp. (2.2')] exists, are finite and do not vanish simul­

taneously. Hence we obtain (2.3), second of (2.4) [resp. first of (2.4)]

and (2.5). Thus (2.7)

[resp.

(2.7')

lim xk (x\, x\, x\) x\ĄĄ-+x\

(ж1-жз)(а:21- хз)<о

jc , ^ { x \ f ^ x l{ B 4 k - B kxl

— #4) ¹

- 2 4 ^ 4 - B ’x i f

= ж

lim xk(t2, x\) = 31 (k — 1, ..., n)].

These values are finite and do not vanish simultaneously.

O sculating circle o f a curve 9

II. Let now exist tlie finite and simultaneously not vanishing limits of (1.5) [resp. (1.9)]. Hence limits (2.7) [resp. (2.7')] exist too. Using the formula for the distance between two points, e.g. between P x and P c, it is not very difficult to check that there exists the finite and positive limit of (1.6) [resp. (1.10)] and so does the limit (2.5). QED.

B e m a r k 2.1. Each of the osculating circles K p (p = 1 , 2 , . . . , 8) exists if and only if the respective limits of coordinates of centres and of radii of circles K a{x\, x\, x\) and K a(t2,x\) exist and are finite. (B) simplifies this condition. Namely, if derivatives xk{xl) are continuous at Xo, then each of K p exists if and only if there exists the finite and positive limit of radii of respective circles K a{x\, x\, x\) and K a(t2,x\).

Lem m a 2 (cf. [1]). The finite limit

(2.8) lim

1» 2» 3 0

v X - v l

x\ — x\ — B k

2 {Jc = 1, 2, ..., n)

exists i f and only i f there exist the finite limits

(2.9) lim ^V

1 Я1 P ’ x 3 X 1xp

1 X, co J

— B2

where {x\ — x\){x\ — x\)< 0.

Lemma 3 (cf. [2]). The finite limit

(2.1 0) lim VkУ r,

(P = 1,2),

д x,

0

= - B h

■x. (k = 1 , 2 , p , s = 1 , 2 , 3 )

exists i f and only i f there exists the finite limit

(2.11) lim ^{tbp dJS}

xp xs = B k.

B e m ar k . If we consider the last limit assuming additionally that x\ = x], we get xk{x\) = B k by (2.11).

From (2.11) it follows that the derivative хк(хг) is continuous in a certain neighbourhood of point x\.

Lemma 4 (cf. [2]). I f the derivative xk{xl) at xl exists and is finite, then the finite limit

™k_Vk i

(2.12) lim ^ = - B k (k = l , 2 , . . . , n ; V = 1 , 2 , 3 )

exists i f and only i f there exists the finite second derivative

(2.13) xk{x\) = B k.

De fin it io n 3. If the function xk(x1) is defined in a closed interval [ ^ y + c ] (c > 0), then by the right-side upper derivative of the function xk{xl) at у we mean

аГ (у) = lim sup +fc, л->о+

xk(y + h) — xk{y) h

Lemma 5 (cf. [2]). The finite limit

(2.14) lim

i>xl ^ s

Vk- x k

l{x\ + x\ ) — x\ B k (k = 1, 2, ...,n )

exists i f and only i f the function xk(x1) is continuous in a neighbourhood (Xq c , x\ -f c) of Xq (c > 0) and there exist the finite limits

(2.15) lim

Vе —xk

1 X,

co ■— X, (V = 2 , 3 ) . Lemma 6

neighbourhood

(cf. [1]). Let the function xk {xl) be continuous in a certain (Xq— c, Xo~j-c) o f Xq (c > 0). Then the finite limit

(2.16) lim

xl * xl

V l - V ko i x\ — x\

о

(k = 1, 2, ...,n )

exists i f and only i f there exists the finite' limit (2.14).

3. The conditions for the existence of the circles K x, K 2, .. ., K 8.

All the osculating circles К г, .. ., K s of L at P 0 exist if the second deriv­

atives xk{xl) (k = 1, 2, ..., n) are continuous at xJ. The last condition is very strong and, in fact, all those circles exist under weaker conditions.

Th e o r e m. Let the curve L be described by the functions xk(x1) (k

= 1, 2, ... , w), continuous in a certain neighbourhood of a?J, and let the derivatives xk{xl) exist at x\ and satisfy \xk{x\) \ < oo for each superscript k.

Then

(A) the osculating circle K 8 exists i f and only i f the limits

(3.1) lim ^V

л x\ — x\

0

к

exist, are finite and £ \Bk\ Ф0.

O sculating circle o f a curve 11

(B) Moreover, if the limits

(3.2) lim Vk2 = xk

xl xl-+xl xl * xl

exist, then the osculating circle К 4 exists if and only i f limits (2.15) exist, are finite and does not vanish simultaneously.

(C) And, i f the derivatives xk{xl) are continuous at x\, then

(i) the osculating circles K 8, K 7 exist i f and only i f the second deriv­

atives xk{x\) = B k exist, are finite and ]?\Bk I Ф 0, (ii) the osculating circles K x, K 2

the limits

(resp. K 5, K 6) exist i f and only if

(3.3)

[resp.

(3.3')

1 1 lim i x[)

CP’X3 ^ X0 r

xi * xl

Tk — Tk

<A/p

x\ B h (P = 1,2)

lim X\Ą-»5

*1**5

/V» ___ r p

i(/j JU ²

.1 x\ — xl

0

= B k \

exist, are finite, and do not vanish simultaneously.

B e m ar k . Limits (3.3') are the special cases of (3.3).

Proof. First we prove (A). I. Let K 6 exists. Then, by Definition 2d there exists the finite and positive limit of (1.10) (where xk = xk). The existence of the finite derivatives xk{xl) at x\ implies the existence of the limits lim F i0 = xk . Hence, first of (2.4) exist (where xk = x\). Since

x\-+xl

the limit of (1.10) is finite, there exists the limit of denominator of (1.10) and is finite and positive. This implies the existence of (3.1).

II. How, let limits (3.1) exist. Since Vk0 — xl in (3.1) is bounded, we obtain the existence of limits lim F*0 = x\. Taking into account

X1— * 1 xi ^ xo

formulas (2.1') we infer that the limits of (1.9) and (1.10) exist and are finite. Hence, by Definition 2d, K 8 exists.

Pr oo f of part (В). I. Let K x and (3.2) exist. Hence (3.4) lim Vp0 = x\ (p = 2 , 3 )

XV^ 0

and from the existence of А 4, by Definition Id, there exists the finite and positive limit of (1.6) (where xk = xk). By virtue of (3.2) and (3.4) there exist the finite limits (2.4) (where x\ = m\ ), and taking into account

that lim {x\ — x\f = O, we infer that there exists the finite and positive

x2~*x0

limit of the denominator of (1.6) (where x\ = xk). Then, by virtue of (3.4) and (3.2), there exist the finite and simultaneously not vanishing limits (2.16), and, by Lemmas 6 and 5, there exist the finite limits (2.16).

II. How let limits (2.15) exist. Hence, by Lemmas 5 and 6, there exist the finite limits (2.16) and we get (3.4). Since V\ — Vk in (2.16) is bounded, we obtain the existence of the continuous derivatives xk(xl) at x\. Now using the mean-value theorem we get (3.2). Taking into account formulas (2.1) we infer that the limits of (1.5) and (1.6) (where xk — x„) exist and are finite. Hence, by Definition Id, there exists H4.

P r o o f of part (i) of (C) for K 3. I. Let K 3 exist. Then, by Definition lc , there exist the finite and positive limit of (1.6) (where xk = xk) and, by (A) of Lemma 1, we get

X

lim Yk' 20 _VkV 10

/уД_ /уД ж 2

Hence, by Lemmas 2 and 3, there exist the finite limits

(3.5) lim

E2~>a'0

x2 — ^{X 0}

/уД /)Д

0/2 O/Q = B k

and, by Eemark after Lemma 3, there exist the finite second deriva­

tives xk{x\) — B k.

II. How let exist the second derivatives xk{x\) — B k. Hence the first derivatives xk (x1) exist and are continuous at x3. Using the mean- value theorem we get (2.3), second of (2.4) and (3.5). Therefore, using Lemmas 3, 2 and (A) of Lemma 1, we can verify the existence of the finite and positive limit of (1.6) (where xk = xk). Hence, by Eemark 2.1, there exists K 3.

For K 7 it can be proved in much the same manner as above.

P r o o f of part (ii) of (C) for K x. I. Let K t exists. Then, by Defi­

nition la , there exists the finite and positive limit of (1.6). Consequently, by part (A) of Lemma 1, there exist the finite and simultaneously not vanishing limits (2.2); by Lemma 2, there exist limits (2.9). Therefore, by Lemma 3, there exist limits (2.11), which are equivalent to (3.3).

II. How let exist the limits (3.3). They are equivalent to limit (2.11) and hence, by Lemmas 3, 2 and part (A) of Lemma 1, there exists the limit of (1.6). Then, by Eemark 2.1, there exists K x.

For K 5 it can be proved in much the same manner as above. How we note that for K 2 and K 6 the theorem is true also, because by con­

tinuity of derivatives xk(xl) the convergence in the case (ж} — x\) {x\ — xl) > 0

O sculating circle o f a curve 13

is not essentially different from that in the case — x\)(x\ — xl) < 0.

This remark complete the proof of the Theorem.

Succesive parts of the Theorem give the conditions — from the weakest to the strongest — for the existence of the circles E x, K 2, . .., K a.

The fact that each condition is really stronger from the preceding one is illustrated by examples in [2].

4. The relations between the conditions for the existence of the circles K x, K &. In formulas (2.7) and (2.6) defining the centre and the radius of the circle K p (p = 1, 2 , ..., 8) for each p appears the quantity B k (for a fixed h). Since B k is defined differently for different p, the question arises whether or not B k is the same for all circles K p. In partic­

ular, if for a curve L some of circles К г, . .., К а exist are they necessarily identical?

An answer to these questions is given by Lemmas 3, 4 and 5 together with the following three lemmas, whose proofs can be found in [2].

L^{em m a} 7. I f in a certain neighbourhood (xl — c,xl~\-c) (c > 0) exists the derivative xk(xl) (Tc = l , 2 , . . . , w ) (finite or not) and i f B k is finite, ‘ then (2.14) exists i f and only i f the second derivative xk(x\) = B k exists.

Lem m a 8 . I f (2.14) exists, where B k (1c — 1 , 2 , ..., n) is fi7iite, then (3.1) exists also.

Lemma 9. I f the finite second derivative xk(x\) = B k (h = 1 , 2 , ..., n) exists, then (2.14) exists also.

From Lemmas 3 - 5 and 7 - 9 it follows that for a fixed h, B k is the same in all conditions of the Theorem. A simple consequence from this are the relations between the conditions for the existence of circles Кx, ..., K a, presented graphically as follows:

K 5+— *~K6 +— ► JCi - K 2

\ f \ /

exist if and only if limits (3.3) exist;

exist if and only if (2.13) exist;

exists if and only if limits (2.15) exist;

exists if and only if limits (3.1) exist.

R eferences

[1] S. F u d a li, O hrsywiźnie Tcrzywej płasM ej, Prace Mat. 8 (1959), pp. 147-165.

[2] Y. J a r n i k , O krużnice Tcfivosti, Ćasopis pro pestovani mat. a fisiky, 74 (1949), pp. D37-D51.

INSTYTUT MATEMATYCZNY U N IW ERSYTETU WROCŁAWSKIEGO