On the osculating plane of a curve in an w-dimensional Euclidean space

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I : COMMENTATIONES MATHEMATICAE X I I I (1969)

C.

and S.

(Wrocław)

On the osculating plane of a curve in an w-dimensional Euclidean space

Van der Waag [7] distinguishes the eight types of an osculating plane of a curve in the 3-dimensional Euclidean space and gives the condi­

tions for their existence and the dependence between the osculating planes of different types. Eadziszewski [5] gives in the terms of the metric of the space the necessary and sufficient conditions for the existence of the osculating planes of the different types in the sense of van der Waag.

In [6] gives that conditions in the sense of Bouliguand [1]. But in the general theory of curve a necessary and sufficient condition for the ex­

istence of the osculating plane of a curve is usually given in the terms of the continuous second derivatives of the functions describing a curve.

However, it is a rather strong condition, because an osculating plane exists also under weaker that.

In the present paper we formulate the sufficient conditions for the existence of the osculating plane without resolving to the second con­

tinuous derivatives of functions describing a curve.

1. Preliminaries. We consider a fixed rectangle cartesian coordinate system in an w-dimensional Euclidean space E n and continuous curve L represented parametrically by the equations xl — xl (t) —

with the orientation corresponding to increasing parameter t. By a regular triplet (P1|P3|P2) of L we mean three not collinear distinct points Pi{x\, ..., Xi), P 2{x\, .. ., x2) , P 3(xl, ..., x3) (where x\ = x%(tr)) of an arc of L such that P 3 lies between Р г and P 2. The unspecified summation will mean the summation over the set {1, 2, ..., n }; the sum ^ will mean the sum over all cyclic permutations of elements of the set {n —1, n, j}, e-g. V4%2 will mean the sum y n- ln-\~ynj' -у у 1п~г.

Let (PpPglPg) be an arbitrary regular triplet of L. The plane containing points P 1, P 2, P 3 is uniquely oriented by the two vectors

(

1

1

P 3P S = £ ( x ls- x l ) k i ($ = 1 , 2 ),

'where kj is an unit vector in direction of i -th ax of coordinate system.

Moreover, H2(tx, t2, t3) can be described by the system of n —2 equations { ^ - Ą ) V n~ln+ {x n- 1- x n b- 1)V n1Ą-{xn- x l ) V in- 1 = 0

(j = 1 , 2 , . . . , n —2), where Vpr = V^Vr2 — Vr2 V[ and Vl = { х ^ - х Ш .- Ц ) and generally F ^

= {ocrm—xrq)l{tm—ta) (m, q = 1, 2 , 3; m Ф q-, i = 1, 2 -,p ,r = 1, 2, . .. ,n ) (cf. [3]). Hence, H 2(tx, t2, t3) is the common part of n —2 (n—^-dimen­

sional hyperplanes Щ ~ 1. Note that (j = 1 , 2 , ..., n — 2) is oriented by the vector Hj with components:

<1.2)

( 0 , 0 , . . . , 0 , Vn~in, 0 , 0 , . . . , 0 , Vя7, F ?W_I);

d - 0 ^{( n - j - 2)}

(1.2) determine the direction of Hj. A square of lenght of Hj is

|H?-|2 = (F m- 1w)2 + ( Ó 2 + (F /,!'- 1)2 = ^ .(7 ^ 2 )2

(cf. remark about the summation) and in consequence of this fact the unit vector in direction determined by (1.2) have the following not van­

ishing components:

у П ~ 1 n у -nj y j n— 1

^ Е

Щ 2

for respective j. The system of n —2 vectors Hj (j — 1, 2, ..., n — 2) gives the orientation of H2(tx, t2, t3) as well as vectors (1.1).

Now consider the other case. By a regular pair (PX\P2) of L we mean two distinct points P x and P 2 lying on an arc of L and such that P x pre­

cedes P 2 on L. Let (PX\P2) be a regular pair of L. The plane H 2(tx, v2) containing the points P x, P 2 and the straight line v2 tangent to L at P 2 is described by the system of n —2 equations

(xf - x i ) 1Vn- ln + (x n- 1- x ^ - 1)1Vn7 + (xn- x y r n- 1 = 0

(j = 1, 2, .. ., n — 2) (where 1Vpr = V fx2 — Vxxf and x% = xv(t2); p , r = 1, 2, .. ., n) if P x does not lie on v2. Then the system of n — 2 vectors lHj (j = 1, 2, . .. , n — 2) with the not vanishing components:

(

/ I f C W ’ V % jC vhi2f ’ V I A 1?*1*2)2 '

for respective j, gives the orientation of H2(tx, v2).

2. Definitions of the osculating planes. Let {P {\Pl\Pl} (a = 1 , 2, ...

always) be a sequence of regular triplets of L. Since each element of {P?|P°|P“} determine the plane H2(tx, t2, t3)a for respective a, hence we have a sequence {H2(tx, t2, t3)a} of planes corresponding to a sequence

I t is known that if points P x,Pl,P% of the triplet (P1IP3IP2) con­

verge along the curve to a point P 0 (indepedently one from another) and the limit H i of planes H2(tx, t2, t3)a exists, then H i is the osculating plane of i at P 0. However, one may expect that the osculating plane Hi depends on the position of P 0 with the respect to triplets (P“ IP3IP2) and , just for this reason four definitions describing 4 types of convergence of (P“ IP3IP2) to P 0 are introduced. We shall prove the existence of an osculat­

ing plane in each of the cases described by these definitions, without refer­

ring to second-order continuous derivatives of the functions describing L.

De f in it io n

1. If (P?|P£|P£) converges along the curve to a point P 0eP, then by the osculating plane °H2r of L at P 0 we mean the limit of the planes H 2{tx, t2, t3)a, if it exists. Thus

°H2r = lim H 2{tx, t 2, t 3)a {r = 1 , 2 , 3, 4) (PM P2>->P0

and we have:

a) °H\ if P 0 does not belong to any arc P?Po;

b) °H22 if P 0 belongs to each arc P\Pl)

e) °Hl if P 3 = P 0 in the regular triplet (P“ IP3IP2);

d) °H\ if Pi = P 0 in the regular triplet.

Thus, °Hl (r = 1 , 2 , 3 , 4 ) exists if and only if the corresponding finite limits of (1.3) exist, the convergence being understood in the respec­

tive sense. These limits define the components of vectors giving the orientation of °H l, °Hl, °H l, °Hl, respectively.

Let {P 1IP2} be a sequence of the regular pairs of L and (Я 2(^, v2)a}

be a sequence of planes corresponding to {P“\P2}.

De f in it io n

2. If {РЦР1) converges along the curve to a point P 0eP, then by the osculating plane xH2r of L at P 0 we mean the limit of the planes H 2(ti,v 2)a, if it exists. Thus

xH 2r = lim H 2(tx, v 2)a (r = 1 , 2 , 3 ,4 ) ( P1 lP2 ) ~> P0

and we have:

a) lH 1 if P 0 does not belong to any arc P axPl) b) lH\ if P 0 belongs to each arc P^Pl)

c) XH\ if P “ — P 0 in the regular pair (P1IP2);

d) lH\ if PI = P 0 in the regular pair.

Boczniki PTM — P ra ce M atematyczne X III 2

It follows that xHl (r = 1, 2 , 3 , 4) exists if and only if the corre­

sponding limits of (1.4) exist, the convergence being understood in the respective sense. These limits define the components of vectors giving the orientation of гН1, 1H l, 1Н\, respectively.

I t is clear that the existence of hH\ or hH\ (h = 0, 1) implies the existence of both hH\ and hH\. We should note that these definitions depend on the fixed coordinate system; it may happen that hHl (h = 0 , 1 ;

t

= 1 , 2 , 3 , 4) exists in one coordinate system and does not exist in another one.

The expressions (1.3) and (1.4) there are undeterminate in the limit always, because there we have fractions containing determinants V412 or with the equal rows, as in the numerator, as in the denominator.

To avoid that we must transform (1.3) [and (1.4)]. Substract the second row from the first one in each determinant Y4%1 [resp. W*1*2], appearing as in the numerator, as in the denominator, and each obtained difference divide by binomial t2—tx. Introducing the designations

Y k _y k y k _;rk

MX2;3 = —--- — and 1MX2;2 = --- - for a fixed к we can write (1.3) and (1.4) then in the form

(2.1)

(

2

2

м пХ2, з Т Г 1 -

M\2,3Vr2 ■Mf2;3r 2

{

u;2^2

H f 2 ’ respectively. It is easy to verify that (2.1) and (2.2) there are not always undetermined in the limit.

3. The sufficient conditions for the existence of the osculating plane*

All the osculating planes hH 2r (h — 0, 1; r = 1, 2 , 3 , 4) of L at P 0 exist if the second derivatives xk{t) (к — 1 , 2 , ..., n) are continuous at t0.

The last condition is very strong and, in fact, all those planes exist under weaker conditions.

Before we formulate the theorem about the existence of the oscula­

ting plane, we shall give some lemmas presented in [2] and [4]. .

Lem m a

1 (cf. [2], L. 2 and [4], L. 3). The finite limit

ф

(3.1) lim Мк12,г ^ = ]-В к (k = 1 , 2 ,

h) <Q

г/ and only i f there exists the finite limit

r fc— Г *

(3.2) ^ lim —--- - = B k.

H>^2 ~^0 ^1 ^2

L

2 (cf. [4], L. 3). The finite limit lim 1Mk2.)2

<1^4 <2

(k = 1 , 2 , n) exists if and only i f there exists the finite limit (3.2).

L

3 (cf. [2], L. 2). The finite limit (3.1) exists i f and only if there exist the finite limits

lim y - = ^-Bk (p = 1 , 2 ; к = 1 , 2 , ...,n )

and is (^—tfjit^ — tfjK O .

Seting t0 instead of t3 in Lemma 3 and applying Lemma 2 we get L

4. The finite limit lim Mk2;o — \Bk (k = 1, 2 , .. ., n) exists

h ^ h ... .

i f and only i f the finite derivative x (L) = В exists at tn (is here Mk12;o = (Vk10- V k20 ) l ( h - t 2)).

L

5. (cf. [4], L. 2). I f the derivative xk(t) at t0 exists (k

1 , 2 , . . . ... ,n ) and is finite, then the finite limit lim [(x2 — Vk

)l(t2 — t0)'] = i-B*

<2-^ 0 ... .

exists i f and only i f the finite second derivative x (t0) = В exists at t0.

De f i n i t i o n.

If- the function xk(t) is defined in a closed interval [y, У + с] (c > 0), then by the right-side upper derivative of the function xk(t) at у we mean

xk(y) — lim sup

h^—>0 h> o

xk (y + h ) - x k(y)

h

L

6 (cf. [4], L. 4 and [2], L. 6). Let the function xk(t) be contin­

uous in a certain neighbourhood (t0—c, <0 + c) of t0 (c > 0). Then the finite limit lim [(L23 — Vk3)/(t2 — £0)] = \Bk exists if and only i f there exists

(Ь—Н ) ( Ч - Ч ) < °

the finite limit

lim xk(t) — xk

t t0 (k = 1, 2, . .. ,n ) .

(3.3)

T

. Let the curve L be described by the functions xk{t) (к = 1, 2, ...,n ) , continuous in a certain neighbourhood of t0, and let the deriva­

tives xk (t) exist at t0 and satisfy |#fc(£0)|< oo for each superscript Tc. Then the osculating plane

(A) XK\ exists i f the limits

FJo-Жо 1 ^

(3.4) lim — --- - = —B k

exist, are finite and B k\

0;

(B) °H\ exists i f the limits

(3.5) lim 7* = xk

о

and (3.3) exist, are finite and ^ \Bk\ Ф 0;

(0) °Hl and lE\ exist i f the derivatives xk(t) are continuous at t0 and the second derivatives xk(t0) = B k exist, are finite and does not vanish simul­

taneously ;

(D) hE\ and hHl exist (h = 0 ,1 ) i f the derivatives xk(t) are continuous at t0 and limits (3.2) exist, are finite and \Bk\ Ф 0.

The osculating plane hH2r (h = 0, 1; r = 1, 2, 3, 4) is described by the system o f n —2 equations

(3.6) (x?'~ xl) (BnXo~1- B n- 1Xo) + {хп- г- a?o_1) {Bjx ^ - B nĄ) +

+ {xn- x n 0){Bn- lĄ - B i xn 0- 1) = О (j = 1, 2, ... , n — 2) and have the orientation given by n —2 vectors with the not vanishing components:

(3.7)

I B nx^-1—B n- 1Xp B jx% -Bnxj B ^ Ą - E ó ą - 1 \

Proof. Since ^ \ B k\ Ф 0, we assume that B n Ф 0 without loss of generality of Theorem.

Por (A) [resp. (B)] write t0 instead of t2 [resp. tt] in expressions (2.2) [resp. (2.1)] and, applying (3.4) [resp. Lemma 6 and (3.5)], converge to the limit with (2.2) [resp. (2.1)].

For (D). Taking into account the continuity of xk(t) we get

(3.8) lim xk = xk

and, using the mean-value theorem,

(3.9) lim Vp3 = Ą

Note, that by continuity of xk(t) the convergence in the case (tx~

— t3)(t2 — t3) < 0 is not essentially different from the one in the case (h —h ){h — h) > 0. Hence, for °Hl and lH\ we can as reason as for °Hl and 1H\.

Now, converge to the limit with expressions (2.1) (for °Hl) [resp. (2.2) for 1BL\] and apply Lemma 1 and (3.9) for^> = 2 [resp. Lemma 2 and (3.8)].

For (C) write tQ instead of t3 (for °Hl) [resp. tx for in expressions (2.1) [resp. (2.2)] and, applying Lemma 4 and (3.9) [resp. Lemma 5 and (3.8)], converge to the limit with (2.1) [resp. (2.2)].

Note, that in the case of °Щ we must to write 0 as the second subscript for symbols V, i.e. F*0, F 2Q an(l M*2.0, instead of the subscript 3, omited in the notation (cf. the remark after (1.2) in [3]). In case of 1H * we have T*2 and 1-М'*2.2 =

= (ж2— V^2)l(t2—tQ) in the respective place.

Then, for (А), (В), (C) and (D), as the values in the limit we get (3.7), which are finite and fully determined and, by assumption that B n Ф 0, simultaneously not vanishing. Hence, by the respective definition, exists hH 2r and is described by the system of (3.6). QED.

Before we finish let us make two remarks. A quantity B k appearing in each part of Theorem is the same quantity. It can be easily deduced from uncited lemmas of [4]. Moreover, the quantity B k plays a role of the second-order derivative of xk(t) in formulated conditions. Hence, if functions xk (t) are of the class G2, then we can write xk instead of B k in formulas (3.6) and (3.7). Taking into account this fact and seting n = 3 and x , y , z instead of xl, x 2, x 3, respectively, we get

(3.10) (X — x0) (у0ё0 — ёоУо) + ( Г — y0) (ź0a?o — x0z0) + (Z— z0) {x0y0—y0x0) = 0 instead of system (3.6). (3.10) is the well-known equation of the oscu­

lating plane in Я 3. Instead of (3.7) then we get the components of the binormal vector of A at tQ.

R eferences

•

[1] B. B o n lig u a n d , Introduction a la geometrie infinitesim ale directe, Paris 1932.

[2] S. F u d a li, O Tcrzywiźnie krzywej p ła skiej, Prace Mat. 3 (1959), pp. 147-165.

[3] — On the osculating circle o f a curve in an n-dim ensional E uclidean space, ibidem (this fasc. pp. 3-13).

[4] У. J a r n i k , О кгйгт се krivosti, Ćasopis pro pestovani mat. a fisiky, 74 (1949), pp. D37-D51.

[5] K. R a d z isz e w sk i, Sur une interpretation metrigue des p lan s osculateurs orientes, Analele §tiintifice ale Universitatii „Al. I. Cuza”, X Ib (1965), pp. 497-505.

[6] — Sur certaines proprietes des courbes admettant les plan s osculateurs orientes, Annales Universitatis MCS, 17 (1963), pp. 105-113.

[7] E . J . van der W a a g , Sur les plan s osculateurs, I , I I , Indagationes Mathematicae 14 (1952), pp. 41-62.