Some osculating &~plane oî a curve in an ^-dimensional Euclidean space

R O C Z N IK I P O L S K IE G O T O W A R Z Y S T W A M A TEM A TY C ZN EG O Sé ria I : P E A C E M A T EM A T Y C Z N E X X I I I (1983)

Stanislaw Fudali (Wroclaw)

Some osculating &~plane oî a curve in an ^-dimensional Euclidean space

A curve L in an w-dimensional Euclidean space E n may be described

П

by the vector equation L(t) = x\t) I f xJ: t->xj(t) for each j e {1, ...

are Cfc-class functions, then at arbitrary point P (t0) e L there exists i=i an osculating fc-plane of L. The condition formulated above for the func­

tions xj is rather strong and for this reason we try to omit this condition or to weaken it. For this purpose we can modify the definition of an oscu­

lating fe-plane of L and show that so defined fc-plane exists if a weaker condition for oP, other than xj be Ofc-class functions (see [1], [2 ], [5], [6]), is assumed.

A fc-plane in E n is uniquely determined by any k - h - j - 1 of its points and h directions, where h is any integer not exceeding (& + 1)/2 ; the points should not lie in any (k — h — 1 ^dimensional plane. In the present note there is considered the fc-plane which contains lc—h + 1 points of L and one straight line tangent to L at one of these k — h + 1 points of L, and is paral­

lel to h — 1 other straight lines tangent to L at other h —1 points of L . These considerations may be repeated without an essential change for the fc-plane which contains k —h-f-1 points of L and is parallel to h straight lines tangent to L at other h points of L.

П

Let L(t) = J? xJ(t) -kj be the radius vector of a curve L in E n, where xJ :t-^-xj(t) are Ck~l-functions and kj are the unit vectors. Assume also 3=1

that the points Px(h)> P 2(^)? • • • > P*(7*) of L are chosen so that P r lies between P r- i and P r_2 for each r e (3, 4, . . . , &} and that.any straight line is tan­

gent to L at P k, whenever it is considered; the sequence of these points will be called regular and denoted by BSP (7c). By hnJc(Ps) we mean the

&-plane through the points P a (s e {1, 2 , . .., к — Щ Щ) (1 < h < k) and through the tangent line to L at P k, and parallel to tangent line to L at P q (g g ^k—h-1- 1, к — Ji-r-2, . .., к —

14 S. F u d a l i

Definition. Let RSP(fc) Ъе convergent to P 0 along L. The limit fc-plane of the sequence {кл к(Р°)}аеХ, whenever it exists, will he called the osculating к-plane of the curve L at P 0 and will he denoted by rV (r 6 {1, 2, 3}, k € {0,1, 2 , (* + l)/2}):

V = lim V ( P ;‘),

RSP(fc)->-P0

where we write

(a) клк if P 0 belongs to none of the arcs P “P 2,

(b) \лк if P 0 belongs to each of the arcs P?P£ and is different from P"

and from P£ for each a e N,

(c) клк if P 0 is one of P “ or P 2'in each sequence of the family RSPa(7c), a e N .

Choose the points P x, P 2, . .. , Р л_л, P*_A+1, . . . , P k on L in some neigh­

bourhood of P 0 e L and let P k be the origin of each vector Р кР{ (г e {1, 2 , ...

— Moreover, let vq — J? xf (tg)kj be the tangent vector to L at the point P 0 for each q e {k — h + l , . .. , к — 1}. Then a 7c-plane generated by vectors P*P* and vg and passing by the points P 1? . .., Р А_Л, Р л is de­

scribed by the system of n — Tc equations:

( 1 ) Л

г е { 1 , , n —k}

n Щ /

^ ^ __^ к (г -п + к )^ д ,г __ ^ r ^ fe y y + 1 ... n in —k + l . . . r - l __ q r = n —k + 1

where we write xp instead of xr(tp) and (2 ) AX r+1- nin—k+1... r— 1

x[+l — xrk+1 г?'» 1 + 1 l~»

л.г+1

•*'*-*+1 • •

h ^к tk-h ~ 4

[& ! S* 1 &

4-n + i • . x i

11

df 4 - 4 4 ~ h- 4 лЛ

^k-h-И *

h ~tk i i

a%-k+l - x nk- k+l „w-fc+l „*-*+1

Xk-h ~ Xk л,п—к+1

хк-к+1 •.. ж Г *+1

tx — tk ^Jfc-Л ~ %

/у»Г“ 1

^л-л-ы •• • ^ Г 1

1<1

for each r e {n — Tc + 1, n — Tc + 2 , . . . , n).

Each equation of (1) describes a hyperplane Hs. A normal unit vector of such hyperplane is

h ~%n—Æ+1...W. h j£ n —k + 2 ...n s

(3) IIS = [ w > , ? 0 ? . . . 7 0 ? n —k

h jf 8 n —k + \ ...n —\ T

’ ’ v ^ x ^ f ï

where £ denotes the summation over all the sequences of superscripts i 1? . . . , i k which are elements of the set {s, n —fc + 1, ...,% }. By this we mean that there are formed all cyclic permutations of the sequence of s, n — Jc + 1, in each of these permutations the first term is omitted and the rest of them is denoted by i x, . . . , i k.

If the points P 17 P k converge to P 0 along L , all components of Hs become indeterminate forms of the type 0/0. To avoid it we must transform each component in (3) in the way which is used in [2]. Then, each determinant hX ll'"lk {i1, . .. , ik e {s, n — Tc + 1 , . .. , n}) appearing in

(3) will be transformed to another form, namely

(4)

i x h ., h y h

•• h^5- h y 4

h -l-A - h- 1 y h

ft - .. \ x tl Лл

®к

h -y^k h y^ k

,. hA . h y^ k

h -\ Æ h -1 y {k

f t - lA - . \ x ik '“ к df

= Ь - Ф 1" ' * ! ,

where the expression for each s e {1, 2 , . . . , Tc}, q e {1, 2, ...,1c — 1}, re {0, 1 ,2 , ...,% } is an abbreviation of divided difference rqX ls[tktk_x... tk_q~\

and is defined recurrently as follows:

ox is X --- ---

1 i x - h

2 -\гга

\x

w i %K/T

t\~tk

for q — I f II

м

tk-q ~~ h-q-1

<5) for any q < le and r e {0 ,1 , 2 , . . . , q — 1},

£ l & W k - i . • • ije-q-^k-q] S[M*-1 • • • ^k~q-2%-q~\\

Ч-q ^k—q—1

for any q < Te and r — q f

= 7-\х\гкгк_х . . • tk_q_2tk_q'\ — Tq-\X 8[^ft_i • • • tk- q-qfk-q-\]

tk-q —tk-q^{- 1}

for any q < h and r = q + 1 .

16 S. F u d a l i

Applying (4), we may rewrite (3) in a new form as

(6) I I =

j- h jg n —fc + l...n j j- ^ h y n —fc4-2...nsj , 0, . . . , 0,

' z r V

n —k

j- A j r s n - k + l . . . n - l y

/ 2 <,h - 1 x h ~ ,l‘ T J

Also the system of n — le equations of (1) takes another form, namely (7) A (a‘ -* D [t_ ÎX » -t+,- * ] +

ге{1,.. .,n—к}

n

+ J T ( - . 1 )*(y-n+ft)(a?r -®^)[AJ Z r+i - n<n“*+1- ,- i ] = 0 .

r —n —fc+l

Now, if BSP(fc) converges to P 0 along L, the components of Hs do not be­

come indefinite symbols.

If x^-.t-^x^t) is a (7r-class function with differentiable derivative of order r, then for a divided difference of function xj constructed relative to points of BSP(fc) we have the equations

(8) ^{, , .} = - X^{1 (r+} '( 0^1).

folr some C e (ij, <2) and r = Je —i (see Theorem 2 in [3]) and

k - 1

. . 1 ( r + l ) . 1 > w . I — Г i ;

(9) ? x u . . . * i+1y = — — * ’ (f) . + 2 2 - ^ > i /

p —k —h + l j = i

for some £ e (b, £>) and Cp e <ft), where 1 < & < r (see Theorem 1 in [3]).

Applying (8 ) and (9) to the expressions in (4), we get the determinant (4) in the form

(1 0) t - Î V i - Ч ] =

P æ ' V ^ + e*1..■ L со ‘' ( f U l + e 1^{(А+1)г- , г-}

1-y*k i n^k

k-i-A т б / * ^ ( Й ) + ^ . ■L * ’* (£ t » ) + G *^{(Л+1). , г-} where.

A = ( l c - l ) l ’ (Л + 1) ! ’ I f =

(h — l)l

and G 9

k-1

2

Хг% pi 2 A-J k-i(p)

p = k - h + l Ц ( tp — t j ) j —i

Lem m a 1 . Let

(i) xl : t->xl(t) for each i e {1, . .. , n} be a Ck~l-class function on inter­

val (a, b },

(ii) at the point tQ e (tx, t2} c= (a, b) there exists the finite derivative

(k)

x l {t0) = B\,

(iii) the points tx, . . . , tk e (a, b), which give the possibility to construct a divided difference k_!{Xl\tk ... ^] (h < Tc) of function хг, form the E SP (h),

(iv) there exists the finite limit

(11⁾ lim

R SP (fc)-*P 0

{* - l6(**-l»**) ip€(tptfk)

к - Up) -

П (tp-b)

= 2D Î

for each p e {h — h + 1 , h — h + 2, . .., Jc — 1}.

Then} at point tQ, there exists the finite limit of the determinant (4) and (12) lim L jZ * !- * * ]

R S P ( f t ) - P 0

- Ir = 1

1 _ 1 rl

• • X ^ + E ^ p 1

(Л) ,

V 1 •• rp A•H

B i k + ] ? D ik . . ihx l)ik+ 2 1 ? } ^{(h) .Х 0 гк . ™lJc}

r = l

к- 1 The symbol appearing in the determinant in (12) always means .

p = k —h + 1

P ro o f. Under the hypothesis of the lemma it is possible to write the determinant (4) in the form of (10) and then to divide the next to last

k - l ( p )

column of (10) by /7 (lp — lj) for each p e {k — h + l , . .., h — 1} and to

i —i

subtract it from each of the first h — h columns. Then in each of the first h — h columns there appears the sum of differences of value of the second derivative of function хг instead of the sum of such values. Each of these sums has the form of the expression under the symbol of limit in (11).

Now, let us find the limit value of the determinant (10) transformed in the way mentioned above if the sequence ESP(&) converges to P 0.

Then by the hypothesis about xi to be a C^-1-class function, the derivatives in all columns, except of the first one, converge to the derivative of the corresponding order at t0, the second item in each’expression appearing in

k—1

the first h — h column, by (11), converges to Dlp for each i e {1, ..., n],

p = k —h + 1

^ R o czn ik i P T M — P r a c e M a te m a ty c z n e t. X X I I I

18 S. F u d a l i

and the first term in each expression appearing in the first column con-

1 .

verges respectively to —--В k (see Theorem 2 in [4]).

k\

Lemma 2. Let

(i) хг: t->x*{t) be a Gk~l-class function on the interval (a, b') fo r each i e {1, .. •, n),

(ii) at the point t0 e <a, by — ftx, tf) there exists the finite limit

(13) lim

h^t-2 ( * -1).

x x l (t2)

tx t2

(iii) the points tx, tk e {a, b), which give the possibility to construct a divided difference k_\X\_tk ... tx] (h < Jc) p f function x1, form the BSP(&),

(iv) there exists the finite limit (11) for each p e {k — h + 1, . . . , k — 1}.

Then, at the point t0, there exists the finite limit o f the determinant (4), which has the form (12), where the C'f appears instead o f the B %k fo r each s e {1, . .. , k}.

A proof of this lemma is the same as that of Lemma 1.

Lemma 3. Let

(i) x{ : t-^xl(t) be a Gk~l-class function on the interval <(«, by for each i e { l , . . . , n ) ,

(ii) at the point t0 e {tx, t2} c (a, by there exists the finite one-side de- rivative x l(t0) = Qlkt o f order к fo r each i,

(iii) the points tx, . tk e (a, b), which give the possibility to construct a divided difference kJ {X l[tk ... tf] (h < k) of function x1, form the ESP (k),

(iv) there exists the finite limit

(14) lim

RSP(A-)—>Pq*

Zpedp>tk)

яЧСк- 1)-*Ч С Р)

k - l ( p )

II Vp-b)

for each p e {k — h-\-1, к — 1}.

Then, at the point %, there exists the finite limit of the determinant (4) and has the form (f 2 ), where the Qkt appears instead of the B ks fo r each s e { l , . . . ,L }.

The asterisk * in (14) and above replaces either of the signs + or — f the same respectively, everywhere. A proof is the same as that of Lemma 1 .

The determinant appearing in Lemma 2 will be denoted by G4 "'4 and the one appearing in Lemma 3 by Q4 "'4 , i.e.

lim

R SP(fc)->P0

c lim [k_]Xiv"ik] = Qh '"ik.

R SP(fc)-î-P0*

Тп е о е е м. Let a curve L in E n be described by the equation L(t)

71

= £ хг(1)к{, where xl: t->xl(t) is a Ck~l-class function on the interval b>

i= l

for each i e {1, ...,n } , and points P f f ) , . . . , P k(tk) e L form the E S P (k).

Moreover, lei there exist the finite limit (11 ) for each p e {k — h + 1, ..., к — 1}.

Then there exists the osculating к-plane клк (where r e {1, 2, 3}) at the point 1\ {f) under the following conditions:

(A) \nk exists i f the finite limit (13) exists fo r each i e { l , and the

П

vector C — ]? Сгкк { together with vectors

i —1

(15) L (t0),L (t0) , . . . ^ L ( t 0) form the system o f linearly independent vectors',

(*). ^. . ‘

(В) клк exists i f the finite derivative x l (t0) = B lk exists for each i e {1, . . .

П

n] and the vector В = В гк { together with the vectors (15) form the sys-

i = 1

tern o f linearly independent vectors ;

(*.).

(С) \лк exists i f the finite one-side derivative x l (t0) = Qlkt of order к exists

П

for each i e {1 , . . . , n} and the vector Q = У Qlk ki together with the vectors

i=l * t

(15) form the system o f linearly independent vectors.

The osculating к-plane клк (r e {1, 2 , 3}) is described by the system of equations

(16) Д (xs - x sk)B

s e { l , . . . , n —k}

,n—k + l . . . n

+ 2 ^

j —n —k + l

4 X l +' -^{n s n —k + 0,}

where j)\l “Ak = ф “Ак, D T "ik = B h "Ak, D*1'"1’* = Qh "Ak.

P roo f. An osculating fc-plane at P 0 e L exists if the components of the vector Hs, normal to hyperplane, exist and do not vanish simulta­

neously. Considering (6), we can see that the problem of existence of the components of Hs which do not vanish simultaneously leads to the problem of the existence of not simultaneously vanishing limits of determinants of-the form (10).

'(A) According to Lemma 2, the finite limit of the determinant (6) exists at the point tQ for each sequence г\ , . .. , ile and it has the form of

(12), where in the first column we have C\s instead of the B%k for each s e {1 I t is necessary to show only that not all these determinants vanish simultaneously.

(fc-b

By the assumed linear independence of vectors C, L{t0), . .. , L (t0) the matrix formed from the components of these vectors has the rank k.

20 S. F u d a l i

I t means that at least one determinant of order Tc formed from the com­

ponents of vectors mentioned does not vanish. By a suitable choice of the coordinate system in E n it is possible to show that such a non-zero determinant has the form as one in (12) for (q, = (n — Je-f 1, n — к -f

+ 2, i.e. that it is just B n~k+u-‘n. A square of B n~k+1,,'n appears in the denominator of each component of the limit form of Hs at t0 and this means that each such component exists. The determinant B n~k+1-”n ap­

pears also in the numerator of some one component of the mentioned limit and this fact together with the previous remark about the existence of all components shows that at least one component of the limit form of H3 does not vanish for each s e {1, . .., n — ty. Consequently, there exist n — Jc linearly independent vectors lim Hs and these vectors determine uniquely a fe-plane. This ft-plane is \nk. I t is easy to see that it is described by the system of (16).

(B) Considerations are the same as in (A) using Lemma 1 instead of Lemma 2 and changing the designations accordingly.

(C) Considerations are the same as in (A) using Lemma 3 instead of Lemma 2 and changing the designations accordingly.

References

Cl] C. F u d a li, S. F u d a li, On the osculating plane of a curve in an n-dimensional Euclidean space, Comment. Math. 13 (1969), 15-21.

[2] S. F u d a li, On the osculating Тс-plane of a curve in an n-dimensional Euclidean space, ibidem 20 (1978), 315-322.

[3] —, Finite differences of type r > 2 (in Russian), Demonstratio Math. 14(1981), 909-926.

[4] —, Some remarks about finite differences (in Russian), Prace Naukowe Instytutu Matem. i Fizyki Teor. Politechniki Wrocl., Studia i Materialy Nr 8 (1973), 75-88.

[5] K. R a d z is z e w s k i, Sur certaines propriétés des courbes admettant les plans oscula- teurs orientés, Annales Univ. MCS 17 (1963), 105-113.

£6] E . J . v a n d c r W a a g , Sur les plans oseulateurs, I, II, Indagationes Mathematieae 14 (1952), 41-62.