Séria I : P R A C E M A TEM ATYCZN E X V I I I (1974)
Staniseaw Fttdali (Wroclaw)
On the osculating /г-sphere oï a curve in an ^-dimensional Euclidean space
In definitions of the osculating к-dimensional sphere (A;-sphere) of a curve in an n-dimensional Euclidean space E n one usually requires that the defining functions xj (t) be of class Ck+1 (i. e. have continuous dérivâtes of order up to fc -H 1, with respect to parameter t). This is a rather strong condition.
Y. Jarnik [3] has shown that, given a plane curve L : y — y(x) of class G1, it is necessary and sufficient for the existence of the osculating circle that the second derivate y" {x0) should exist at the limit point x 0.
The present paper comparises a generalization of Jarn ik ’s result to the case of arbitrary n and к < n, the dimensions of the Euclidean space and the osculating sphere. However, the generalization is not complete since it gives but a sufficient condition for the existence of the osculating k-sphere. A necessary condition cannot be obtained by means of employed methods if к > 2. The appropriate generalization for the case к — 1 has been given in [1].
1. General remarks. Let a coordinate system in an ^-dimensional Euclidean space E n be fixed and let
П
(1.1) Z(i) =
i = 1
be the parametric equation of a curve L in E n, where x'{t) are (7w+1-class functions of parameter t and kt denotes the г-th unit vector.
Definition 1.1. A point P0 = (a>J, . . . , a£) = (ocl {t0), . . . , xn(t0)) of the curve L of class Gm+l will be called point o f inflection o f order m iff
( m )
the vectors L (t0) , . . . , L (t0) are linearly independent yet the vectors
( m + l )
L (t0), . . . , L (t0) are linearly dependent.
I t follows from the definition that if P 0e L is not a point of inflection of order m, then it is neither a point of inflection of order m —r for any r = 1, 2, . . . , m — 1.
1 2 S. Fudali
A ^-sphere in E n is uniquely determined by any of its & + 2 points which do not belong to a dimensional plane. I t may be also determined by some k — h-\- 2 of its points and h tangent directions, where h is any integer not exceeding (l+7c)/2; the points should not belong to a ( k - h ) - dimensional plane. In the sequel we shall consider only ^-sphere determined either by
1° Jc-j-2 points and 0 directions, or by 2° 7c+ 1 points and 1 direction;
the points will always be points of the given curve L and the direction will be tangent to L at one of the chosen points.
Select 7c —T& + 2 (h = 0, 1) points P s = (x\, . . . , х3) = х г(у з), . . . , x n(ts)) (s — 1, 2, . . . , k —h p2) of the curve L given by (1.1) which do not belong to a (k — h)-dimensional plane and consider h directions tangent to L at the point P k_ h+2 (whenever tangent directions are considered, the case h = 1 is meant). Join the point P k_h+2 with each of the remaining points P r (r = 1, 2, . . . , 7c — h + 1 ). We obtain к — h + 1 vectors
71
'(1-2) P k- k+2P r = (xl ~~0t)k - h+2)kj-
i = 1
If к = o, the selected к + 2 points P s {s = 1, 2, . . . , 7c+ 2) of L deter
mine a 7c-sphere which will be denoted by Sk({P s}) and whose centre will be denoted by P c = (£*, . . . , £”). The coordinates of the centre are given by the formula
( 1 * 3 ) £c — \ { x k + \ + ж&+г) +
+ « ( у ([*: + l ] f » j ) ( - i r 1 n +2 [* + 1 W # * )) (2 Â ([ft + l ] f +1))!)
i(k) r — 1 i ( k +1)
(p = 1, 2, . . . , n) and the radius is given by П
(1.4) B ( {P S}) = i ( W . - W 2)2 y ( îV [ it+2(t+1])3 +
' i = l
+ Ê ( [ f c + i ] f +i) 2) " ’ ) ' * ,
i(k) r = 1 i ( k + l )
where
(1.5) (7 = ( _ 1)(*+D»+1
П
and £ denotes summation over all increasing 7c-tuples i(k) of indices m
i ! , . . . , ik admitting values 1 through n\ i(k ) p denotes the multiindex
i i, Further symbols are defined by induction:
o - v i r ^ 4 i d f X k + l ~ X k + 2 . 1 - V K 4 4 л df l h + i h l ~ x l + l 1 ^ - \ f k + 2 l k + l J — . , ? ] Л l l k + l ffeJ — , , j
X+l ^fc + 2 % + l
[tk + 2-htk+l-hh--fl] —
(h = 0, 1) and, in general, (1.6) qX* [tk+2_h. .. tr+1tr]
df ] X J’ [ t k + 2 - h h - h l ~ }] X j [ t k + 2 - h ^ k + l - h ^ Lk —h l k + l ~ h
df l - l - X ? [Ьс + 2-А * • • % + 2 У q - 1 ^ ' [ X + 2 -A * • • ^ r+ 2 ^ r + ll
tr tr + 1
(q = 1, 2, . . . , 1c + 1 — /i ; r = 1, 2, . . . , & + 1 — Д) for j = 1, 2, . . . , n. Further
n k —r+ 1
(1.7) 3TP = 1 1 (k-r+°s^ [У * • • • ^r+s—1 ]0s^ '[ W f X + S — ] • • • У )
1 = 1 S = 1
(/8 = fc + l , & + 2), and finally
( 1 .8 ) [ft + l ( * ) ] f > df
fc+;z ** [...]•• • fc+JX1* [••..]
...
к_г+1Х{'[...-] ,_ г+3° х гЧ . . . ] - ,_ г+? х гЧ ...]
If h = 1, the selected fc + 1 points P r (r = 1, 2, . . . , fc + 1) of X and the vector
П
Q k + l = Z *1, æ ( X + l ) ^ t 1 = 1
tangent to L at Рй+1 determine a fc-sphere which will be denoted by Sk({P r], Qk+1) and whose centre will be denoted by P c = {rjl, . . . . ??”).
The coordinates of the centre are given by the formula (1.9) T)ç = 0Ck+i +
n к n
+ o [ i ( [ h + i f k» i ( 1 ) + &"k+i [ f t + l ( f ) ] f >))(2 2 " (tft + l ] f +1,)! ) '
Цк) r = 1 i [ k + 1)
14 S. Fudali
(p = 1, 2, n) and the radins is given by (1.10) B ( {P r},Q t+,)
where notations (1.5), (1.6), (1.7) and
(1.1 1) [ft + l(r )]? ‘
I X * * Ч ...] l x h 1 Х Ч --Л
k - r . l X h [ . . . ] * - r + l X h [ • • • ] . . . u - r + \ X i k l • . ]
1 * 4 - . ] 1 х г* с ...] ..
bk+l X\’*+1 Jük + 1
are employed.
I t is evident that the structure of formulae (1.3)* (1.4) and (1.9), (1.1 0) does not depend on the choice of ft —ft+ 2 points and ft tangent vectors. I t is also independent on their arrangement.
Definition 1.2. A finite sequence . . . , tk_h+2 of values of para
meter t will be called regular provided tm lies between tm_2 and 1т_г for m = 3, 4, . . . , /г — ft+ 2 (see Fig. 1).
4---1--- 1— H ---- 1---1—I— I---1--- (-
*1 *3 4 fk~h+2 *6 U 12
Fig. 1
A regular sequence of ft — f t - f - 2 numbers will be denoted E S (ft — ft + 2).
In the sequel we assume, for simplicity, that the selected points of the curve L are arranged in such a way that the corresponding sequence of values of argument t is a E S (ft —ftp 2). In this case the sequence of the selected points P 1, . . . , P k_h+2 will be called a regular (ft —ftp 2)- tuple and will be denoted E S P (ft — ftp 2).
2. The osculating ft-sphere. Let A be a curve in E n defined by (1.1).
To each regular (ftp2)-tuple of its points (E S P (ftp 2)) there corresponds the ft-sphere S k ({P s}) whose radius and coordinates of the centre are given by formulae (1.4) and (1.3).
Let (E S P a(ftp 2 )} (a = 1 , 2 , . . . ) denote an infinite sequence of regular (ftp2)-tuples {P “} selected in a neighbourhood of a point P 0e L and let |$*({P“})} be the corresponding sequence of ft-spheres. We say
that the sequence (R SP a(& + 2)} converges to the point P0 along the curve P whenever it converges pointwise (i.e. P°s P 0 for all s) pre
serving the relation of “lying between” established by Definition 1.2.
The osculating ^-sphere of L at P0 is defined as the limit sphere of the sequence {$ *({P “})} when {R S P “(& + 2)} converges to P0 along L . The point P0 may be situated in several positions with respect to the sequence {R S P a(Zc + 2)}. We consider the following situations:
1° P0 belongs to none of the arcs P\Pl.
2° P0 belongs to each of the arcs P “P£ and is itself none of the points of any regular (k + 2)-tuple.
3° P0 is a point of some regular (k -h2)-tuple.
In each case the conditions concerning the defining functions xj (t) required by passing to the limit are somewhat different. In connection therewith we introduce various symbols to denote the osculating A:-sphere.
Thus:
Definition 2.1. Let R S P a(A; + 2) converge to P0 along P ; the limit /с-sphere of the sequence |#*({P“})}, whenever it exists, will be called the osculating к-sphere of the curve L at P0 and will be denoted by
°r8 k (r = 1, 2, 3):
°rSk = lim Sk({P as}),
R S P a(A: + 2)-»JP0
whereby we write
(a) j8k if P0 belongs to none of the arcs P “Pa,
(b) 2$* if P0 belongs to each of the arcs Р\Р1 and is itself none of the points of any regular (fc + 2)-tuple,
(c) 38k if P0 is a point of some regular (As+ 2)-tuple.
Obviously, the sphere °r8 k (r = 1, 2, 3) exists iff the corresponding finite limits of expressions (1.3) and (1.4) exist, and the latter is positive.
The following definition concerns the case h = 1; let (R S P a(A; + l) } be a sequence of regular (jfc + 1)-tuples (P"} selected in a neighbourhood of P0 and let {$ *({Р “}, Q£+1)| be the sequence of corresponding spheres.
Definition 2.2. Let R S P a(fc + l) converge to P0 along P ; the limit /с-sphere of the sequence j$ fc({P “}, Q*+i)|, whenever it exists, will be called the osculating k-sphere of the curve P at *P0 and will be denoted by ^ 8k {m = 1, 2, 3):
i S k = . lim Sk({P ar},Q ak+1),
I tS P a(ft + l)-HP0
whereby we write
(a) j/S* if P0 belongs to none of the arcs P "P £,
(b) g8 k if P0 belongs to each of the arcs P “Pa and is itself none of the points of any regular (k + l)-tuple,
1 6 S. Fudali
(c) \8k if P0 is a point of some regular (& + l)-tuple.
I t is clear again that the sphere ft8 k (m = 1, 2, 3) exists iff the corre
sponding finite limits of expressions (1.9) and (1.10) exist, and the latter is positive.
3. Sufficient conditions for the existence of the osculating ^-sphere.
Write
(3.1) [Б [fc + l (/)]]*<*> =
(r = 2 , 3 , . . . , k) and
B l1
(k\
K 1
(f c -r+ 3 )
nnh ■1 Xq
(Л-Г+1) J 1
B 'l (fc + 1)! (Л + 1)!
(ft)
< k
(fc -r+ 3 ) X,
П
m3=2 (k — ^ + 2)!
(A-r+l) fk-'o
X n
X,} k
П к r + l (fc —Г —S + 2)
(3.2) x%2 у у 4
(r = 1,2,
(s!(jfc —r —s + 2)!)П- 1 i=i s=i
We assume further:
1° the curve L is given by (1.1) and is of class Ck, 2° the point P 0e L is not a point of inflection of order k,
3° the points P 1, P k_h+2e L form a regular (k — h + 2)-tuple (h = 0,1).
Theorem 1. The osculating k-sphere 2Sk {h = 0 ,1 ) o f the curve L at (fc+i)
the point P0 exists provided there exist fin ite dérivâtes x j (t0) = B { (j — 1, 2, . . . , n), not all equal zero. Then the coordinates o f the centre o f 2Sk may be expressed by the form ulae
n к
(3.3) v = + < * ( y (L® ( —i r +1* ':[ 5 [ f c + i( r ) ] ] i,,:>))x
i(k ) r = 1
n
x(2 У (|B[& + l]|i(*+1>)2) ' 1 г(/с+1)
(p = 1, 2, . . . , n) and its radius 2B k is given by
n к
(3.4) %& = è ( y i y t - i r ' ^ l B t f c + if r ) ] ] ’' » ) ^ ' г(&) r=l
n
x ( ^ ( И ^ + ч Р 44)2) " 1) ;
for the summation index r = 1 suppress the first row in determinants (3.1) which occur in both form ulae. I n form ula (3.3) notation (1.5) has been used.
The proof is based on two lemmas :
Lemma 1. Let xj (t) be a Gk+h-class function (h — 0, 1) in an interval (a , b} and let G, . . . , tk_h€ (a, b) be a regular sequence o f points BS(ft — h).
Then fo r every point ts€ (tm, tk_h) (m = —1, 0, letting t_1 = a, t0 = b) with ts Ф tp (p = 1, 2, . . . , k — h — 1) there exists a point %e(tm, h-m) su°h that
1 (ff+A)
J Z * - - - X* (x) ( q < h - h -1 )
(of. [2]).
Lemma 2. Let xj (t) be a Ck-class function in an interval {a, 6 ); let
(fc+i)
a fin ite derivate x* (t0) = B { exist at a point t0e <(a, by and let a, b, b , t2, . . . . . . , t k_h (h = 0,1) be a regular sequence o f points E S (ft — ft-f-2) in the interval by. Then there exists a fin ite limit
lim i_ti+fc-î7' \fk-h- • L l ab]
RS(fc—7t + 2)->^0
---В1
(ft + 1 ) !
klj
at the point t0(cf. [2]).
P r o o f of T h e o re m 1. The coordinates of the centre and the radins of 2 Sk [resp. lS k~\ are defined as the limit values of expressions (1.3) and (1.4) [resp. (1.9) and (1.10)] when E S P “(ft — ft+ 2) converges to P 0; and P 0e PjPg for each a. Let us pass to the limit, then.
The limit of the first term on the right-hand side of (1.3) [resp. (1.9)]
equals x% and the limit of the second term exists whenever a limit of the numerator and a non-zero limit of the denominator (the expression (.. .)-1 ) exist. This denominator is a sum of squares of (ft 4-1) x (ft + 1 )-deter
minants; those determinants have the form
(3.5) [ft + l ] f +1) (ft = 0 , 1 ) (see (1.8) and (1.11)), and their elements are
(3.6) kX W[tk + 2_h. . .tk+2- h -il
(i = 1, 2, . . . , ft + l - f t ; = h , i2, . . . , ik+1).
There are n\[(n — ft — l ) !( f t4- l)!)_1 such determinants since this is the number of all sequences ги . . . , ik+l chosen out of the index sequence 1, 2, . . . , n . Passing to the limit, we obtain from each determinant (3.5), by Lemma 2, the finite-valued determinant
(3.7) [P [ft4 -l]]*(fc+1) (see (3.1)).
2 — R oczniki PTM — P r a c e M atem atyczn e X V IH .
18 S. Fudali
(3.7) is a subdeterminant of the n x (fc + l)-m atrix whose rows are
(fc+i)
vectors L (t0), . . . , L (<0); and since P0 is not a point of inflection of order h, so at least one of the determinants (3.7) is not equal zero. Its square is positive and so the limit of the denominator is positive.
The numerator in question is a sum of the following expressions:
(& + 1) x (& + 1 ^determinants of the form (3.5) (with ik+l = p), k x k - determinants of the form
(3.8) [^ + l ( r ) l (fc) {h = 0 , 1 ) (see (1.8) and (1.11)) with elements (3.6) (w ranging i 1, ik) and sums of the form
k —r + \
(3.9) k_r+8X? [tk_h+2tk ... tr+s_ilg.57 [tk+1tr+3_ 1 ... Ir] .
s = 1
The limit of the (fc + 1) x (& + 1)-determinant is determinant (3.7) by Lemma 2. The limit of the Tc x k -determinant is a subdeterminant of (3.7), i.e. a finite-valued determinant (3.1). In the sums (3.9) products of expressions of the form (1.6) (with h = 0 and q ^ Jc ) occur. So the limit of such a sum is the finite valued sum
k - r + l (S) ( f c _ r - s +2)
(3.10) y j x30 [s\(k — r — s + 2)!)-1 S = 1
by Lemma 1. Consequently, the limit of the numerator has a finite value and thus the limit of the right-hand side of (1.3) [resp. (1.9)] exists for every p = 1, 2, . . . , n. The value of this limit is given by the right-hand side of (3.3).
Now we pass to the limit with expression (1.4) [resp. (1.10)]. This limit exists whenever a limit of the numerator and a non-zero limit of the denominator exist. (The limit of the first term in (1.4) is zero.) The expres
sions occurring in the considered numerator and denominator do not differ essentially from those which occur in formula (1.3) [resp. (1.9)].
In view of this remark we may assert that the limit of the denominator is positive and the limit of the numerator is finite non-negative. Thus the limit of the right-hand side of (1.4) [resp. (1.10)] exists and is given by the right-hand side of (3.4). This latter expression is positive whenever its numerator is positive, i. e. when the expression
к
(З.П ) £ { - . 1 ) г+1аГ0[В[1с + 1 {г)]у м (see (3.1) and (3.2))
T — Ï
is different from zero for some index-tuple гг,г 2, . . . , i k. We shall show that this is the case.
Suppose that (3.11) is zero for any index-tuple i x, i 2, . . . , ik . This means that either
1° the determinants of the form (3.1) are equal zero for every r and every index-tuple i x, i 2, . . . , ik , or
2° the sums of the form (3.10) are equal zero for every r and all j ’s, or
3° determinants (3.1) and the sums (3.10) are not all equal zero, however, the sum of the occurring products of them is zero for every index-tuple i x, i 2, ...,% .
We consider those three cases: 1° means that each Tc vectors out of (k+l)
the system L {t0) , . . . , L ( t 0) are linearly dependent, i. e. P0 is a point of inflection of order Tc, contrary to the assumption. 2° means that L {t0) —0 and P0 is again a point of inflection of order Tc.
In the case 3° some of the summands in (3.11) are different from zero, say аГр Ф 0. This means that the determinant [В [Тс + 1(г0)]]г(А:) (see (3.1)) is a linear combination of the remaining determinants (3.1). Note that determinants (3.1) with r = r 0 are just subdeterminants of (3.7). Hence it follows that (3.7) is equal zero for any index-tuple i x, i 2, . . . , ik+x and
(fc+i )
linear dependence of vectors L (t0) , . . . , L (t0) results contradicting the assumption.
The above considerations yield the conclusion that expression (3.11) is different from zero for at least one index-tuple i x, i2, . . . , ik and so the limit of expression (1.4) [resp. (1.10)] is positive. Hence, in view of the fact that the limits of (1.3) [resp. (1.9)] are finite for every p , the existence of the osculating ^-sphere 2Sk follows (by Definition 2.1.b [resp. Defi
nition 2.2.b]).
Corollary. The osculating Тс-sphere \8k (h — 0, 1) o f the curve L at
(A+J)
the point P 0e L exists provided there exist fin ite dérivâtes xj (t0) = B }k (j = 1 , 2 , . . . , n), not all equal zero. Then the coordinates o f the centre o f the sphere %Sk and its radius may be expressed by form ulae (3.3) and (3.4).
The proof is the same as that of Theorem 1. We have also
Theorem 2. The osculating Тс-sphere \Sk {h — 0, 1) o f the curve L at the point P0 € L exists provided there exist fin ite limits
(3.12) lim
q,<2-><o
(A) (A)
■X3(t2)
tx t2 = 4 (j = 2> •••>*&)>
not all equal zero. Then the coordinates o f the centre o f xSk and its radius may be expressed by form ulae (3.3) and (3.4) with Ck in place o f B { .
2 0 S. Fudali
If the defining functions xj (t) are not of class Gk+1, no necessary condi
tion for the existence of kSk (h = 0, 1; r — 1, 2, 3 ; к ^ 2) may be obtained.
In fact, the existence of hrSk means that finite limits of expressions (1.3) [resp. (1.9)] and a finite positive limit of (1.4) [resp. (1.10)] exist; those expressions are quotients in which terms of the form (1.6) with the same q occur both in the numerator and in the denominator. I t may happen th at neither the numerator nor the denominator has a limit though a limit of the quotient exists. Such a difficulty does not occur if class Gk+1 of the functions xj (t) is assumed and thus this condition cannot be replaced by a weaker one.
In the case Тс = 1 the situation is exceptional. Namely, terms (1.6) occur in the denominators of the quotients (1.3), (1.4), (1.9), (1.10) with a larger q than in their numerators and class G1 of xj (t) suffices for the calculation of the limit of the numerator. Thus the existence of a finite limit of the quotients (1.4) and (1.10) implies the existence of a finite limit of the denominators; hence, in turn, it follows that either finite limits (3.12) or the dérivâtes xj (t0) = B{ exist, not all equal zero. This fact has been exploited in [1].
References
[1] S. F u d a li, On the osculating circle of a curve in an n-dimensional Euclidean space, Prace Mat. 13 (1969), p. 3-13.
[2] А. О. Гельфонд, Исчисление конечных разностей, Физматгиз, Москва 1959.
[3] V. J a r n i k , О Jcrûënice Jcrivosti, Casopis pro péstovani matem. a fisikÿ 74 (1949), p. D37-D51.
IN S T Y T U T M A TEM A T Y C ZN Y U N IW E R S Y T E T U W R O C LA W SK IEG O
