I. A necessary and sufficient condition that a curve lie on a quadric surface; II. A necessary and sufficient condition that a curve lie on a hyperquadric

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a ieqess&ix aid S B m cistrr ccutsmoif th a t a coeve

bit OUA QUADEIG SUEFACK

by

LGOXS G. GMUE

Sv’G_\n Subaitted to th© Faculty of the Graduate School In p a r tia l fu lfillm e n t of the requirements fo r the degree, Booto r o f Philosophy, in the Department o f Mathematics,

Indiana University

ProQuest Number: 10295191

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i

IHTBQBUCTI01S

Tb© problem o f finding a necoseajy and s u ffic ie n t condition th a t & curve l i e on n cmr d ric su rf see m s suggested to w ^ Professor Elevaiy because, a t the time, he m s presenting the theory o f sp h erical curves in our D iffe re n tia l Geometry ©ours© and th is problem should be & nice extension o f the known necessary and s u ffic ie n t condition th a t a curve be sp h erical, k% f i r s t the problem m s thought to be a simple on© end th a t w© could obtain the r e s u lt by merely considering contacts o f a quadric and a curve as m s don© In the case of a sphere, I worked on th is method o f attack fo r quit© m m time m d found th a t although on© could th eoretically obtain the r e s u lt to th is manner i t would be very mess^y and geometrical in te rp re ta tio n s o f the d iffe re n t contact® were d if f ic u lt i f not impossible to fin d ,

I became discouraged

but then professor Hl&vaiy suggested introducing

11

& siew m%ri& which, as i t turned out, led to & very elegant method o f solving the problem. Xu section X X have found a necessary and su ffic ie n t condition, In tern© of quadric curvatures, th a t a curve l i e on a quadric hosiothetie to a given quadric surface*

Having ovt&ined th is

resu lt X was faced w ith the problem of tran s­ fe rrin g th is condition on the qiu-drlo curvatures to a condition on the crdinaxy curvatures which would be independent o f the given quadric and thus would give a necessary m d s u ffic ie n t condition fo r a aurve to l i e on any quadric* At th is p o in t X again had d if fic u lty In obtaining m elegant method*

Here again X owe the ales

to P rofessor Slavsty f o r hi® suggestion of constructing the invariant© which appear in sectio n IX* Xa ©action X we nade the essuaptions th a t th e quadric tangent and normal are not p a ra lle l to e lin e of the asymptotic ©one of a quadric homothetlc to the given quadric*

In ©action III

tit

I have shown which curves have been excluded by these assumptions* For refferen ce, m f a r as th is pspcr is consumed, there i s nothing b e tte r to be suggested than V* H lsvety's D iffe re n tia l Geossetrie* I wish to express my sin c e re st appreciation to p ro fesso r HL&vnty fo r both id s assistan ce and laep riatio a* LOUIS C; GRAUE April, 1949

X

Simon x X* fbaadamental Concents Xa th is section w© sb: 11 bo concerned w ith the problem of finding s. necessary m d s u ffic ie n t condition th a t a curve l i e on a quadric hoisothetic to a given quadric outface.

Ifhere ty a quadric

we always neon a re a l c e n tra l quadric whose equation may be w ritten in the fo m M

2 »bit*b * ©bX *e - sc) ♦ Y0^a * 0

w ith given re a l c o e ffic ie n ts a ^ and cen ter pfc^,®^*©25) where the determinant ( a ^ J • l t Y© * P *9 and d I s a re a l number. w ith the vertex pm

I f d « 0 then (1,1) i s a cone I f d j# 0 then the number |dj

i s c a lle d the rad iu s of the quadric (X*X). th e same number i s also ca lle d the radius o f a conic sectio n o f (1,1) by a plane in cid en t with i t s center p. Hots also th a t Y~ *0 * X ch aracterises the nonruXed re a l surfaces and yc 38 ~1 ch aracterises the ruled hyperboloid.

a

Three functions o f the m m argument t

n *X(b), # * x ^ ti),

(1,2)

xs » xs(t)

define a curve i f they posses the following p rop erties (a)

They are uniquely defined m d continuous ✓*\

fo r a l l values of t In an in te rv a l S L ^ (b)

la

th e re e x is ts aa dnterv&l S L guch

th a t the functions (1,2) have continuous d e riv e tie s o f a t l e a s t order k 1 1 fo r a l l t in JT1 • (o)

the Wittolx p * ( ^ * ||? »

does not

have the rank h * 0 fo r e l l values o f t in SL- , Each number t la th e curve*

I s ca lle d a p o in t of

Each point of the curve in the

in te rv a l J X fo r which the m atrix $ has the rank h » 1 I s ca lle d a regular point of the c la ss k» I f ©very point in m in te rv a l about^is o f c la ss k thsb the p o in t t i s said to belong to an in te rv a l o f c la s s k .

Unless the contrary i s sta ted

we sh a ll assume any point we speak o f to be regular# V® introduce th® radius vector r o f & p o in t of the curve w ith coordinates x ^ ( t) , x ^ (t) and la place o f the equations (1,2) w rite r * r ( t ) fo r the equation o f a curve*

Without

I

horn of generality w© my assume that a point of the curve under Investigation corresponds to th® value i « 0 and express by the symbol ( .« ,)0 the value of an expression (..#} for t » 0* In p a rtic u la r we m y designate the point t * 0 IT V

D efinition 1,1 I f u and v are two vectors w ith u * (u^,u*#u^) and v # (v^t v^,vs ) then th e in n er product u*v i s defined by the re la tio n ( 1 ,5 )

* V .U =

2 -a fc c u V .

D efinition 1«2 Th® vectors u and v are said to be p a r a lle l I f u « ev f o r some constant c jf 0* Tit® vector® u and v are said to bo oonjug&te i f u*v » 0. 'r/ ^ >>■I f we have b th ir d vector v * (v ^ v ^ v * ) then I t I s said to be conjugate to the plane detersdned by u and v I f w*u * 0 and v*v a 0

.

4

D efinition 1.5 th e expression ( l f 4)

u .u »

jla ^ u ^

i s ca lle d the modal of the v ecto r u and the p o sitiv e square ro o t o f the absolute value ( I . e . i s c a lle d the len g th o f ih@ vector u«

Bemrk sSSSSsASS, ¥e v i l l o ften use the symbol

and sh a ll

always saean th© p o sitiv e square ro o t unless th e contrary i s sta te d .

D efinition 1*4 $ v ecto r whose length I® equal to one I s c a lle d a u n it v ecto r.

1 HStiUL I t m y be e a sily ©hovn th a t the d e fin itio n s (1>1) to (1,4) do not depend on the choice of the coordinat© system.

%

Sm SLM I f u , v and v are t-hre^ veotore then we understand by [u#v,w] the determinant [ua >vftfwa] , Hot® th a t I f u&, vm m d v& are the cov&ri&at component© o f u, v and w we here [**&#▼&,,%} * wh*r« [*te ] a 1 ,

AMBatptlon In th is sectio n we w ill consider only those curves fo r which dr*dr ^ 0*

Hence we

exclude the case o f d r being a d irec tio n p a ra lle l to a lin e of the asymptotic cone of a quadric hosiotheiic to (1,1)*

D efinition l.S th e pammeter S defined tgr ( l fb)

dS « vjfdr.drl a

\[

?l

w ill be c a lle d e quadric a re .

g » i& Henceforth we put 1r{t) * H(8) and dendte the d eriv a tiv es w ith respect to S by accents (i*e* R *

etc.)*

6

Definition 1»6 Assume Mq belong© to an in te rv a l o f cl&g$ 1, For a ourve E * E(S) the i n i t i a l point o f the ire©tor dfL w ill elvsyB be taken In the p oint TL. m

•

The u n it v ector j g w ith It© i n i t i a l point 00 fixed as 1® ca lle d th e u n it quadric tangent and v i l l be designated by T.

I t s coatravarifent components

e re I s" * i2% i t s cov&rlant cospoiaente are f e * abo^b» dS

I seS Throughout th is section v© v i l l confine ourselves only to the case

0.

Hence we

- /

exclude th e case o f T being a d irectio n p a ra lle l to a lin e of the asymptotic cone of a quadric

feomothetio to (!» !)• Ve denote by end

y2 ^

Y^ 38

36 ®P* T*T

D efinition 1.7 Assume th a t

belongs to an in te rv a l of * //

c la ss 2 sad th a t R0 5®0 I s a vector not p a ra lle l —/

to Bp# Then the u n it v ecto r of the m m d irectio n

7

and orient*tioa a© the vector ? o which bee it * i n i t i a l p o in t in E0 i s ca lle d the f i r s t quadric aor&al and is designated ty K0 *

Theorem 1*1 (1,6)

M* 5

*

K«1

and v© have y2 * ITjf *

Proof*

By definition 1*7 Ma

tL. and since ? «

then ? * Jl'm A ty d e fin itio n 1.1

MM* ^ d g 2 dS2

By the l a s t rem&xfc above we have y„ a sgsf #f (& hence y *

D efinition 1.8 *—/

th e len g th of the v ecto r f » namely &i « i® c a lle d the f i r s t quadric curvature o f the curve E a R(s) and quadric curvature*

i s the radius of f i r s t Botlce th a t

^

*

.

a

theorem 1.2 th e plan© f 0 defined by T end ¥& has the following p ro p erties: (a)

I t i s the osculating plane of our

curve at H© . (b)

I t i s n o t p a ra lle l 'to a tan g en tial

plans o f the asymptotic cone of the quadric (1*1). Proof,

The osculating plane is defined tgr d r

and d$F ^ dr. at* dt

Since dir « MS. dt dt

the plane defined by d r end d^r 4e the seme as the plan© defined by S^w T and which proves asse rtio n ( a ), Because T .f j* 0 , t i s not p a ra lle l to a s tra ig h t lin e of the asymptotic cone of the quadric (1,1) and fey the eeote reason the ean& is tru e fo r T * %N, Since f S 38 0* the vectors t and H are conjugate. Hence the plane defined by ? and H can not be p a r a lle l to a ta n g en tial plane of the asymptotic cone.

9

D efinition i>9 knmma th&t \

belongs to an in te rv a l of

c la ss 5 and th a t

a**)

nC C i^ i ^ o .

then the u n it v ecto r B0 which in conjugate to t Q and S0 w ith It# i n i t i a l p o in t a t 1^ and o rie n ta tio n so th a t , v

(1*3)

„_______ ,

-/

-H ~///

#W» tt0,Ho,B0l * sgti [Rq, Bq,

] *y

i s c a lle d the quadric binoim lp

Theorem I»S th e d ire c tio n cosines of the biaoxm l B a t R(S) are given tjp

where

10

Broof*

If

are the eontr&v&rient

components o f the' vectors T, ¥ f ¥ we have by virtu© o f d e fin itio n X«9 • b c t V - 0 , &te * V - 0 eo wo

any writo fjB?’ * TjB2 + fgB® » 0,

"i®1 ♦ "z®2 * MgB® * 0 .Hence B*i B2 i B8 « B^*©2*©8 th is (1.10)

# * $>#, B2 a pB2, B8 * pB8.

Because B i s conjugate to f and H and because the plane defined by f and H I s not a ta n g en tial plane of the asymptotic cone of the quadric (1,1) we have S*B / 0# know th a t

Bo in order to determine p we * 1 hence J p ^ s ^ B ^ 0! 38 1

thus

(MW where we choose the si^ a to s u it (1,9)* (1.11) in (1,10) w ith Y

S u b stitu tin g

the appropriate

sign m d replacing T^, and

fcy th e ir expressions

m d xjj we obtain (1,9) *

In

T h eo rem !,! The formulas fo r the d eriv ativ es of N and B ere (1.12)

I 7*

(i,is) k2 «

♦ KgB, 1

ts/, t 5, i i f .

where y$ 5 B*B,

11

Proof*

** mmL mmUmmIII The ds^ im im at [ l #B,8 ] * f t , ^ ^ - H K j I tp i

J i ~ J U fi

1

« ./

~2[K, R, R 1 j* 0 so 8 Is not & lin ear combination o f T and 8 alone*

fhen we m y w rite

F * /LT •*> A2¥ + KjjB wher* K2 ^ 0 and from th is and the equations ?.H » 0, K .l » y ve obtain mmfmm mmmtJ mmmm yj& l * n *f 38 ~H*T 38 m

mmmm^ YaAa « *.8 * 0

hence th e f i r s t o f equations (1 ,1 2 ).

How Ib t f ' * Cf + B» * iS

and since

B .f * 0f ¥ ,¥ « 0 , B*B * v we have /

Ylc = . i : F . - f . F = . . I . l c 1 » o YgB * M

* -E .8 » - I . ( - S XT * S„B) » -S2y 5

Y«® * i l l ■ 0

th erefo re we obtain the second equation in ( 1 ,1 2 ) • Sow to determine K„,* Using the f i r s t o f the mm/// /mm mm xml equations (1,12) we have E * ft^S * K^(-y^y^K^T * K^B) * ^ 1^ 2^

*

since

[ r1 K r 1 * [T, Kj®. *3*2?! * 4 S2 ^ » ®* R„ a pS* S»' ~ ■)■k *

[* f 8 , B ] 1

and thus I s p o sitiv e .

hw>c# Sow ly

_ _ _

m ultiplying th e two determ inants o f T, S, B one

546743

13

expressed In ©oatramrteat components and the other In eovariant components wo obtain » # j Y2Y8 * [Xb,lltB bl 2 « 1 * y #o Kg “ ^|£® » *>

h»nee

p»naitiQ B j..ao ?h© function

» i given fcy the equation (1,15) fig I s called the second quadric curvature o f the curve I * H(S)

and R i s the radius of second

quadric curvature.

Theorem 1JB The system of lin ea r d ifferen tia l equations (1,14)

- -y8_lYaK|l_ iq - l + V « 4

(a * l,a,8, t 0 1 I ( 3 0)

admits a solution o f three isuteXiy conjugate unit vectors T^, I g, Tg. Proof* l e t (1,16) f a. f b * Then 0 and I2 * K2(S) > 0 are given*

Then there exixt® a unique curve

with the following properties* (a)

B 1® it®

arc md Kp&g are i t s curvature®*

(to) I t goto® through m arbitrary given point* («r) I t posses a t th is arbitrary point the prescribed mutually conjugate unit vectors

t &9 80, ©0 m I t s tangent, nors&l and biaonaal.

14

Proof* (1.18)

Consider th© system of d if fe re n tia l ©cu&tiong

7-

f p l ' - "Ya_iYasft_i^a-l * V a+i (• « 1.2.8. *„ =*g = 0)

According to theorem 1*6 the constants o f in teg ratio n o f the In te g ra ls ( P, 1^, X^, £g) o f th is system may b© chosen so th a t the condition of conjugate u n it vector© fo r T& i s s a tis f ie d fo r a l l S. We sow choose the i n i t i a l conditions

fo r P

so th a t P(0) i s the radius v ector of

the fixed

prescribed fcy condition (h)*

point

Then w© choose

th e i n i t i a l conditions fo r

according

to the equations 1^(0) « T0# 1^(0) *

WQ>lg(0)

m

th is same

is possible ly theorem 1*5 and by

35BQ

theorem lj( 8 ) = T(S), I 2(S) *M (S), I«(S ) = I ( S ) . Wow l e t P{$) * B{$) so ve may writ© the system (1.18) as follows (1.19) B* f , ¥'» Sjf, f'= -YjYgKj? + K2B, l a Since T(S), ¥ (S ), B(S) are eonijug&t© u n it vectors from I » t v« gee th a t & i s the arc o f the curve whose p oints are given iy the radius vector R(S) • From the r e s t of the equations (1,19) which represent fre a c t formulas, we see th a t S^,

are the curvatures

o f the curve m 4 th a t the vector® T ,$, B are l i e tangent, noxn&l m& binormal. According to Cauchys axistance th e o r y the curve m constructed la the only curve which s a tis f ie s the conditions o f our theorem.

theorem 1*7 *2^®) * °

defined fcy (1,15) Is

a necessary and s u ffic ie n t condition fo r a plan® curve. Proof.

I t i s obvious from

For a p o in t H(8) s u ffic ie n tly close to S(o) o f the curve R * R(S) we m y w rite (1,20)

1(8) - I 0 « T0 (S -

- S**(Ki4)oYiYa ♦ •••! 5•

* Proof.

d,

+

< s 2 h +-

Using the f a c t th a t R * T and then taking

four d e riv itle e o f R making us® of formulas (1,12) then su b stitu tin g these values in the expansion we obtain (1,20)

1

16

2 . Contact of Curves

D efinition 2«1 Consider tvo curves R * E(s) &ad IP1 * 8*(S*) Vith a point R0

common. Then we say that

the curves have at IL « I?0 a quadric contact of 0 a t le a st the q’kh order, or that they have there an a tle a st (q*l)-point quadric contact I f for B0 « s j « 0

(p a

Theorem 2.1 A necessary and su fficien t condition that the two curves above have a quadric contact o f a t le a st order q at the point

a R* is

«? * 1 . . . * 0 - T *

« » 2 ...

*

« i* . ( q ) 0 « (Kf)0

1?

Proof*

For the curve R* « li*(S®) there te a

sisdl&r

developemeai to ( 1 ,2 0 ) and for SQ = S*

we have I ( ^ - R*(S*) • Sq{T ~ T*)0 + g ^ I - S ^ * )e

'* lffriY2Mfr + sf**) + ( # * * ^ F > + (V i5 - s^;p»)]0+... Bi fo r a quadric contact of a t le a s t order one we m e t have (2 ,2 ,1 )

Um*(aa) - B*(S*) .

s0®°

s0

f _^

O

th u s obtaining the f i r s t of the equations (2*1). For a quadric contact o f a t le a s t order two besides (2,2,1) we must have

* \ (KXH - *fe*)0 " 0 .

s jo

Assuming S t ^ 0 we obtain the second of the se t o f equations (2 ,1 ). F in ally fo r a quadric contact of a t le a st order th ree In addition to the conditions (2,2,1) and (1,2,2) we m e t have 1(11m

80a°

R(S )

s|

IS

Slue® F0 * 5® are conjugate to B0 and B* the

© - S & . - < w v - «?*>. must both vanish*

Sauce the th ir d s e t of equations

of (2,1) ere to fee s a tis f ie d .

th eorem 2*2

A quadric contact of a t le a s t order q i© tli© same m m ordinaiy confect of a t le a s t order q fo r q « 1, 2* Proof*

Consider fee two curves r{») * 1{S), 9*(®*) a B*(g*)

(2,B)

t *

,

t* -a ?*dS* di*

d®

t 0 - t* iapXies ?0 - ** ifflj>u e e *0 *

then

hence

(g )o . (| g ) o •

and thus fo r q * 1 they are the same fcy theorem 2 .1 . faking d eriv ativ es o f (2,5) we have -* jp c s ) as

w ith stare*

2

2

+®B. ob

and a sim ilar expression

Since a confect of a t le a s t order two

involves a contact of order one we have

10

Since f 0 * f * Is conjugate to H0 * HJ then the right side o f (2*4) fcqpal to sero implies

(% )0 * (s f)0r 0O *

( ^ |) o * (|g|g)© #

Hence an ordinary contact of a t le a s t order two Implies a quadric contact of a t le a s t order two. Conversely we m y write (2*4) in the form C Mo

t V - « P l * t f i 6 - 6 > .V » . ■ (mi - «£*!.(*}> Since t 0 ® tjj i s orthogonal to nQ and a* then the rig h t side o f thi© expression equal to sero

implies

(kj)© * (**)0, “o * “©* ( fp l* . * ^ 3 e V Hence a quadric contact of a t le a s t order two im plies an ordinary contact of a t le a s t order two. So fo r q a Z the contacts ere the seme.

B— wfe

We could continue the process of theorem 2.2 sod show th a t the quadric and ordinary contacts are the m m fo r la rg e r values o f q.

20

D efinition 2*2 A ncnsingul&r conic mde by any section of a quadric homotbeUc to (1*1) i s called a privileged conic.

Theorem 2*g Thera i s only one privileged conic which lie® In the osculating plane o f a point RQ o f the curve R * R($) and has at tM e point contact with the curve o f a t le a s t order two* This conic i s not a parabola. Proof*

The privileged conic® are hoiaetbeti©

end thus have the same two points at in fin ity in ootmen* The three point contact determines three conditions and these together with the point® a t in fin ity uniquely determine a conic.

It

must l i e in the osculating plane because fey theorem 2*1 i t m e t have the same tangent and normal vector® as the curve since i t has a contact o f order two,

iy the same reason i t s f ir s t

quadric curvature 1® not aero and consequently

21

i t i s not a singular conic,

According to theorem 1.2

the oeculatiag plane i s not parallel to * tangential plane o f the asymptotic com of a quadric houothetic to (1 ,1 ).

Hence the osculating plane in tersects

any such quadric In a eonlc which i s not a parabola,

B e fta ttlo a 2.S The privileged conic which goes through the point EQ o f the curve E * E(S) and has at th is point a contact o f a t le a st order two i s called the osculating conic of the curve in the point 3^.

I t s center i s called the center of

f i r s t quadric curvature of the curve at the point Eq*

The radius o f a p riv ileg ed c e n tral conic i s also i t s radius of f i r s t quadric curvature. Proof.

Let G fee a p riv ileg ed ce n tra l conic

and ?(*£* *£» * |) it*

T h e* , i s only one

r e a l quadric through C hossothetic to (1,1) with ce n ter p .

I t s equation i e

2j*fee^xb ~ S^ ^ JC° * *S^ 3

where y 0 I s a su itab ly chosen sign*

Without

22

lo s s o f .generality we m j suppose that the plane o f 0 I s (2,S)

x | # Hence tb s equations of G are

( f - p).(R - 5) * Y0^>

38 3e|,

Taking the f i r s t d eriv ativ e of ( 2 ,t) we have T.(E - p) « 0 hence (2#6)

B - p * iff *

S u b stitu tin g (2,6) in (2,S) we get Coaeequently pd « YgY©*^ S© we m y put p * -r*

a

* Y0r ^* and y2yo 28

Taking the d eriv ativ e o f (2,6)

we have t a jrtP»

v irtu e o f th is

equation p a - y ^ i ^ a - r ,

^ * y ^ r.

Theorem 2*B The rad iu s of the osculating conic in th© p o in t

o f a curve E s |( g ) i s given by the

radius o f f i r s t quadric curvature i \ ) Q and i t s ce n ter i s given by (2,7) Proof •

% » \

* Y!Y2( » i ) o V

% theoreis 2*1 the curve and the

o sculating conic «ust have the satae radius of

f i r s t quadric curvature luad since tbs radius o f th is conic is the same as i t s radius of f i r s t quadrto curvature (theorem 2*4) the radius of the osculating conic i s (% )0 whore (% )0 designate® the radius o f f i r s t quadric curvature of the curve in %he point I t .

Therefore

slues these two curves m ist also have the same quadric normal vector one of the two points of the radius v ector

£ (R^)0S0 isust be the

cen ter o f the o sculating eoaie*

To determine the

sign in th is expression we use the equation p * ~YxY2% ®°

E - Bq 3 ~Y£YpHqH« Hence

YlY ^ i* o rien ted towards the center of the conic and hence only the sign y^y^holds* us equation (2,7)*

This gives

24

S» Contact o f a fei&drle and a Curve

Let

be the radius vector o f a variable

point in space*

Let p be the radius vector o f

afljr fixed point and r any real numerical constant* Then any quadric bonothetlc to (1,1) and having center p s* (x*,

x®> and radius r has one of

the equations (Spl )

(R* * p ) . ( ^ ~ p) + yo»^ 58 0 where y0 « ± 1 •

¥e confine ourselves to real quadrlce ($,1) which restrict© the choice o f vTo * I f r ^ 0 the© v*o * 1 characterises the aonruled res! surface© and Yq * -1 characterises the ruled bgrperbolold*

d efin ition S*1 f he power P of an arbitrary point R in relation to the quadric (Bpl ) i s defined ty the expression

(8,2) F * (I - 5) .(1 - 7) *■Y(>r2-

2b

B eflntttoa S>2 Consider & quadric (8*1) which goes through o f the curve R * E(S) and F(EfO) the

the point

power of the point E{5) o f the curve ¥ » ¥($) with respect to the quadric (S#l)» {&*$) l i » m

Then i f

K,g.O> * O (p * 1,2,•.*,%) W

wo say that the quadric (S#l ) has at the point ¥

o

a contact with the curve I « E(S) of at lea at order q or that i t has with the curve at th is point m a t leant (q*l) -point contact#

Eewgqfe Henceforth we sh all never take the point R o f the curve ¥ * E(8) a t the vertex of a cone# o

Theorem g»X

A neeeseaxy and sufficient condition that a quadric (5,1) through the point I

of a curve

0 R * S(S) has at th is point a contact with the

carve o f a t le a s t order one i s that I ts center H e in the plane of the vectors

and B0 #

Froof*

The power F(0 f0 ) o f t b s p o in t %

w ith re s p e c t to th e q u a d ric ( S ,l ) v a n ish es tgr h y p o th esis s in c e th e q u ad ric goes through t h i s p o in t* F or th e d e v elo p m en t o f th e fu n c tio n F(SfO) we can th e r e for© w rite

(8,4) P(S,0) * W* ♦

+ lV " + ... 2(

S' °

f$e® a n e ce ssa ry m d s u f f i c i e n t c o n d itio n f o r a c o n ta c t o f a t l e a s t o rd e r one 1© th e re f o re acco rd in g to d e f i n i ti o n 8*2 m d th e eq u atio n ( 8, 4) t h a t 2^ * 0 * From equation ($ #2) we have

(5.5.1) p'(s,0) * 2(f(S) - p) J f a 2(R(S) - p).T and hence th e shove n e ce ssa ry and s u f f i c i e n t c o n d itio n can be w r itte n in th e fo ra

(8.6.1)

» 2(R(0) - p).f0 * 0.

Bine© th e q u a d ric goes through th e p o in t 8^ which i s not th e v e rte x o f a cone, then R0 - p ^ 0 . fh e re f© re by eq u atio n ( 3, 6 , 1) ^

- p i s conjugate

to th e v e c to r T0 end we m y w rite

(8,7,1)

^

- p a vS q * 780

where u and v a re a r b i t r a r y number®, n o t bo th z ero .

27

According to ib is equation the p o in t p, th a t I s , th e c e n te r o f the quadric l i e s in the plane of th e vector© !f0 m d B0 #

D efin itio n £m% fh e lin e through the cen ter of the osculating conic a t the point Ec conjugate to the plane o f th is canto, in called the quadric p o lar lin e o f the point 1 > 0«

anw w m . 8 .2 The equation of the quadric polar lin e ©f the p o in t Eq may he w ritten in the for* (5,3) * R * 1 ^ + where

•*&*

I s the radius vector of a point of the

p o lar lin e and ~u i t s d irected distance from the o scu latin g plane o f the point 1^* Proof*

I f 1© i s the cen ter o f f i r s t quadric

curvature f o r th e p o in t SQ then according to the d e fin itio n o f the quadric polar lin e the vector - I j must be p a r a lle l to the quadric biaora&l.

28

hence

- I© * -uB0 where ~u 1® the d irected

d istan ce o f a point o f the polar lin e from the o scu latin g p icas o t the point RQ* How bgr theorem 2*5 + YjY^R^Jo^o m d waking th is su b stitu tio n we obtain (5 ,3 ).

Ite m M i necessary and s u ffic ie n t condition th a t the quadric of theorem 5*1 have w ith the curve E * 1(0) a t the p oin t 10 a contact o f a t le a s t

order two i s th a t e ith e r of the following two equivalent conditions i s s a tis f ie d . (a)

The cen ter of the quadric lie® on the

po lar lin e o f the p o in t Bq. (b)

The quadric goes through the osculating

conic o f the point RQ. Proof*

A necessary t-ad s u ffic ie n t condition

fo r a contact o f a t le a s t order two according to (5#4) i s th a t the equation (5 ,6 ,1 ), which has as a consequence (8 ,7 ,1 ), be sa tisfied and th a t pf * o* v

Prom equation (5,5,1) we may get

2*

(3.6.2) P /J( B t O) ■ 2(1(8) - p ) . T * 2 l ( s ) .T a 2(1(3) - p)jCjf + 2yr

flmn using (5,7,1) ve get ( 8. 6 . 2 )

Fg ■ 2 ( ^ , - p).(&XS)0 * 2 yx

« 2(K1S)0.(a ^ ) + vH0) + 2yx * Z( V & * * Yl)o * 0 * So ve haw

(5.7.2) v -=• ^ ( V o Fro® equation ($ ,7 ,1 ) v© obtain the following equation fo r the cen ter |> o f the quadric (8,9) ? < * \ + Y]Y2 (^JD « ■ “V therefor® th e cen ter of the quadric l i e s on the p o lar lin e of R * th is quadric cuts the osculating plane in a p riv ileg ed conic whose cen ter i s the in te rse c tio n (u ~ 0) of the lin e (8,9) w ith the osculating plan©* Therefore th is c en ter of the conic i s the center of curvature fo r th e p o in t 1^*

Since the quadric also goes

through B0 th is conic also goes through th is point and accordingly must be the osculating conic o f the point

m

itemsrk

Since & quadric (£,X) which has a contact of o rd er two w ith the curve E « R(S) a t Eo has an a rb itra ry radius we m y prescribe i t to be r « 0*

theorem 1.4 A necessary and s u ffic ie n t condition th a t a cone (S 91 ) ( i . e . (S 91 ) with r * 0 ) have w ith the curve R ® 1(S) a t the point Eo a contact of a t le a s t order two Is th a t i t s vertex p i s (V-G) P »

Proof.

- (%) c (y / 0

Prom theorem B.S equation ($,9)

we have th a t a necessary and s u ffic ie n t condition fo r a contact of a t le a s t order two i s S , - f * *YjY2 (*x*)o + u®°* S u b stitu te th is la (S ,l) with r - 0 and we hi*ve ( “YxY2 ^ % ® ^ + '% > * ( - Y 1Y2 (Kx®>e *

* 0

Y a ^ o + Yg»2 3 0

a * ± (%)* Y2 3 -YS» Yj. 3 -1 and hence su b stitu tin g th is value of u in (-,9 ) we obtain (£*10 ) .

31

D efinition 3*4 Tfc* 0m m of theorem 3.4 are called the osculating cones o f th e point Bq. 4Remark SS8SSSSSSSS»

Ordiaarly a cone homothetto to (1,1) has a contact o f a t le a s t order two but i t doe© not always have © contact of a t le a s t order three w ith the curve*

We w ill now answer the question

when i t can have a contact o f a t le a s t order three*

A necessary m d s u ffic ie n t condition th a t the o scu latin g cones have w ith the curve R » R(&) a t the point RQ a contact o f a t le a s t order th ree i s th a t ( s ,ii) Proof*

(-R a iY g ltfo * o . A n ecessary end s u f f i c i e n t c o n d itio n

f o r a c o n ta c t o f a t l e a s t o rd e r three according to ( 3 *4 ) i s th e fu lfillm e n t o f equation© (H ,6 ),

which have equations (3,7) ^nd th u s (5 ,1 0 ) a s a

52

consequence, and in addition th a t

G.

From equation (5 ,6 ,2 ) wo got ( 8 , 6 , 8 ) f w/( 8 , 0 ) - 2 (1 ( 8 ) '5 i.(-Y iV 2Ki* + *1* + K ^ B ) which by using (610) becomes K

* 2( V o ( y / o ± I , ) .(-Y jY ^f* > K£? > t xt j b 0 • 2 (% )a ( ^ ±

0 * 0

nad since (B^)0 j* 0 we have (5 ,1 1 ).

The saa-etugul& r quadric (5,1) which 1b tm o th e tle to ( 1 , 1 ) and has a t the point BQ o f a curve 1 * B(S) a contact o f a t le a s t order three i s ca lle d the osculating quadric o f the point 10 *

&W& A nonsiogular quadric bomoihette to (1,1) I s determined uniquely by four points*

Hence

tb s quadric mentioned in d e fin itio n 5*6 may fee r ig h tly c a lle d m osculating quadric*

m

Theorem 3.6 Assume th a t

ji 0 m i (y2l f + y 84 2^ )

f 0,

then a aecess&ry and s u ffic ie n t condition th a t tb s quadric o f theore© 3.5 hsv® a t the point E0 o f the curve E * B(S) a oont&ot o f a t le a s t order thro® w ith th is curve and th erefo re represent th e osculating quadric i s t h a t .i t s ,cen ter *p 1 ® (5 ,1 2 ) p =. d

Proof.

YiYg^S ♦ Y rY s^ Jy^ o-

k necessary and s u ffic ie n t condition

fo r a contact of a t le a s t order th ree according to (3,4) i s ih© fu lfillm e n t of equations (3 ,6 ), which have equations (5,7) as a consequence, and in addition th a t P^#» o 0* S u b stitu tin g (3,9) In to (3 ,3 ,5 ) we obtain

(8,6,5) P* - 2{-y1Y2a1i' + uDo.C-YiY/ff + sj» + Kl s^ ) c * 2(Ki )0(Y14 * ySK2u) o * 0 and sine® wo assume (&2) ^ 0 we have ( 5 ,7 ,8 ) u a -YjYgt®!^)©*

S u b stitu tin g th is in (5,9) we obtain (3,12). I f we s u b s titu te (3,12) in (3,1) the square o f the radius of th is quadric s a tis f ie s the equation

34

(Bf 18) -y ^r 2 « (y 2b | * Y$4 2 i f ) 0, vh»p » - y 0 i s tb s sign o f tb s rig h t sid e.

Since we have

assumed th is quantity to be d iffe re n t from sen© th is quadric is non singular and hence dees rep resen t the osculating quadric,

Remrk According t& th e o r y 5,6 the osculating quadric 1 ® uniquely determined provided

equations (S , 12 ) and (5,15) w© g et (8.16) p ' * B ( y j l

Y jt^ / )

•*2

-Y0( r2 /* m d hence p » const,

♦ Y ,^ ) ' ) ~ const,

therefor© the ©enter and the radius of our o scu latin g quadric are independent of i t e contact point of the curva since the parameters Is missing. So a l l the points of the curve belong to the same o sculating quadric and hence the curve lie s on i t .

57

Theorem 5*9

A necessary and sufficient condition that a curve l ie on a cone (5,1) ie that a t least one of the equations

(t*17)

£ y J S l * 0 Is satisfied a t each of

% th e points of the curve*

Proof*

I f the curve lie s on a cone the the cone

has & c o n ta c t with i t of a higher order than two a t every point of the curve and hone® a t avery point nt le a st one of the equations (5,17) m at hold iy theore® 5.1. Conversely I f a curve 1m a t le a s t one of

the equation© (5,17) satisfied at each on® of i t s points then t'rox the equation p * B - {E1 ) ( y J ‘ ± B)

a t le a s t on of the equation® ?'=* (“Yz®! ± & ) * * (-Yzll t ®2 ®2

*4.)® 3 0

Therefor© the vertex of at least one of the

ofcuI bting

cones Is independent of the contact point of the curve since the jpe.ruaeter S is missing. So a ll the points of the curve belong to at least one of the osculating cones end hence every point of

m

the curve vould h*&v© the seme osculating cone und thus would l i e on I t .

UmsLTk In order th a t the curve l i e on both osculating cones both o f the ©cue lion© (8*17) oust be s a tis f ie d a t evQr/ point of the curve and hence S2 = 0 .

So such a curve i s & plane curve v lth

the equation

^ constant and thus I s & privileged

oonio l f h ** °* *“* * tbeorKS 2*2 lfc is not a parabola*

m

In th is section we s h a ll be concerned w ith the problem of finding a necessary and s u f fic ie n t condition th a t a curve l i e on any re a l c e n tra l quadric surface# We s h a ll w rite r * r(s ) when the parameter used to describe ih© carve is the ordinary arc len g th given by the equation

A ll d eriv a tiv es w ith resp ect to th is parameter w ill be denoted by dots ( i . e .

» y , e t c .) .

The tangent, normal, hi normal, f i r s t curvature, second curvature, etc* w ill be denoted by small le tte rs ( t , n, b, k^, k^, etc*) • When the parameter used to describe the curve is the quadric arc defined in definition 1*4, w© shall write E » E(S) to denote the curve, f he notation for the derivation of “E with respect to B and for the quadric tangent, nonsal, tdnormel, etc. w ill remain the mme &s that of section I .

40

Vhen ve ©ay th a t tm expression i s lava r ia n t ve ©ball m m th a t fo r the curve r(s ) =» E(S) The expression i s th e ©»®@ when represented la terms o t e ith e r the ordinary arc » o r the quadric arc 6 ,

theorem 1*1 For the curve r($) - 1 (S) ve have the in v arian t

P roof.

^

3 |||

and hy theorem 1*4 equation (1,1$) of section I [S*t

E^'] »

Similarly from the theory

o f ordinary d if f e r e n tia l geometry *»•

[t , r

**

, r

But since

**

. ~

.

_*•

—

] * Y*fcfk 2 vhere y* 58 ®ga[r , r

is oosltive y * y* so we have ds equation (1,2)

, r

41

Definition 1,1 The parameter defined fcy

(1,5)

I

1

dd « $d» « «$dS, where $ « ( k ^ ) ^ | *

w ill be referre d to as the in v a ria n t parameter*

ite a a a jy t The

vector®

a.^t eg defined by

(1*4) ox a v t, I 2*&L, «5 = ^ 2 wl»r« V = 9~l a re In v arian t m d lin e a rly independent* Proof*

« v t 28 /JT, where^w# hence they

are invariant* To show them lin e a rly independent *1 *

mm

(X l ) #

*mm

mm

« W t * v tn

* ( w #a * v^v** - v^kjyt *

+ v%c£)n

* y^k^b and the determinant o f thee© three vector® becomes

(1,6)

[ I j , a2, asl « - A f k jf t, n, b) * y* ^ 0,

vh»T» y* * »gn{t, a , b j, eluoe v 6 * ( k ^ ) d e fin itio n 1*1*

Hence since th is determinant

i s d iffe re n t from aero the vector are lin e a rly independent.

fro a

42

flotation TaldUf the d eriv ativ e of

as given tgr

the th ird equation of {3.f t) ve have ||& « (w*^ + d v ^ v V * 4 v^v*** •

- fiv*kjk£)t +

(7v*V*k^ * dv5v #*k^ * 6v^v*k£ + V^k£* - v 4k | - v \^ k ^ )n ♦ ( f e V ltjk g ♦ 2v^t£k 2 ♦ v ^ k j j b . This equation and the equation®- (l,S ) v l i l be w ritten in the fo m W Vt

(lfT )

J* *8 * dU&®i df *

-

*51* **82" * *Wb #*iH **"* *■** *41* ♦*42“ * *48b

theorem X«8 The expressions p^ and p^ defined is? \

(1.8)

H

-

(1.9)

^ - y-I y

where y R *

n § & ,

v *s] «d g e . «5],

% h i,

are invariant*

45

¥voof*

Since

ag are lin e a rly independent

we say w rite ( l f 10 ) |||g » l^x^x * ^ 2a2 * **SeS* th u s i t i s obvious i t e t

jig are invariant*

Talcing the d e riv a tiv e of the determinant ( 1 , 6 ) we have a2*

38 I®1 * a 2 *

38

P fP l»

a 2 »a $ l 35

Hence pg « 0 and conation (1,10) become® (1 , 11 )

Jj® *

f t^ Ig .

S u b stitu tin g the value® o f

fro® ( 1 , 11)

in the determ inant ( 1 , 6 ) m ultiplied by obtain (1,3)* of

we

Likewise su b stitu tin g the value

^ro® ( l , l l ) in the determinant ( 1 , 6 )

m u ltip lied by ^ we obtain (1 ,9 ).

Since the functions *jp

are in v arian t

I f we find th e ir expressions in tanas of the parameter s then they are equal to the expression® obtained from the® by replacing s by S.

44

theorem 1*5 The functions ^ and of

are functions

and kg and th e ir derivative® w ith respect

to a and ve may writ® *41 *42 *45

j*a *22 0

(1,18) ^(B*. fijj) *

*51 *82 *58 (1*14)

Bj) * v

‘42 *48 i

52 *85 Proof*

From (1#7) and (1*3) w# teve

»*1 * ^ f f 8* *2*

* Y"[*4 i t , *22a , Xjjb] *

Y "t*4^» * a > , *58®] ♦ Y"l*45^» * 21* . * g ^ l ♦ Y*£*4 g^» * 22»» *5!®] * (1*18) . From (Xt 7) and (X>$) we few® 5% - Y 't ^ , ^ 8 , »81 - Y "[vt, *42®* *55^ * Y «[vt, *45b, Jtjgo) * (1 ,1 4 ).

Bea^yk expanding (Xfl&) and ( l f 14) we find th a t .// Bp ts fli *. i» a function of B p B£f B£, Bp «,/ B£, n.ff Bp « an^ ^2 i s e function o f Bp B p B£* !t,# Hi,

45

Theoron. 1*6 flie functions Proof.

and

define a curve*

Using th e d e fin itio n s o f

and ag

equation ( 1 , 11 ) » y be w ritten In the fo ra

br virtue o f ( l f15)f m

~ |1

* const*

I f © Is the in v a ria n t parameter o f d e fin itio n 1.1 then th is constant Is y* which i s not zero* Sov

* J&£ end r - jludd i s a curve and ve may dd writ® jig * {^(kijkg) whar®

fc**■ and k-j£ are the curvatures of this curve* Hence p.^ and j*g define a curve up to the i n i t i a l conditions*

fheore® 1*7 A necessary and s u ffic ie n t condition th a t a curve w ith

^ 0 and HfgR^ + YgC^j&p^ ^ ^

l i e ©a any nonsingularf r e a l, c e n tra l, quadric surface i s th a t

46

(1 .1 6 )

Y,Jjl

v g(^® 2^

* 0

ftt eeeh o f the points of the curve whore end

are to be computed by neaa# of the equations

(1,15) and (1,14) * Fro® theory® Z,B equation (1,16) 1®

Proof.

a necessary mad s u ffic ie n t condition th a t the curve l i e on a given quadric surface (1,1), since and Eg contain the coefficient© a ^ o f the given quadric. mud (1,14) we

How by using equations (1,19)

mj obtain

mu expression involving of the coefficient®

fro® equation (1,16) and Se^ w ith none appearing*

Hence we

obtain a aeeess&ry and s u ffic ie n t condition th a t e curve l i e on may nonsinguXar quadric*

fheore® 1*3 A n ecessity and s u ffic ie n t condition th a t u curve l i e on any re a l quadric cone i s th a t a t le a s t (1 .1 7 )

am of

the equations

- » i ± Ygjk * 0

47

i s s a tis f ie d a t each of the points of the curve, where

and B^. are to be eos^uted fcy mesas

o f the equations (1,13) and (1,14). Proof.

Theorem 3*9 and the same argument

need In th e proof o f theorem 1 . 7 *

48

s m im x ti

fn section X we assumed th a t d r. d r $ 0 and

^ 0 fo r the curve r ( t ) * E(S).

Xu th is section we w ill fee concerned with the problem o f finding what curves E » ft(t) hare th e following relation® holding a t e&eh point* (a) v '

JttWtt * 0 and 4 0 dt di d P dSBT r w

(fe)

**

|* 0 «

«

«

(o)

#

» 0 *

*

s* 0

o

Whenever we speak of a cone in th is section we sh a ll always mmn the asymptotic con® of a quadric feomothetic to (&*&) o f section X* t he i n i t i a l point® o f a l l vector® w ill fee taken on the curve I * i ( t ) and the vertlele® of a l l cones w ill always b© taken on th is curve.

49

1 « Ib e F ir s t Case

Ve s h a ll assume th a t the curve E » R(t) has (a) holding a t each I t i t s points m d we wish to dind what curve© can have th is property. V© s h a ll suppose th a t t 1© a parameter such *p»«* th a t S*j| i s a u n it vector and ©hall denote derivative© w ith respect to th is parameter by accents*

th e ordinary arc w ill he denoted by ©

and d eriv a tiv es with resp ect to i t w ill be denoted by d o ts.

Furthermore, we assume

th a t each point o f the curve ft * R(t) belongs to an in te rv a l o f c la ss th ree.

D efin itio n 1.1 For th e curve E * E( t) the vector ifS w ill dt be denoted by U. It© eontravariant component© are a® * j r . i t s covariant cos^oaeat© are CL * ©bJB^, dt th e u n it vector d^B w ill be denoted it?

'ey ? .

60

Theorem 1*1 The following re la tio n s are tru e fo r the vectors W and 7 *

f '« 7, f .7 » o, ?*? * -i, vC?* o, ?(? * x P roof.

Vc have by d e fin itio n

hence 0 / « iL j| « f .

dt

Thlo prove® t m f i r s t o f

th e equations. Taking d eriv ativ e o f ¥ •¥ » 0 we have 0*0 a P .0 a

which proves the second

re la tio n * The in n er product o f any vector w ith I t s e l f has the sane sign fo r any vector o u tsid e o f & cone and we only consider real ooaea ~«As^ **

» 0 * The v ecto r ( 1 , 0 *0 )

lie® outside th is ©one m d i t s inner product give® us

thus ® negative sign*

low sine®

V.U a 0 we know th a t f l i e s in a tangent plane

of the cone and banco is outside the cone* Sine© 7 i s a u n it vector

» -1#

equation i t follows th a t v l7 38 0#

we have ?{? » -7,7 * 1.

Froia th is From b*Y ~ 0

a

D efinition 1 .2

Let V fee the vector with the following properties*

¥.? ?s o, v.v * o, v.v * o. feet the quantity 1C fee defined fey the equation 8 a

Bemg.rk Sote th a t i f ¥ ^ coincides with ¥ the X 38 0 * However we sh a ll assume th a t K ^ 0*

theorem 1.2

aunt

w eU P 4

We m&j choose the vector ¥ so th a t ¥ .¥ * 1* P roof.

* p¥ s a tis f ie s the conditions of

d e fin itio n 1 . 2 . tta t if

¥*.¥ » p(W.O) * i im plies y(?.W ) our theorem 1 ® tru e.

theorem l.B tb s vector® ¥ , ¥ and ¥ are lin e a rly independent. Proof.

M iltplying the determinants of covariant

and co n trav arian t components we have ftl# V, W]^ - 1. th e re fo re [IF, ¥ , W] » y*9 where Y* * * 1#

since

th is determinant i s not sero they are Xiaeorly independent.

52

fhaorea 1*4

The formula® fo r the derivative® o f V and W are (1,1)

? #* Ktf + V, ? * = » ? .

Proof.

Wg know from theorea 1.1 thni "v *

l i e s in the plane o f U and W* So we my w rit#

? ' 3 X« + yV V*.W * K ■ X0.V * z, banes > > i ? #.0 » 1 * yW.O 3 y, hence y * 1.

Thus w®

haw ? '= KB * V. To prove the second equation w rite i ' w a f 4- b f * oV. ¥ ' . ! * *¥*¥# « -¥*¥ * 0 a cw .f w c, hence o » 0 . F . ? * 4F*?# * -* * # . ? * ~b, hence b * K. ¥*♦¥ » 0 « &¥*W a a , hence a « 0 .

Thus we

have ¥ * * &?.

Theorem 1*5

I f X ^ 0 then the absolute value of & 1® equal to o»e~half the square o f the length of the vector ¥• Proof*

Pro® the f i r s t o f equations ( 1 *1 ) ve have

3 (KB f 5).(SB + 1) = 2OT.V * 2%

th e d eriv ativ e of the parameter t w ith respect to the ordinary *ro * 1 © given by the equation (X,2) | i Proof*

* ( 1tfk2) f For th e curve E(t) « r(s ) we have

f*»' I ? W * [f# y , ? '1 - [ f ,

r Y* «* (3 i)

t r

7) =» y * »

1 * (3 4 )^ * ^ ,

or

But since (JjS^k^kg is aXwsys

p o sitiv e y* 3* y S|*^ hence w® obtain ( 1 , 2 ) .

Theorem 1.7 A curve R * B(t) which has

dt* at*

* 0,

|| q and I ^ o a t e&ch of i t s ooints

m e t be a so lid curve whose curvature® s a tis fy the equation ( 1 , 8)

F* * 2v V % - 0 * 0 , where

f » |v * V 2

* 2 v~V*

* R /k J 5^ + ^

0 * v S * fiw lc* - S v ^ k j >

* | ^ k2 - f ^ l

1 V ■ (kfk 2 )~ 5 .

Proof.

I t i* & so lid curve since [E^ E* B;//] ~ y* ^ 0 .

le t V a | | a

2

«*1

(kjfcg) % * ^l3®° *o r

curve

E ft) w r{») by d iffe re n tia tlc n and use of the formulas ( 1 , 1 ) we have the following* E « Vt I » vH + l« ?

• IE * (v* -

- vX)t ♦ (S w 'k^ + vsk{)n + v^k^k^b

¥ y a ( v " '- 8v V k | - av^kjkj - v'K - v £ '- Sv^v'k* - v ^ k j f t (w'*k^ -

- v2kiS + Sv'^ki + 5 w