Curve in R3

LF.

,'^ ;*"Qa"-*{- & t*-*

,-S- J' c L1r

L zr .* )

ea,ù<^-

l. f':

, e**

g g l t I g g g g s s s t l

g g g g g E E

g

f t r l g z : L

l @

t n

= ( . " . r t * t ) , a - étR'

4(t) 1 - t ) ) ; " ,

. ) ; t

.[ .

*to 4t) p*-

(nco .r,*

À,,-1+ r-

lt'r.^ a..- : .-^,)-t p,*-

7{ ,

-J- ,!-Ca- ,*A'&-

'L<-- ^-è.-

.

/- f/'z'

| ,9 .a'/.

| \ -,

L^

Ì e ,to-

1 ' . ,.n

= L-'&','""oJ

.,

.l | t*>/ ,

| ,

| 21qL.

l. r c*-1,^*-

î; "**

"A

&w\r pL.]l^ Ct (-'za

,

( <-

".

.,.t ,t<.4 td^o^1' ú

*u e

,^

é4vz-,ú:- 7.*,

,

, f"-..!L

ti\qA!-

^ 1 t l

t'L- |

r,\.t^/- at/-\

ìèr (,,)

x(-.)-- ,-,(j)

f

I

l

( = r r',;)

.l

cn

r 0t) ,"-

=,t

{ ,l'"1-"+.

Ad. "/"'""?,", * G)

(=Í,ÌétK, L(^:n'.*-

ARailrÈ(él);.

J

'PtÈA 7q- '

l*,-g.t- fo ..J

. L cuL,<

\"

-.1-J^, ..,"^,r t^

obb,,-o l--

^ > a

- - 5 .r'\.L-fc

i: ii\ rÌ)!i!

c <,"-1'.""; c-.-.pL.--i,-

it< a.U

; -L Dj.,)r.-3 <-';R!A .1 a,v;,

Èe J"l-.

L,ot-rt- L+

J- '"u r}*f ,,

È-.

J . *

L tR' -a

J- Llc..;... -i4]

l, +a-;|,.i*-

^.f! {rL av.*,..!-l

= (ù :Lr-pÈL.4!à1r

4] (*r,"ú._

= C",41-[2

\ .-

Í,,- ia-t-...^i*\-

i--,1+

r-z dt c't a

* i-a g.-.*o,

!- "-^*

J

u,ìe,!' i Crr -..-:1,/,..t!

c.*r.

l\-u'r',;*t-,;!ùr a..1 -l

{-rc é llL <.r

3--q'r Úa(^,/Jri'r'

-&Pj,!.'-s .,i

r:,a a-l.r_

, .,., -s

'. => lR; .'L i J

e /v "7-4",

f-:l-.----, E.,bl

a, ,f R€

.ì óo LAAé

,z'(t)+e.

(c-"t |

.--11^.,o-)

I^ ( \ \ .

( > ( 1

( t ) =

. t ) , a , î r,1,

n,I_" l(-

é.

;J ì /

. u . . . l.

J.I.- l. ...-

,

; r . 4 u,{o

-

'?; )^

I,l

g g g . g

g g g g g g g g g g g I

s s { s

{ f ; [ I

A'(l) *'ttíow

Dl ;,,t.- ao p?,,",<.-..t-

! J

c-u*-;(,.",

p*f" (vat-oc,,r*'

=

=

"tG) xtry.-

n'(t) ll

U z'AF a

/ ,\r)'- r--4 ee

sa )

n-()

io tt /

.4. ì t.

.*"*- +o e î .

,

X-Gu

-'

--É -

,jì n'G\ .Là

ir-.- 5-pp''-- ck I

Al.,t-r-y^ €

\;

l+JF

J tr

ttt), \in:;

é- ay-t. :

L. nìl- po.,<,,.n.1n^c* ,.\:-ìS ( . 7

(t,O=x(L)+1-x'cr-)

I $(t)* t\'l:'-) ÉerR

6-

4 ^ 1

I

[3cal= &(L)+ tz'ft-) À(t.h)

'"

r. '

\+ .l"

,,., *,

4 & rrClc

t r . : & l o

P P f c1

= fc)

t-

&1;.

?-o'. cuulo'"^-

c+.1!e dor,t-

*o- p,e*! f-*z;.-* i(.1)

la^g^t^ li",

*

z1".o n<tl* *

8 x' uta.c/" J*J:t^

o^^1, È,.*-lo

7-t-r,^

.*n-r-i; :

./; "o-- .t("

-J-. = x"*t

e<t)

f ( t'u-^ r'e' <

r'A ?AN'É )

I ^

= 9 - + F ' ( x " )

t " l ( t ) t

v

(t) - x.

I

L -

I

l 4 ( t ) :tooc,ne

- (

(x.) r"l .^12,e;,,a,u^)

5.+ | tee>-

. _#*

r(É)

' ' : l - . . ._ 5,*.-

(?)

= e

R,

e4*L.

,-.,',-n L ti:. -- it?-(t-) t

d.^,- ,t2

^ f

r

Òz- òL I - - . ? z t t - >

( r , )

t c

tlr T , ; ' í - .

T. .L IrJ

], t - t "

r Yo [z

*{o,. ,rf2. L

ft , ct"-

" t: t. o J

pa'- l"u

- Ya-

9 : d'.

Z o"^ "1.

7 -

. SA: ;-t.

..,- yL"- t--

-.

=,trcz"1

ALQ")

^-(

,Y. ?*t- ,d

fJ

<{--":t

=

.-a({) (o<1< rf \

,^j' tÀ) r"(") z ,1{t,)

g g g g f l r

f I g I l t g g l t s s s r t r u

f l r g

''z--4:'(-.1- ^-<r .n

C^' ,l3r.t+" tc

C'

"<- o^ )

c*.

-

7- .r'^^ .,\.^4t

'' (t

o.-.

^-fi I

,r'G ; )

,a lj -{

) J r-{Li.,Èi e< r!€i

/

{.^.f"

= ..r(t)'+ (Acce

q-G) - one

teulz, :n ^a^r-)

vtn x,,(t)'* è"

zu [t)'

L t ) ! 1

l. l]rn ",t, (

=

(t) (<), (t) zr

c-) 4i- 1a

c-" ;o,

) J

a i ,*

?r.r.*ul^zta.z:, o."^t cj< éctcn--..< .1.

^

*, (."^,

.^4<-\-...,/t ^"tt

-'-

(t),

('z:^ ''-i"t"-

=

tt'(t) c-r(t))

- - = -

.

(-^.-Gù'

{ 1

2 ' ( t ) i { tl

. 4 r ( t \ C*'11))' 4

" - - - s

-- - -zG)

-^.- (t)) = - t d .

(t), c , c c . ,

( r.^ rLn *

ec

2."(t) <tlt <,

=

ù (

n " G ) f ( Z "

c ( t ) - zT/*

- (<-r (*t), -+- t e

) J

b ) l.-t)

1ft\ L-o,

,

(-u: ,^t <*t(ut))

4-(*t)

a'G\- , --

= "^,

( t ) . c - r a ; ' ( r ) . * ' ( - r ' G )

l , {

^

r " - r

G ) t t - , . ' , t ' ( ' t

( - " , - t \ )

e ' L * t )

,'e)- c - , , , '

'-P - 4".:

p,\t*

^-1a

^u-\. ì N(

cu.^. é" .

Gr'o (:.- f

?.*U-'

(o t"-pl J c<r'utJ*1e' ì

i , '* U)

oo-o

cu^6i

,r^o Pr.4

, tz*P' -

a^uLt o *-s*.

c- *.a6u,- ?*

l--

,

{*& \e ,

cl.-,, lÀ{.-r j , ! e -

r , è ,

f ì , " A , 1 ^ , , :

s

C..:,t ? :

* a a A

;:,-',v-*---., 3

''

--j' r-ri

(t) Y

L a ; I

-.u- 1,:

,'-..- Lo c* 4

4^^,rr -1...t.

L", Y ( ,^.; F, .*'n-

'! t= -

-l--'"] ft;*c'- )

7o.u;*-tt-

=

* -

| ;- ,r(*t)

4tt) .î'{ù -

1 , t ' " " $

r

u.\e-

r (lí) J

^<'t't'.

(f,,t) at .9

v"^."

" " .4-ÀY",

c*1r'''

Lc .t<."1 -1^.

t, " .!

,*11- ,1{c..r:

f-,*yr,,.1 r

,.-c:.>e

-. ú p*1.

(.j,.ta-.),vu,,ÀZè ;

- ^L<.r -l-e^ct-

| ir.i\)-'t'a!oa'

,u*b:.^ ì J-.L ..l A"-' |

,,^, 1o; g'{-s

g g sr{-f{{{{{'n

E{-{-.g

{-g{

I{

T <

'a a^ra"Fiè

4

(f,î) t^ì,l'.

(f,,2) Òo-f;

o

c-^-<

ù

Lt 't't"'-

L' o'' +la-

J*('

-ai*;^oG-"'1'

,-,.,^r,

. 7e''*

P'*f" zrJ

(t)) t f',

é

(r*'(t), z\r-

-

nk)

) )

€u* = u'-t

a

<on-L; c p,ar,o*z,lo- '7 J

("tz)) #

a

( . - ' ;",

= (t^rz) ^^

- -

.L@ )

/ L ( c 4 ' T ) / "

"

'7^- f

L 4-Jf '

'L"^ '^' n1"'*'

(-t,,-*b c'.4-',^r<

-l'

" t* :

j /^'<-)2"è

ÙuYli n'

@41\

'^Zo-

Jt

,"<t.z- 7*fi-

Pt/lcdÙ\r ('Lz)Î*c'"(-Qi

= - 'aa

7è)

n

- - . 5 . ^ "-o&)

G ) l

T G ) * î i =.

' ' " t ( 7 )

(t) 4'k)=

T

n'(!)- i -,^"TG)

-

fr't'z)

-rt&) i

n"&)= nR,

1' i

f

" !-'

( Y , , ' t ) , , 2 : J - R ' ' z - y G - .'

5 , o . - . 6:q' a'Lz

c'vu"7-L',<tn

Vz<í t

<ec4(i\ 7<'.:r)+":

t,í [z-o+12)-t7

6- &--)

òr

rRrèaA?(o^r6

->F\t lAe^rYé

(z) , /^J''rv4

" I i

n

í &\= nry-e,-

J

- " =

V

f

(t) ' ('€(Ò\ Ll,

(n" z

{)

ig)= /L'(<

fta- @) no

fi

ìr l e { i

I i è a l = f i n ' ( v e . ) ) i l

v , \ d z r

l . {

c c &1]

'L ti--l:"-'r

'\t^:-

(Í, -t L-.*-

(n,Y [

o

) l>"1-'4'?.1"'

)

'

.-> -,1 <:> z'=V-act)

-lt-{G) h "^^^'

-} lz^ ì'Ècri'r-:

>o A

1n'>^L.

s^

{'

9"

:' ,to- ^

-u-f al. rt

o,t- o+ :gp*-f'

<>1ì

:) ;,-t-

tucl

5,- \'zo ho*^. ( ' 1 ^ t' t

r '(^-

' )\! t

(À, ao-a é''vrÚAr€^rrr

' (l,l Y)

"

) ' 't""' "

t c^-fz*' L"

otloo

!'Li

?'o'^J .

Éf{2 |

f \tè

fl-4 Xa^

I *

f-t;

lr- qFFryW+#FYE glFE

ÍTg+ 't|''-':l'l

t .'.:t

{_+{- .,.'.

/tc-l (b,l

: I

A

-l g

- .z:L-Lrb-] .tr lR- <o-1,nu ---t* , \ \

cy'..d1-t i 9eGA,(€Nlr '

/ta-b / \

r . I l

g;r^y

a-e=;l@lx"fl 7 :/ I,

----_-,È

- =

f,'lrro^ J&^,.,.i f"

o \,^.^È D

o,^, tz, 6

I L, to, t^=

^é .-., | Jg

t " " . - , A t - f - , b l * ^(tî)=Pt

p o ^ - . o ^ ^ ^ "

p

. * .

J=a.4-...i,14 ,

i i^À:^'t

PF A^,-,,o .*'

f *a.,ut{.- ,,-.Lto

p,^-l p.( <^^".^r* oq

,

6-^-(,.

.a^