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We are

booking

for

parametricequatirepreseuh.mgsurfaces.

" lui D= ?

Y

( u

, u (^) ) =? (^2) ( u , u (^) ) = (^)?

( u , u ) = xlu, u)^ i^ + ylu,^ r) j

( u , r) K

☒ :^ a Sphere of radius (^) a { ✗ =^ asinclus y = (^) a soin ¢ sino 2 =^ acos ¢ ✓ (^) (¢ , G) = asinocoso-i-asi.no/sin0-j-acosQk

j'

Ë of radius^ a Ex '

: a

Sphere

• centered (^) at (^) P " (% , (^) yo / 27 { ✗ =^ asinclus^

• x. y = (^) a soin ¢ sino^

Yo z =^ acos ¢^ + Zo

Ex Plaines passing through^ a point P . and perpendiculaire to^ avatars ×"

minimis

#¥Ës. » L P (^) is perpendiculaire to (^) B Pis perpendiculaire to any

line in

B " any rector Jing

vi. F- o

for any^ rector in^ B J' • (F- (^) F) =o V-P-lx.y.pe/ → Lui (^) , riz , riz

• (x-^ Xo^ ,

y

_

Yo ,-2--47=

→ n,(x-xo)-na(y-yo)-nz(←Zo-

EI : P=

(l

, 2 , 3) À = < l ,

o

, o (^) > À =^ U , I , o> Find (^) the paramétrisation

of

the plane passing (^) through P and^ containing

the

rectus I and^ Î . ① (^) T' ✗ (^) v7 should^ be normal to^ that

plane

= | i

j

k p I (^) O (^) O I I^ I^ O I

✓(u

, u^ ) =^ alu

, v) il y

/u

, u) j

• (^) Ku , u)^ K

O

O rv =

Luo (^) , vo) i +2¥ Luo , volj-F-ulu.is) te Rende : tangent curve of (^) grid curve^! ru =

JI

Luo (^) , vo) i + 0 £ (no , volj-F-ulu.is) k & If^ ru^ ✗ (^) vu -1-0 (^) S is (^) smooth the (^) tangent plane is^

the

plane containing ru, ru and (^) ru x^ ru is a^ normal^ vector^ to^ the^ tangent

plane.

: r=^ n' i-iij-u-zdkru-2uii-oj-k.ru -0 i (^) +2 ✓ (^) +2k The normal^ vector^ ni to the^

tangent

plane is : " (^) ' (^) "» ru x^ ru^ =/ i j k

Zu O^ I

I o^ zu^ o / = (^) - zu i^

• Yuj +4Wh P = (^) (I , I^ , 3)^

, ✓ =/ → normal vector - Zi - 4J -14k quatar (^) of tangent plane^

d-

Pis -2(x

• l ) - 41g - 1) (^) -1412 - 3) =^0 x +^ 2g

22+3 =^0

EI :

Sphere of

radius (^) a { x-asinocosoy-asinosi.no D= z =^ a^ ces § ¢ rolxvo =p ' ' j

le

acosdcoso-acosdsmo-asi.no//-asinOsin0asin0/casO-^0

a2sin20coso-i-is.in?cfsinO-j-a2sin0coscfk/rqXro-/--a4sin4#s0-- l-a4sinkfces.RO/--/a2srn0l Als) = sino / (^0 ) =

21T a

Z

GU-surfaaareao-fgrap.hu { ✗ =^ x y = y 2 = f- (x , (^) y) rx =

i +^

¥

le ry = j

¥yk rx ✗ ry (^) =/

i

j k l'o?^ ˧ / =

• ¥ i - ¥ :

+ le

lrxxryt-F.IE?--yiT like (^) we (^) saws in section (^) 15.

Tangent

ædors DX tu = -2g

qui

auj

le = - Sin (^) u (^) Cos u i - sinus invj

• (^) cos u k vu =^

• ( Itosu) sinv (^) i + (Itosu ) cosuvj

• (^) Ok .

→ Normal vector is

ru x (^) ru = |

i

j te

• sinucosv - sinus

inv Cos

| ]

• (Itosu)sinv l'+as (^) a)

asv

=

• Sin u (Itosu) le SÇ as -5¥

o o / (^) tu✗^ tu / dudu = ¥[ /^ sinn^ ( Itosu)^ / dudu