WTW258: LU 3.6: PARAMETRIC SURFACES AND THEIR AREAS Lecture notes

3. 6. Parametric surfaces -1
their areas
Describe a curve in ! different ways :




y=V4

}
a) Explicit :
all these

y 2=4 describe
't
b) Implicit : x
o≤y≤ 2 the
c) Paramo trite Flt) =
< z cost zsint >
same
,
cUr#
↓ 1- c- Co,ñ]



The surface :
Variables
2
parametric equations .




S Fly ,V) :<
KLUM ) 2- ( UN )>
ugly ,v)
:

, ,



( UN ) t DCI >
region in UV -



plane
surface !
* exactly the same
Just describing in another way .




Parametric representation of a PLANE

.
.


through
_
a point Pocwith position vector ro )
containing 2 non parallel vectors
-

d- and b- is :
FCUN ) =
For y d- a v5 ( U , V) C- R2

•
If 2- =
fcxiy ) -


use ✗ and
yds parameters :



✗= K

y=y
2- =

ftxiy ) ,
(x
,y)
C- D


s :[ ( x ,y) c ≥
y f- I≥ , y ) >
• =
, ,

, example r
/ / \
1. S :
FLY / b) =
< ZCOSU Zsihy ,v >
, 'c 2-
y
d) 41-112 ,
✓ c- IR 11 11 11
↳ can be ANY number ! U V u v u v
=




Kley
' =
4 ② → radius .




N
22
2- =
V EIR .




radius 2
cylinder ,




b) ◦ ≤ att ◦ ≤ v13 → restricted
,


it ↓ =

right hdlfot
cylinder .




in
'




C) f :
FLY / V ) :< 213414 144-4412 ui3V >
point ← ↳ constants -
Takeout

p -


going :
<2
, 1,4 > + <3
, 1,234+51 , -1,3 > ✓
through =
Plane parametrization .




1