12 6
.
Cylinders & Quadric Surfaces
Cylinders : sufface that consists of all lines 1) to a
given line &
pass through a
given plane curve
(in 3 space >only two
variables)
Ex #1 :
z =
x2 Z
-
I
parabolic Cylinder
>
y
x
Ex # 2 :
y2 + z2 =
1
Y
Z
Quadric Surfaces :
graphs of 2nd degree equations in 3 variables
AX2 By 2 2z2 Dxy + Eyz + Fxz Iz J 0
+ + Gx Hy + =
+ + + +
Ex #3 :
z =
x2 + y2 zI0
Traces :
1 1 z =
0 :
0 =
x + y2
-
z =
1 : 1 =
x + y2
I
z =
4 :
4 = x +
y2
, 4
1
Intercepts Traces
:
+ + = 1
20 , =c)
#Zellises
0,
ellipsoid Co I b
, ,
0) s
c = a
,
0
. 0
· -
E E 03
=# I hyperb
2 =
+ 10, 0 , i
3 Laxis z =
#ellipse
C
a cone
A
R
O
T 4 e
(0 0 03
3
#
z = + , ,
x =
⑤ Laxis x + y2
y =
# - Parabolas
z z = #
ellipse
: =
- ⑰
elliptic
paraboloid
-
3
- = #2
4 +
hyperba e
-
Hyperboloid ellipse
O
z =
# >
-
of one
- sheet
N
, +
same as 4
>
-
is
& hyperboloid
of 2 sheets
↓ !
⑥
E
# parabaSola
z = -
2
Hyperbolick)
Paraboloid C
&I&
I
<
I
-
&
, Techniques : exercises to show how to determine general shape
4y2 x
+ 0)
Ex # (0
intercepts 0,
=
: ,
Laxis
Traces :
Y = 0 : 0 =
x +
4
=:= + elligee
N
T
D. ellistie D
L
~
Ex#5 : 462 + 4y2 + 22 +
8y -
47 = -
4
1 .
complete the square
46 + 4(y2 + 2y + 1) + z2 -
42 + 4 = -
4 + 4 + 4
4x +
4(y + 1)2 + (z- 2)2 = 4 * we want this to =1
x +
(y + 1) + (z -
2) = 1
4
intercepts : (0, 0, 2)
Co, -1
, o) all traces are ellipses
·
ellipsoid
· ·
-
·
.
Cylinders & Quadric Surfaces
Cylinders : sufface that consists of all lines 1) to a
given line &
pass through a
given plane curve
(in 3 space >only two
variables)
Ex #1 :
z =
x2 Z
-
I
parabolic Cylinder
>
y
x
Ex # 2 :
y2 + z2 =
1
Y
Z
Quadric Surfaces :
graphs of 2nd degree equations in 3 variables
AX2 By 2 2z2 Dxy + Eyz + Fxz Iz J 0
+ + Gx Hy + =
+ + + +
Ex #3 :
z =
x2 + y2 zI0
Traces :
1 1 z =
0 :
0 =
x + y2
-
z =
1 : 1 =
x + y2
I
z =
4 :
4 = x +
y2
, 4
1
Intercepts Traces
:
+ + = 1
20 , =c)
#Zellises
0,
ellipsoid Co I b
, ,
0) s
c = a
,
0
. 0
· -
E E 03
=# I hyperb
2 =
+ 10, 0 , i
3 Laxis z =
#ellipse
C
a cone
A
R
O
T 4 e
(0 0 03
3
#
z = + , ,
x =
⑤ Laxis x + y2
y =
# - Parabolas
z z = #
ellipse
: =
- ⑰
elliptic
paraboloid
-
3
- = #2
4 +
hyperba e
-
Hyperboloid ellipse
O
z =
# >
-
of one
- sheet
N
, +
same as 4
>
-
is
& hyperboloid
of 2 sheets
↓ !
⑥
E
# parabaSola
z = -
2
Hyperbolick)
Paraboloid C
&I&
I
<
I
-
&
, Techniques : exercises to show how to determine general shape
4y2 x
+ 0)
Ex # (0
intercepts 0,
=
: ,
Laxis
Traces :
Y = 0 : 0 =
x +
4
=:= + elligee
N
T
D. ellistie D
L
~
Ex#5 : 462 + 4y2 + 22 +
8y -
47 = -
4
1 .
complete the square
46 + 4(y2 + 2y + 1) + z2 -
42 + 4 = -
4 + 4 + 4
4x +
4(y + 1)2 + (z- 2)2 = 4 * we want this to =1
x +
(y + 1) + (z -
2) = 1
4
intercepts : (0, 0, 2)
Co, -1
, o) all traces are ellipses
·
ellipsoid
· ·
-
·
