ÆM⌃3–12.6 Cylinders and Quadric
Surfaces
⇥˘ Mao-Pei Tsui
˙'xx˚
Feb. 15, 2022
Plan of the this lecture
In this video, we study "Cylinders and Quadric Surfaces".
We sketch the graph of
I a plane
I a sphere
I quadric surfaces
.
We live in three dimensional space. We use the coordinate
(x, y , z) to describe a point in the space R3 .
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Plane
A plane in R3 is described by the equation ax + by + cy2- = d,
that is P = {(x, y , z)|az + by + cz = d} represents a plane is
R3 .
For example: x y + 2z = 4 represents a plane that goes thru
the following three points. (4, 0, 0), (0, 4, 0) and (0, 0, 2).
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✗
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Yao
,
too
*0,7=0
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+
2-o
z
A surface in 3D is defined by a 3-variable equation
f (x, y , z) = 0. For example, we have seen that the above
equation defines a plane in 3D when
f (x, y , z) = ax + by + cz d. In this section, we attempt to
sketch and visualize more interesting surfaces.
-
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CKY
plane=/
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in
ix.
T.nl
axtbytcz D=
-
if
surface
a
IRS
.
Sphere
A sphere with radius R and center (x0 , y0 , z0 ) is described by
the equation (x x0 )2 + (y y0 )2 + (z z0 )2 =Rt
r 2.
Example: The equation of a sphere
with radius 2 and center
i
(1, 2, 3) is
" "
Its graph is the following.
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kid 3)
,
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a
Here we investigate two other types of surfaces: cylinders and
quadric surfaces.
A cylinder is a surface that consists of all lines (called rulings)
that are parallel to a given line and pass through a given plane
curve.
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.
.
Example: Sketch the graph of the surface z = x 2 in R3 .
fcxis.tl/z---xJ
-
-
Solution: In x z plane, z = x 2 is a parabola in x z plane.
Since the equation is independent of y , we still have z = x 2 for
each y . So there is a parabola curve on each plane y = c
where c is a constant.
The graph z = x 2 is a surface, called a parabolic cylinder, made
up of infinitely many shifted copies of the same parabola. Here
the rulings of the cylinder are parallel to the y-axis.
2-
z=XZ
on
X Z
-
plane
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2-
-
plans
✗
In
✗
←
All the
I
i
i
I
1
1
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Sana
7=5
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y
In previous example, the variable y is missing from the equation
of the cylinder. This is typical of a surface whose rulings are
parallel to one of the coordinate axes. If one of the variables x,
y or z is missing from the equation of a surface, then the
surface is a cylinder.
EXample: Identify and sketch the following surfaces in R3 . (a)
x 2 + y 2 = 1 (b) y 2 + z 2 = 1
y
-
z
¥04
,
✗
circle
indepofz
-
with
radius 1
.
✗
z
¥¥
I
y
t
circle
fizzy
.
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y
✗
y
Quadric Surfaces
A quadric surface is the graph of a second-degree equation in
three variables x, y and z. The most general such equation is
Ax 2 + By 2 + Cz 2 + Dxy + Eyz + Fxz + Gx + Hy + Iz + J = 0
where A, B, C, · · · , J are constants.
Using translation and rotation, we can simplify it as one of the
two standard forms Ax 2 + By 2 + Cz 2 + J = 0 or
Ax 2 + By 2 + Iz = 0.
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+
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+2×+8
EE 5+2×+8
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e
=
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ka ¢ YE
-
f
4
We can first use rotation to simplify it as
Ax 2 + By 2 + Cz 2 + Dx + Ey + Fz + G = 0.
it
j-z-nx-xy-ba-8-ocx-ui-IY-a.fi
-
wie f
Suppose A 6= 0, B 6= 0 and C 6= 0. Then we can use translation
to rewrite it as Ax 2 + By 2 + Cz 2 + J = 0.
A-tui.BY 0
If A or B or C is 0, we may assume C = 0, Then we have
Ax 2 + By 2 + Dx + Ey + Fz + G = 0 and use translation to
rewrite it as Ax 2 + By 2 + Fz + G = 0.
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2- +
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If F 6= 0 then we can do a translation to get Ax 2 + By 2 + Fz = 0.
Ax 2
If F = 0 then we have
+
Ax 2 + By 2 + Cz 2 + J = 0.
By 2
¥
+ G = 0 which is of the form
-
¥
+
g.
To determine the graph of a quadratic surface, we first review
quadratic curve. Up to an appropriate change of variables (e.g.
translations), conics (=quadratic curves) can be classified as
follows. These curves are called conics because they are
sections on a double cone.
conics
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¥+¥=o←> ye
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Lax
After reviewing curves in 2D, we will investigate how to sketch
quadratic surfaces in a three dimensional space. There are 6
different types
The following software is useful in understanding 3D pictures.
Geogebra 3D : https://www.geogebra.org/3d Also
available on iOS and Android.
Let’s start with the ellipsoid.
Example: Sketch the quadric surface with equation
y2
x2
z2
4 + 9 + 25 = 1.
2 v
Zu
20
Solution: First, we notice that
, |x|  2, |y |  3 and |z|  5.
x2
4
 1,
I
Fe 4
 1 and
I
IYIE }
⑦
1×182
y2
9
.
z2
25
t
1-21
 1, that is
5
,
¥+¥+
The surface intellect with the horizontal plane z = k is a curve
y2
x2
k2
4 + 9 =1
25 .
E-
If k = 0, then it is a ellipse
x2
4
+
y2
9
→:÷É÷
= 1.
2
2
If k = 2 or k = 2 , then it is an ellipse x4 + y9 = 1
which is a smaller ellipse than the previous one.
E- +8=1 E-
4
25
=
21
25
-
If k = 5 or k =
If |k | < 5, then
5 then it is reduced to a point (x, y ) = (0, 0).
x2
4
+
y2
9
=1
If |k | > 5, it is an empty set.
k2
25
is an ellipse.
ki z
Keo
kit
Example: Let us sketch z 2 =
x2
4
+
y2
9 .
← Each cross
section
q
is
an
ellipse
,
E-
¥ -1¥
¥-1 ¥=o
2- =o
z=
.
Cky )-40,0)
¥ -1%5=4
2- →
2-
¥+4 izz
:*
±÷÷:*
2- = -1
or
⇐
2- = -2
4
ellipse
-1k
same
becomes
ellipse
.
larger
-
¥+Y¥=K
¥+8
-
i
¥ ¥
.
+
.
=
I
⇐ 5+1%5=1
-
Example: Let us sketch z =
x2
4
+
y2
9 .
2- =
☒ Fg
When
¥-1¥
¥ -1¥
2- =K
" 7-
°
k=o
Kou
ellipse
2-
zu
¥ -1¥
20
=K
¥- ¥=o
+
IX. 9140,0)
¥+¥=kµ¥¥+¥=1
①-÷
¥.it#ri-t
Example: Let us sketch the surface 4x 2 + 4y 2
z 2 = 1.
4×445=2-7-1
E-
IK
4×445=1+14
is
a
x4j=¥=(¥fY
circle
-1-33
.
From 4x 2 + 4y 2 z 2 = 1, we have 4x 2 + 4y 2 = 1 + z 2 , so if
2
z = k then 4x 2 + 4y 2 = 1 + k 2 , i.e. x 2 + y 2 = 1+k
4 . So it is a
q
2
circle of radius 1+k
4 when z = k .
If k = 0, then it is a circle of radius 12 .
If k = 1 or k =
1, then it is a circle of radius
q
1+1
4
=
q
1
2.
If |k | increases, then the radius of the circle also increases.
2--0
7=1
2-
=
2
,
Example: Let us sketch the surface z =
x 2 + y 2.
E- K
z=k
,
×7y
?
2- =
-
K> o
"
x4yIk
⇐
:* .
-
Koo
*F-
,¥÷¥
x-t-KTK-v.es j☒
,
-
~
y=o
-
"
,
Keyes
✗=3
ur
✗a
-
J
.
2-=
when
x4y
"
-
✗
you
z
✗
✗
"
ya
⇐
z=
-
y 2-
E
plan
¥¥
-
2-
plan
.
"
;
1¥
fA
×
⇐*
Example: Let us sketch the surface
x2
4
y2
9
→
+
z2
25
= 1.
¥+¥=¥
=ÉÉ5zo
-1
-
£-2520
12-125
¥,+¥=¥¥z
2-2-2520
z=5orZ=¥
K> 5
z=K
or
Iz / 25
¥ You
+
¥-1 #
ellipse
=
-
as
-
cxiyklo.it
¥-7114
¥
2- = K
.
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become
> g-
layer
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