10.6 Surfaces in Space
Graphs we know and love already:
1. (x-x0)2 + (y-y0)2 + (z-z0 )2=r2 is a sphere
2. ax+by+cz+d=0 is a plane
Another type is called a "cylindrical surface" or cylinderNot what you think: We are used to cylinder meaning a
right circular cylinder, but a general cylinder is less defined,
really.
Def Cylinder
Let C be any curve in a plane and L be any line not in a
parallel plane. The set of all lines parallel to L that intersect
C is called a cylinder. C is the generating curve and the parallel
lines are called rulings (but I'm used to calling them generators).
The purple lines in the picture are the rulings (generators) and
the black circle is the generating curve in the right circular cylinder:
Ex 1 sketch the surface: z = x2
f
2
f ( x , y) := x
Ex 2 sketch the surface z = cos y
f ( x , y) := cos( y)
f
Quadric Surfaces
These are the 3D equivalent of conic sections
(what are conic sections you've studied??)
The equation for all of these in general is
2
2
2
A⋅ x + B⋅ y + C⋅ z + D⋅ x⋅ y + E⋅ x⋅ z + F⋅ y⋅ z + G⋅ x + H⋅ y + I⋅ z + J = 0
There are 6 basic types: ellipsoid, hyperboloid of one sheet,
hyperboloid of two sheets, elliptic cone, elliptic paraboloid,
and hyperbolic paraboloid. The names are similar to the conics,
and have to do with the trace of each surface in each of the
coordinate planes. My software won't draw these, but sample
drawings are in all Calculus books. I can do a few by solving
the equations for z
2
x
Ellipsoid
a
2
2
+
y
b
2
+
z
c
2
2
=1
2
Hyperboloid of one sheet:
x
a
Hyperboloid of two sheets
2
2
+
y
b
z
a
2
2
2
z
−
c
2
=1
2
2
−
y
b
2
2
−
x
c
2
=1
Elliptic cone
2
x
a
Here's a sample:
f ,g
2
2
+
y
b
2
−
z
c
2
2
f ( x , y) :=
=0
1x
2
2
 + y 
4 9
4 
1x
2
2
y 


g( x , y) := −
+
4 9
4 
2
Elliptic paraboloid
z=
2
x
a
+
2
y
b
2
2
f ( x , y) :=
Here's a sample one of these:
f
2
Hyperbolic paraboloid
z=
x
a
And a sample of this one:
f
2
2
−
y
b
2
2
f ( x , y) :=
y
9
2
−
x
4
x
16
2
+
y
9
Ex 3 Do one by hand: sketch and identify the quadric surface by
identifying the trace in the coordinate planes and other planes
parallel to them if necessary:
2
2
z −x −
2
y
4
=1
Trace in xy plane: none
Trace in yz plane: hyperbola
Trace in xz plane: hyperbola
x and y int: none
z int + or - 1
 1 

  

2
 1 ⋅ 4 ⋅ x2 + y2 + 4   
 2



 1 

 
 2 
 −1
2
2
⋅
4
⋅
x
+
y
+
4
 2



I solved for z to get
f ( x , y) :=
f ,g
(
2
1
2
2
⋅ 4⋅ x + y + 4
)
(
)
(
)
 1
 
 2
g( x , y) :=
−1
2
(
2
2
⋅ 4⋅ x + y + 4
)
 1
 
 2
Ex 4 Identify and sketch the quadric surface:
2
2
2
x + 2y + z − 4x + 4y − 2z + 3 = 0
1. Complete the square as necessary:
2
2
2
x − 4x + 2y + 4y + z − 2z = −3
2
(2
)
2
2
2
x − 4 ⋅ x + 4 + 2 ⋅ y + 2 ⋅ y + 1 + z − 2 ⋅ z + 1 = −3 + 4 + 2 + 1
2
( x − 2) + 2 (y + 1) + (z − 1) = 4
2. The center is at (2,-1,1), now divide by 4
(x − 2)
2
4
+
( y + 1)
2
2
+
( z − 1)
4
2
=1
3. This is an ellipsoid, centered at (2,-1,1) and
fatter in the x and z directions than the y
The trace in the xz plane is a circle of radius sqrt(8),center
at x=2,z=1
The trace in the yz plane is a point (0,-1,1)
The trace in the xy plane is an ellipse with major axis sqrt(3)
centered at x=2, y=-1
x int are (3,0,0) and (1,0,0)
y int is none
z int is none
You should be able to finish the sketch.
The fifth special type of surface we will look at in this section
is the surface of revolution (see Calc II notes for finding the
volume of one). We will attempt to find the equation of such a
surface.
First, think of a generating curve in the yz or xz planes.
For this example, say x = r(z) in the xz plane is revolved
about the z axis. Cross sections in the plane z = z0 will be
circles of radius r(z0 ), so the curves each have equation
2
2
( ( 0) )
x +y = rz
2
Substitution of z for z0 will produce an equation valid for all
values of z:
2
2
x + y = ( r( z) )
2
Surface of Revolution
If the graph of a radius function r is evolved about an
axis, the resulting equation is one of these:
1. Revolved about the x axis : y2 + z2 = [r(x)] 2
2. Revolved about the y axis : x2 + z2 = [r(y)]2
3. Revolved about the z axis : x 2 + y2 = [r(z)] 2
Ex 5 Find the equation of a surface of revolution
a. if y=1/z is revolved about the z axis
x + y = 
2
2
1
2

z
This is in the book, but not graphed. Solving for z gives
1
z=
2
2
x +y
and
z=
−1
2
2
x +y
Here are the graphs of the upper and and lower surface.
I had to add a small number to each radical to get the
denominator to not be zero. The graphs should be fairly
accurate. There is an obvious asymptote at the origin
1
f ( x , y) :=
2
2
x + y + .0000001
f ,g
−1
g( x , y) :=
2
2
x + y + .0000001
You can't always do this, but under some circumstances,
it's possible to find a generating curve and an axis of
revolution from a surface.
Ex 6 Find a generating curve and an axis of revolution:
2
2
x + z = ( sin( y) )
2
The graph of x = sin(y) is revolved about the y axis.