Surfaces in Space
Terminology:
1.
Coordinate plane - there are 3the xy plane, the xz plane, and the yz plane.
2.
Traces
Ex: The trace of the graph of         in the
xy plane is the graph that results in the xy plane when we
set    We get     . Therefore the trace of
        in the xy plane is
4
y
x
4
The trace in the xz plane is the ellipse      
z
(0,0,2)
(-4,0,0)
x
We may also look at traces parallel to the coordinate
planes such as    or   
z
z=4
y
x
z = -2
3.
Right Cylinder
This surface is generated by 1st taking a graph in the
plane. We can assume it's in a coordinate plane. Take
     in the xy plane. A right cylinder is generated by
combining all the lines perpendicular to the xy plane that
intersect the circle. What is generated is an ordinary
cylinder. It's equation in space is       can
be any number.
y
x
However take the curve    in the xy plane. This
cylinder looks like.
y
x
Note that these 2 cylinders only use 2 of the 3 variables.
The missing variable z can be any real number.
3.
Quadric Surface
This is a surface generated by an equation of the form
               
       where  , or   
There are 6 types of quadric surfaces.
We will mainly look at those surfaces which are
symmetric to 2 or possibly 3 of the axes.    
and most likely  ,  and  will be 0.
The 6 types are Ellipsoid, Hyperboloid of one sheet,
Hyperboloid of 2 sheets, Elliptic Cone, Elliptic Paraboloid,
and Hyperbolic Paraboloid.
The simplest graph, for example, is the graph
                
which is the graph of a sphere.
                  
     


a  b  a  b  a  b  
This is the graph of a sphere centered at a   b
with radius 5.
If we changed the term   to   the graph would be
an ellipsoid.
However, for the most part, we will consider surfaces
which are symmetric to 2 or 3 of the coordinate planes.
Ellipsoid





  is an ellipsoid going thru the



points a  b a  b and a   b
The graph of
Elliptic Paraboloid
Example: Sketch the graph of      






  
 

Take a fixed value of  . For ex., if we let    we get
the circle      This is called the trace of the
graph in the plane    As  gets larger, the circle gets
larger.
z
circle (or maybe ellipse)
parabola
(0,0,0)
Note that the trace of the graph in the yz plane (where
   is the parabola     The trace of the graph in
the xz plane is the parabola    



is an Elliptic Paraboloid centered


about the z axis with lowest point a  b
In general,  


Similarly,      would be an Elliptic Paraboloid centered


about the  axis
x
The most complicated graph: The Hyperbolic Paraboloid .
Example:     
The traces:
xy plane:          
xz plane:    , a parabola opining upward
yz plane:      a parabola opening downward
To get an idea what this graph is, look at traces with the
plane    which is        a parabola
opening downward with vertex a   b As k get bigger,
the vertex of the parabola gets higher and higher.
Note these parabolas are parallel to the yz plane.
z
Saddle point
y
x
Here is a better graph.
0
-2
-4
2
4
20
0
-20
-4
-2
In general  
0

2
4




is a hyperbolic paraboloid




     is also a hyperbolic paraboloid.


Look at the traces with the plane    which are


parabolas      opening downward. These


parabolas are parallel to the xz plane
The next 3 surfaces, like the ellipsoid, will involve
    and   , however with 1 or 2 negative signs.
Hyperboloid of One Sheet and Hyperboloid of 2 Sheets
These surfaces can be formed by rotating a hyperbola 360°.
z
Hyperbola
in xz plane
x
y
If we rotate the hyperbola (in blue) about the z-axis we
get a hyperboloid of one sheet.
Example:        
If    we get the trace on the xz plane:      
z
z
x
x
This surface results when this hyperbola is rotated about
the z-axis. We can generate the equation
        as follows:
(0,0,z)
z
dist. =x
x
Every pt. a   b on this solid must lie a distance 
from the pt. a   b The hyperbola is      
   È    
The distance from a   b to a   b is
a   b   a   b   a    b       
                
To get this graph from        
look at traces with the plane    which is
        For each k we get a circle with
radius at least 1.
Hyperboloid of Two Sheets
Example:        
The xz trace is again      
z
x
However now rotate both branches 360° about the
x axis. This gives a surface in 2 parts.
We can derive the equation of the surface where we
rotate this hyper.       about the z-axis.
Any point surface a   b on the surface must lie a
distance   È   from the pt. a  b
z
dist. =z
x
(x,0,0)
This means
a  b  a  b  a  b     
                 
To get this graph, look at the traces with the plane
  
You get          These are circles which get
bigger and bigger. They begin when  or    which
gives a point at a  b and a    b
z
x
z
hyperbola
x
Elliptic Cone (only 1 negative sign, but 0 on right side)
Example:        
          È   
One point on the surface is a  b
There is one part where    and one part where   
The 2 parts are symmetric with respect to the xy plane.
For any     , the trace is a circle or radius 
As  increases, the circle gets bigger and bigger.
circle
z
y
z=x
x
z=-x
Note that the trace in the yz plane is       
    2 straight lines thru the origin.
The trace in the xz plane is also 2 lines thru the origin.
Note: If you have 2 negative signs such as         , the
graph is still an elliptic cone, but these 2 cones now go along
the x-axis.
View point is a  b
Shown below are some graphs of quadric surfaces.
Hyperboloid of One Sheet a        b
Hyperboloid of Two Sheets a        b
Elliptic Cone a        b
Hyperbolic Paraboloid a      b
20
z
4
0
2
-20
0
-4
-2
-2
0
x
2
4
-4
y