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Quadric surfaces
Quadric Surfaces
In the previous two sections we’ve looked at lines and planes in three dimensions (or 3 ) and
while these are used quite heavily at times in a Calculus class there are many other surfaces that
are also used fairly regularly and so we need to take a look at those
...
Quadric surfaces are the graphs of
any equation that can be put into the general form
Ax + By + Cz + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0
2
2
2
where A, … , J are constants
...
Ellipsoid
Here is the general equation of an ellipsoid
...
If a = b = c then we will have a sphere
...
Clearly ellipsoids don’t have to be centered on the origin
...
Cone
Here is the general equation of a cone
...
Note that this is the equation of a cone that will open along the z-axis
...
This will be the case for the rest of the surfaces that we’ll be looking at in this section
as well
...
For instance, a cone that opens up along the x-axis will have
the equation,
y 2 z 2 x2
+ 2 = 2
2
b
c
a
For most of the following surfaces we will not give the other possible formulas
...
Cylinder
Here is the general equation of a cylinder
...
If a = b we have a cylinder whose cross
section is a circle
...
The cylinder will be centered on the axis corresponding to the variable that does not appear in the
equation
...
In two dimensions it is a circle, but in three
dimensions it is a cylinder
...
x2 y 2 z 2
+ 2 − 2 =1
2
a
b
c
Here is a sketch of a typical hyperboloid of one sheet
...
Hyperboloid of Two Sheets
Here is the equation of a hyperboloid of two sheets
...
The variable with the positive in front of it will give the axis along which the graph is centered
...
They are exactly the opposite signs
...
x2 y 2 z
+ 2 =
2
a
b
c
As with cylinders this has a cross section of an ellipse and if a = b it will have a cross section of
a circle
...
Here is a sketch of a typical elliptic paraboloid
...
Also, the sign of c will determine the direction that the paraboloid opens
...
Hyperbolic Paraboloid
Here is the equation of a hyperbolic paraboloid
...
These graphs are vaguely saddle shaped and as with the elliptic paraoloid the sign of c will
determine the direction in which the surface “opens up”
...
With the both of the types of paraboloids discussed above the surface can be easily moved up or
down by adding/subtracting a constant from the left side
...
Here is a couple of quick sketches of this surface
...
The sketch on the right has the standard set
of axes but it is difficult to see the numbers on the axis
...
In most
sketches that actually involve numbers on the axis system we will give both sketches to help get a
feel for what the sketch looks like