12.6: Quadric Surfaces
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- Identify a cylinder as a type of three-dimensional surface.
- Recognize the main features of ellipsoids, paraboloids, and hyperboloids.
- Use traces to draw the intersections of quadric surfaces with the coordinate planes.
We have been exploring vectors and vector operations in three-dimensional space, and we have developed equations to describe lines, planes, and spheres. In this section, we use our knowledge of planes and spheres, which are examples of three-dimensional figures called surfaces, to explore a variety of other surfaces that can be graphed in a three-dimensional coordinate system.
Identifying Cylinders
The first surface we’ll examine is the cylinder. Although most people immediately think of a hollow pipe or a soda straw when they hear the word cylinder, here we use the broad mathematical meaning of the term. As we have seen, cylindrical surfaces don’t have to be circular. A rectangular heating duct is a cylinder, as is a rolled-up yoga mat, the cross-section of which is a spiral shape.
In the two-dimensional coordinate plane, the equation
A set of lines parallel to a given line passing through a given curve is known as a cylindrical surface, or cylinder. The parallel lines are called rulings.
From this definition, we can see that we still have a cylinder in three-dimensional space, even if the curve is not a circle. Any curve can form a cylinder, and the rulings that compose the cylinder may be parallel to any given line (Figure

Sketch the graphs of the following cylindrical surfaces.
Solution
a. The variable

b. In this case, the equation contains all three variables —

c. In this equation, the variable

Sketch or use a graphing tool to view the graph of the cylindrical surface defined by equation
- Hint
-
The variable
can take on any value without limit. - Answer
-
When sketching surfaces, we have seen that it is useful to sketch the intersection of the surface with a plane parallel to one of the coordinate planes. These curves are called traces. We can see them in the plot of the cylinder in Figure
The traces of a surface are the cross-sections created when the surface intersects a plane parallel to one of the coordinate planes.
Traces are useful in sketching cylindrical surfaces. For a cylinder in three dimensions, though, only one set of traces is useful. Notice, in Figure
Cylindrical surfaces are formed by a set of parallel lines. Not all surfaces in three dimensions are constructed so simply, however. We now explore more complex surfaces, and traces are an important tool in this investigation.
Quadric Surfaces
We have learned about surfaces in three dimensions described by first-order equations; these are planes. Some other common types of surfaces can be described by second-order equations. We can view these surfaces as three-dimensional extensions of the conic sections we discussed earlier: the ellipse, the parabola, and the hyperbola. We call these graphs quadric surfaces
Quadric surfaces are the graphs of equations that can be expressed in the form
When a quadric surface intersects a coordinate plane, the trace is a conic section.
An ellipsoid is a surface described by an equation of the form
Sketch the ellipsoid
Solution
Start by sketching the traces. To find the trace in the

Now that we know what traces of this solid look like, we can sketch the surface in three dimensions (Figure

The trace of an ellipsoid is an ellipse in each of the coordinate planes. However, this does not have to be the case for all quadric surfaces. Many quadric surfaces have traces that are different kinds of conic sections, and this is usually indicated by the name of the surface. For example, if a surface can be described by an equation of the form
then we call that surface an elliptic paraboloid. The trace in the xy-plane is an ellipse, but the traces in the
Describe the traces of the elliptic paraboloid
Solution
To find the trace in the
The trace in plane
In planes parallel to the

A hyperboloid of one sheet is any surface that can be described with an equation of the form
- Hint
-
To find the traces in the coordinate planes, set each variable to zero individually.
- Answer
-
The traces parallel to the
-plane are ellipses and the traces parallel to the - and -planes are hyperbolas. Specifically, the trace in the -plane is ellipse the trace in the -plane is hyperbola and the trace in the -plane is hyperbola (see the following figure).
Hyperboloids of one sheet have some fascinating properties. For example, they can be constructed using straight lines, such as in the sculpture in Figure
Energy hitting the surface of a parabolic reflector is concentrated at the focal point of the reflector (Figure

Solution
Since
Seventeen standard quadric surfaces can be derived from the general equation
The following figures summarize the most important ones.
Identify the surfaces represented by the given equations.
Solution
a. The
Dividing through by 144 gives
So, this is, in fact, an ellipsoid, centered at the origin.
b. We first notice that the
This is an elliptic paraboloid centered at
Identify the surface represented by equation
- Hint
-
Look at the signs and powers of the
, and terms - Answer
-
Hyperboloid of one sheet, centered at
.
Key Concepts
- A set of lines parallel to a given line passing through a given curve is called a cylinder, or a cylindrical surface. The parallel lines are called rulings.
- The intersection of a three-dimensional surface and a plane is called a trace. To find the trace in the
-, -, or -planes, set or respectively. - Quadric surfaces are three-dimensional surfaces with traces composed of conic sections. Every quadric surface can be expressed with an equation of the form
- To sketch the graph of a quadric surface, start by sketching the traces to understand the framework of the surface.
- Important quadric surfaces are summarized in Figures
and .
Glossary
- cylinder
- a set of lines parallel to a given line passing through a given curve
- ellipsoid
- a three-dimensional surface described by an equation of the form
; all traces of this surface are ellipses
- elliptic cone
- a three-dimensional surface described by an equation of the form
; traces of this surface include ellipses and intersecting lines
- elliptic paraboloid
- a three-dimensional surface described by an equation of the form
; traces of this surface include ellipses and parabolas
- hyperboloid of one sheet
- a three-dimensional surface described by an equation of the form
traces of this surface include ellipses and hyperbolas
- hyperboloid of two sheets
- a three-dimensional surface described by an equation of the form
; traces of this surface include ellipses and hyperbolas
- quadric surfaces
- surfaces in three dimensions having the property that the traces of the surface are conic sections (ellipses, hyperbolas, and parabolas)
- rulings
- parallel lines that make up a cylindrical surface
- trace
- the intersection of a three-dimensional surface with a coordinate plane