**12.6** Cylinders and Quadric Surfaces

### Cylinders and Quadric Surfaces

We have already looked at two special types of surfaces:

planes and spheres.

Here we investigate two other types of surfaces: cylinders and quadric surfaces.

In order to sketch the graph of a surface, it is useful to determine the curves of intersection of the surface with planes parallel to the coordinate planes.

**These curves are called traces (or cross-sections) of the **
surface.

### Cylinders

### Cylinders

**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.

### Example 1

*Sketch the graph of the surface z = x*^{2}.
Solution:

*Notice that the equation of the graph, z = x*^{2}, doesn’t
*involve y.*

This means that any vertical plane with equation

*y = k(parallel to the xz-plane) intersects the graph in a *
*curve with equation z = x*^{2}.

So these vertical traces are parabolas.

*Example 1 – Solution*

Figure 1 shows how the graph is formed by taking the
*parabola z = x*^{2} *in the xz-plane and moving it in the *
*direction of the y-axis. *

The graph 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.*

cont’d

*The surface z = x*^{2} is a parabolic cylinder.

**Figure 1**

### Cylinders

*We noticed that the variable y is missing from the equation *
of the cylinder in Example 1. 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.

**Note**

When you are dealing with surfaces, it is important to
*recognize that an equation like x*^{2} *+ y*^{2} = 1 represents a
cylinder and not a circle. The trace of the cylinder

*x*^{2} *+ y*^{2} *= 1 in the xy-plane is the circle with equations*
*x*^{2} *+ y*^{2} *= 1, z = 0.*

### Quadric Surfaces

### 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

*where A,B,C,…, J are constants, but by translation and *
rotation it can be brought into one of the two standard
forms

*Ax*^{2} *+ By*^{2} *+ Cz*^{2} *+ J = 0 or Ax*^{2} *+ By*^{2} *+ Iz = 0*

Quadric surfaces are the counterparts in three dimensions of the conic sections in the plane.

### Example 3

Use traces to sketch the quadric surface with equation

Solution:

*By substituting z = 0, we find that the trace in the xy-plane *
*is x*^{2} *+ y*^{2}/9 = 1, which we recognize as an equation of an
*ellipse. In general, the horizontal trace in the plane z = k is*

*which is an ellipse, provided that k*^{2} *< 4, that is, –2 < k < 2.*

*Example 3 – Solution*

*Similarly, vertical traces parallel to the yz- and xz-planes *
are also ellipses:

*(if –1 < k < 1)*

*(if –3 < k < 3)*

cont’d

*Example 3 – Solution*

Figure 4 shows how drawing some traces indicates the shape of the surface.

**It’s called an ellipsoid because all of its traces are ellipses.**

Notice that it is symmetric with respect to each coordinate plane;

this is a reflection of the fact that
its equation involves only even
*powers of x, y, and z.*

cont’d

The ellipsoid

### Example 4

*Use traces to sketch the surface z = 4x*^{2} *+ y*^{2}.
Solution:

*If we put x = 0, we get z = y*^{2}*, so the yz-plane intersects the *
*surface in a parabola. If we put x = k (a constant), we get*
*z = y*^{2} *+ 4k*^{2}.

This means that if we slice the graph with any plane

*parallel to the yz-plane, we obtain a parabola that opens *
upward.

*Similarly, if y = k, the trace is z = 4x*^{2} *+ k*^{2}, which is again a
parabola that opens upward.

*Example 4 – Solution*

*If we put z = k, we get the horizontal traces 4x*^{2} *+ y*^{2} *= k, *
which we recognize as a family of ellipses. Knowing the
shapes of the traces, we can sketch the graph in Figure 5.

Because of the elliptical and parabolic traces, the quadric
*surface z = 4x*^{2} *+ y*^{2} **is called an elliptic paraboloid.**

cont’d

**Figure 5**

*The surface z = 4x*^{2}*+ y*^{2} is an elliptic paraboloid. Horizontal
traces are ellipses; vertical traces are parabolas.

### Example 5

*Sketch the surface z = y*^{2} *– x*^{2}.
Solution:

*The traces in the vertical planes x = k are the parabolas *
*z = y*^{2} *– k*^{2}*, which open upward. The traces in y = k are the *
*parabolas z = –x*^{2} *+ k*^{2}, which open downward.

*The horizontal traces are y*^{2} *– x*^{2} *= k, a family of hyperbolas.*

*Example 5 – Solution*

We draw the families of traces in Figure 6, and we show how the traces appear when placed in their correct planes in Figure 7.

cont’d

**Figure 6**

Vertical traces are parabolas; horizontal traces are
*hyperbolas. All traces are labeled with the value of k.*

*Traces in x = k *

*are z = y*^{2}*– k*^{2} *Traces in z = k *

*are y*^{2} *– x*^{2} *= k*
*Traces in y = k*

*are z = –x*^{2}*+ k*^{2}

*Example 5 – Solution*

_{cont’d}

*Traces in x = k* *Traces in y = k* *Traces in z = k*

Traces moved to their correct planes

**Figure 7**

*Example 5 – Solution*

In Figure 8 we fit together the traces from Figure 7 to form
*the surface z = y*^{2} *– x*^{2}**, a hyperbolic paraboloid.**

Notice that the shape of the surface near the origin resembles that of a saddle.

cont’d

*Two views of the surface z = y*^{2}*– x*^{2}, a hyperbolic paraboloid.

**Figure 8**

### Example 6

Sketch the surface Solution:

*The trace in any horizontal plane z = k is the ellipse*
*z = k*

*but the traces in the xz- and yz-planes are the hyperbolas*
*y = 0 and x = 0*

*Example 6 – Solution*

**This surface is called a hyperboloid of one sheet and is **
sketched in Figure 9.

cont’d

**Figure 9**

### Quadric Surfaces

The idea of using traces to draw a surface is employed in three-dimensional graphing software.

*In most such software, traces in the vertical planes x = k*
*and y = k are drawn for equally spaced values of k, and *
parts of the graph are eliminated using hidden line removal.

### Quadric Surfaces

Table 1 shows computer-drawn graphs of the six

basic types of quadric surfaces in standard form.

All surfaces are

symmetric with respect
*to the z-axis. *

If a quadric surface is

symmetric about a different axis, its equation changes

**Table 1**

Graphs of Quadric Surfaces

### Applications of Quadric

### Surfaces

### Applications of Quadric Surfaces

Examples of quadric surfaces can be found in the world around us. In fact, the world itself is a good example.

Although the earth is commonly modeled as a sphere, a more accurate model is an ellipsoid because the earth’s rotation has caused a flattening at the poles.

Circular paraboloids, obtained by rotating a parabola about its axis, are used to collect and reflect light, sound, and

radio and television signals.

### Applications of Quadric Surfaces

In a radio telescope, for instance, signals from distant stars that strike the bowl are all reflected to the receiver at the focus and are therefore amplified.

The same principle applies to microphones and satellite dishes in the shape of paraboloids.

Cooling towers for nuclear reactors are usually designed in the shape of hyperboloids of one sheet for reasons of

structural stability.

Pairs of hyperboloids are used to transmit rotational motion between skew axes.